Migrate to VitePress

These include molarity and activated complexes

.gitignore

@@ -1,2 +1,151 @@
-site/
-docs/resources/
+docs/resources
+
+# do not commit lockfiles
+package-lock.json
+yarn.lock
+
+# Logs
+logs
+*.log
+npm-debug.log*
+yarn-debug.log*
+yarn-error.log*
+
+# Serverless directories
+.serverless/
+
+# Runtime data
+pids
+*.pid
+*.seed
+*.pid.lock
+
+# Directory for instrumented libs generated by jscoverage/JSCover
+lib-cov
+
+# Coverage directory used by tools like istanbul
+coverage
+
+# nyc test coverage
+.nyc_output
+
+# Grunt intermediate storage (http://gruntjs.com/creating-plugins#storing-task-files)
+.grunt
+
+# Bower dependency directory (https://bower.io/)
+bower_components
+
+# node-waf configuration
+.lock-wscript
+
+# Compiled binary addons (http://nodejs.org/api/addons.html)
+build/Release
+
+# Dependency directories
+node_modules/
+jspm_packages/
+
+# Typescript v1 declaration files
+typings/
+
+# Optional npm cache directory
+.npm
+
+# Optional REPL history
+.node_repl_history
+
+# Optional eslint cache
+.eslintcache
+
+# Optional REPL history
+.node_repl_history
+
+# Output of 'npm pack'
+*.tgz
+
+# Yarn Integrity file
+.yarn-integrity
+.yarn/cache
+.yarn/unplugged
+.yarn/build-state.yml
+.yarn/install-state.gz
+.pnp.*
+
+# dotenv environment variables file
+.env
+.env.test
+
+# See http://help.github.com/ignore-files/ for more about ignoring files.
+
+# ignore server configuration
+env/development.js
+docker-compose.local.yml
+*.azcli
+
+# DynamoDB Local files
+.dynamodb/
+
+# parcel-bundler cache (https://parceljs.org/)
+.cache
+.parcel-cache
+
+# compiled output
+/dist
+/tmp
+/out-tsc
+
+# IDEs and editors
+/.idea
+.project
+.classpath
+.c9/
+*.launch
+.settings/
+*.sublime-workspace
+
+# IDE - VSCode
+.vscode/*
+!.vscode/settings.json
+!.vscode/tasks.json
+!.vscode/launch.json
+!.vscode/extensions.json
+.vscode-test
+
+# TernJS port file
+.tern-port
+
+# misc
+/.sass-cache
+/connect.lock
+/coverage
+/libpeerconnection.log
+npm-debug.log
+testem.log
+/typings
+
+# e2e
+/e2e/*.js
+/e2e/*.map
+
+# vuepress build output
+.vuepress/dist
+
+# vitepress build output
+.vitepress/dist
+
+# Next.js build output
+.next
+out
+
+# Nuxt.js build / generate output
+.nuxt
+dist
+
+# Diagnostic reports (https://nodejs.org/api/report.html)
+report.[0-9]*.[0-9]*.[0-9]*.[0-9]*.json
+
+# System Files
+.DS_Store
+/.DS_Store
+*/.DS_Store
+Thumbs.db

CONTRIBUTING.md

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+# Contributing guidelines
+
+If you would like to add or fix any content in this project, you will need to ensure that your changes retain the same style as the rest of the repository. The Eifueo project consists of multiple (mostly) standard Markdown documents.
+
+## General Markdown
+
+Even Markdown has multiple implementations, and prior knowledge of the markup language is required for contributions. For the following stylistic elements, the following symbols should be used:
+
+ - `**Bold text**`: **Bold text**
+ - `*Italicised text*`: *Italicised text*
+ - `***Bold and italicised text***`: ***Bold and italicised text***
+
+If a phrase would be bolded, do not bold any surrounding punctuation. If a phrase would be italicised, italicise any surrounding punctuation.
+
+Tables must contain exactly one space between content and the vertical bars.
+
+```
+| Heading 1 | Heading 2 | Heading 3 |
+| :-- | --- | --: |
+| Left justify | Normal justify | Right justify |
+```
+| Heading 1 | Heading 2 | Heading 3 |
+| :-- | --- | --: |
+| Left justify | Normal justify | Right justify |
+
+Lists must use hyphens with a space before and after. There should be newlines before and after lists.
+
+There should be a newline before and after headings, except for headings that start a file. There should also be a space after a heading number sign.
+
+```
+### Heading example
+
+Pomme de terre!
+```
+
+### Heading example
+
+Pomme de terre!
+
+Links to images must be either from Kognity or available for free non-commercial use. They should be sent to the site administrator's email to be loaded in as a static local asset. Images should be linked using HTML, have a maximum width of 700 pixels, contain a reference to the source organisation, and fit the overall theme of the site. The link to the image path is `/resources/images/image.file-extension`. If possible, PNGs are preferred. Images should be relevantly named in lowercase with hyphens separating words.
+
+`<img src="/resources/images/velocity-time-graph.png" width=700>(Source: Kognity)</img>
+
+## Special Markdown
+
+Math is written in LaTeX and is rendered using MathJax.
+
+`$Inline math: 1+2=3$`: $Inline math: 1+2=3$
+
+Display math should always be on its own line. Multi-line math should have dollar signs on their own lines. There should always be a newline after display math.
+
+```
+Proof that 1 + 1 is 2:
+$$display math: 1+2=3$$
+
+More proof:
+$$
+1 + 1 = x \\
+1 + 1 = 2
+$$
+```
+Proof that 1 + 1 is 2:
+$$display math: 1+2=3$$
+
+More proof:
+$$
+1 + 1 = x \\
+1 + 1 = 2
+$$
+
+Admonitions are provided by `mkdocs-material-extensions` and documentation for them can be found [here](https://squidfunk.github.io/mkdocs-material/reference/admonitions/#usage). The types used in Eifueo are `example`, `definition`, `note`, and `info`. **The use of admonitions should be minimised, aside from example admonitions**.
+
+## Writing style
+
+This repository uses, unfortunately, my own personal style of English that is a mix of British and American English.
+
+ - Titles should be written in sentence case
+ - List items should have their first letter capitalised
+ - "-ize" words should be written in the British form ("-ise")
+ - "-or" words should be written in the British form ("-our")
+ - "-ph-" words should be written in the American form ("-f-")
+ - Punctuation follows quotes ("'content', more content")
+ - Quotes should follow the American style ("'It was a potato,' he said.")
+
+
+Most areas should be written in paragraphs, when possible, using a mostly formal tone.
+
+The primary focus of this project is organisation and brevity. This is not a textbook; there should not be unnecessary fluff nor examples in the main body text if it can be avoided. Examples should go in admonitions instead. Especially long admonitions should use `???` to make them collapsable.

README.md

@@ -2,9 +2,7 @@
 
 A "competitor" of sorts to magicalsoup/highschool.
 
-The LaTeX formatting in this repository uses `$...$` for inline math, and `$$...$$` for multi-line math. MathJax is used to render this LaTeX.
-
-Admonitions can be added with documentation available [here](https://squidfunk.github.io/mkdocs-material/reference/admonitions/#usage).
+Please see [CONTRIBUTING.md](/eggy/eifueo/src/branch/master/CONTRIBUTING.md) for guidelines and formatting information.
 
 ## Dependencies
 

docs/.vitepress/config.ts

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+import { defineConfig } from "vitepress";
+
+export default defineConfig({
+  title: "Eifueo",
+  description: "Rewritten notes!",
+  lang: "en-US",
+  lastUpdated: true,
+  themeConfig: {
+    nav: [{ text: "HL Chemistry 2", link: "/sch4uz.html" }],
+    sidebar: getSidebar(),
+    repo: "https://git.eggworld.tk/eggy/eifueo",
+    docsDir: "docs",
+  },
+  markdown: {
+    config: (md) => {
+      md.use(require("markdown-it-mathjax3"), {
+        loader: { load: ["[tex]/mhchem"] },
+        tex: { packages: { "[+]": ["mhchem"] } },
+      });
+    },
+  },
+});
+
+function getSidebar() {
+  return [
+    {
+      text: "Grade 12 IB",
+      children: [{ text: "HL Chemistry 3", link: "/g12/snc4mz" }],
+    },
+  ];
+}

docs/.vitepress/theme/Layout.vue

@@ -0,0 +1,15 @@
+<script setup>
+import DefaultTheme from "vitepress/theme";
+const { Layout } = DefaultTheme;
+</script>
+
+<template>
+  <Layout></Layout>
+</template>
+
+<style>
+main > .container {
+  /* love that */
+  max-width: 54rem !important;
+}
+</style>

docs/.vitepress/theme/index.ts

@@ -0,0 +1,7 @@
+import DefaultTheme from "vitepress/theme";
+import Layout from "./Layout.vue";
+
+export default {
+  ...DefaultTheme,
+  Layout,
+};

docs/chw3mz.md

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+# HL History - 1
+
+The course code for this page is **CHW3MZ**.
+
+## Command terms
+
+The following terms are regularly used by IB and have specific meanings:
+
+ - **Analyse**: Break down an idea into its essential elements
+ - **Compare**: Identify and justify the similarities between ideas
+ - **Contrast**: Identify and justify the differences between ideas
+ - **Discuss**: Provide a balanced review with a range of justified opinions or conclusions
+ - **Evaluate**: Appraise an argument with strengths and limitations
+ - **Examine**: Consider an argument or concept, revealing its assumptions and interrelations
+ - **To what extent**: Consider the merits and demerits of an idea or argument with justified opinions or conclusions
+
+## Historiography
+
+Historiography is the study of how history is developed and how historians have viewed history. It helps develop the ability to view events from multiple perspectives and reveals how others might view events in multiple perspectives. The **historical method** outlines the process of making history from start to finish:
+
+ - **Questions are asked** about any of the following from controversies, new perspectives, and gaps in history:
+	- major/minor long-/short-term **causes** of an event
+	- the **nature** of an event — the purpose of the event, the status of the event
+	- the immediate/long-term/short-term **impacts** of an event
+	- the **ethics** of an event
+	- a hypothetical **counterfactual** version of an event
+ - **Data is collected** to help answer the questions
+ - **Data is analysed** to identify patterns and trends via statistics, make judgements using OPCVL, and make comparisons of the main message (content) and PERMS (political, economical, religious, military, social)
+ - **Arguments are created** with a thesis and critical analyses per argument
+ - and the **findings are communicated** via any means, including essays, opinion paragraphs, debates, presentations, music, etc.
+
+### Lenses
+
+There are seven main schools of thought historians usually fall under that attempt to explain why history happens.
+
+ - The **great people** lens views history with the perspective that individuals create changes, and without those individuals history would be drastically altered. This school focuses on their individual motivations, relationships, etc.
+ - The **structuralist** lens views history and events as caused by changes in economic structures, technology, ideology, and social norms.
+ - The **decisionist** lens views history with the perspective that it is the decisions of people that are the cause of events.
+ - The **Marxist** lens focuses on class conflict — the exploiter and the exploitee and how their conflicts affect history.
+ - The **Toynbee** lens regards the response of people to crises as the largest cause to historical events.
+ - The **post-modern** lens states that history reflects the time period it was written in and the intent of the author. This school asserts that there is no historical truth and focuses on how history is shaped and manipulated to serve the agendas and needs of various civilisations.
+ - The **feminist** lens states that history should reflect the experience of both genders equally. This school particularly focuses on the treatment of and lives of women.
+
+### Historical concepts
+
+There are six key historical concepts that should be taken under consideration in the analysis of history.
+
+ - **Change**: how people and events create change, the differences before and after an event, and the nature and pace of the change
+ - **Continuity**: what doesn't change and to what extent things don't change
+ - **Causation**: why an event occurred (social, economic, political, etc.), and what factor(s) caused it
+ - **Consequence**: the long- and short-term effects of an event on society
+ - **Significance**: why the event matters and the importance of some details over others
+ - **Perspective**: the different perspectives, why there are different perspectives, and how those perspectives affect the interpretation of history
+
+### Data collection
+
+The abundance of **diverse** data allows for greater perspectives to be gleaned and thus more solutions to a problem. Data and their sources can fall under one of three categories:
+
+ - **Primary** data is from an original source with no sources under it — e.g., newspapers, memoirs, photographs, diaries, etc.
+ - **Secondary** data is an interpretation of a primary source(s) with its own argument — e.g., articles, thesises, documentaries, etc.
+ - **Tertiary** data compiles secondary sources and should only be used for context — e.g., wikis, textbooks, encyclopedias, etc.
+
+### Logic
+
+!!! definition
+    - **Logic** is the study of rules of inference and the analysis of arguments.
+    - A **conclusion** is a proposition that follows all others.
+    - An **inference** is a connection that acts as a logical leap between a premise and a conclusion.
+    - **Logically consistent** statements follow the three laws of thought and do not contradict.
+    - **Logically contradicting** statements do not follow the three laws of thought.
+    - The **validity** of a statement is its correctness of reasoning via the laws of thought.
+    - A **sound** argument is of valid form and has a true premise.
+    - An **argument** is a simple statement or disagreement that attempts to reach a conclusion by proving something true with evidence. Good arguments are sound, valid, clear, and avoids hasty conclusions.
+
+**Aristotle** laid the foundations for the principles of formal logic via the three laws of thought.
+
+ - **Law of identity**: everything is identical with itself.
+ - **Law of noncontradiction**: contradictory statements cannot both be true.
+ - **Law of excluded middle**: any proposition must either be true or false.
+
+!!! example
+    - Identity: A football is a football.
+    - Noncontradiction: If water and oil do not mix, and substance A mixes in water, it must not be oil.
+    - Excluded middle: The Nintendo Switch must either be or not be a potato.
+
+**Deductive** arguments connect a general statement to a more specific statement based on laws, rules, and/or widely accepted principles.
+
+!!! example
+    As monkeys like bananas and Lucy is a monkey, Lucy must like bananas.
+
+**Inductive** arguments connect a specific statement to a more general statement based on **empiric** data.
+
+!!! example
+    As three of the eight billion humans on Earth are mortal, all humans must be mortal.
+
+### Logical fallacies
+
+ - An **ad hominem** argument attacks the arguer or anything else instead of the argument.
+    - e.g., *"You're a Nintendo fanboy; of course you think that."*
+ - Arguments that **appeal to authority** use the opinion of an authority on a topic is used as evidence to support an argument.
+    - e.g., *"The President of the United States said that we should inject disinfectant into ourselves, so it must be a good idea!"*
+ - Arguments that **appeal to emotion** manipulate the recipient's emotions typically via loaded language to win an argument.
+    - e.g., *"Those island devils have robbed us of our sleep at night — they must be eradicated!"*
+ - Arguments that **appeal to force** use threats to win an argument.
+    - e.g., *"I'm right, aren't I?" said the jock, flexing her biceps threateningly.*
+ - Arguments that **appeal to ignorance** assert a proposition is true because it has not been proven false.
+    - e.g., *"My laptop must secretly have chips in it that no one can detect from aliens because we don't know if there* aren't *undetectable chips in there."*
+ - **Bandwagoning or herding** arguments assert that a conclusion is true because it is accepted by most people. This is a result of confirmation bias.
+    - e.g., *"Ma, everyone else is jumping off that bridge, so why shouldn't I?"*
+ - A fallacy of **accident** wrongly applies a general rule to a specific exception.
+    - e.g., *Since surgeons cut people with knives and cutting people with knives is a crime, surgeons are criminals.*
+ - An argument that **begs the question** has circular reasoning by having premises that assume its conclusion.
+    - e.g., *Acid must be able to eat through your skin because it is corrosive.*
+ - **Cherry picking** occurs when evidence that supports the conclusion is pointed out while those that contradict the conclusion are ignored or withheld.
+    - e.g., *"Look at these perfect cherries — their tree must be in perfect condition!"*
+ - A fallacy of **converse accident** wrongly applies a specific exception to a general rule.
+    - e.g., *As the Nintendo Switch, a game console, is portable, all game consoles must be portable.*
+ - A **complex question**, also known as a trick question,  embeds a proposition that is accepted when a direct answer is given to the question.
+    - e.g., *Have you stopped abusing children yet?*
+ - Arguments with a **false cause** incorrectly assume a cause to an effect.
+    - e.g., imagining correlation implies causation.
+ - **Hasty generalisations** appear in inductive generalisations based on insufficient evidence.
+    - e.g., *Since the first seven odd numbers are prime or square, all odd numbers must be prime or square.*
+ - Arguments that **miss the point** provide an irrelevant conclusion that fails to address the issue of the question.
+    - e.g., *"Is it allowed?" "It should be allowed because it's nowhere near as bad as alcohol."*
+ - A **non sequitur** is an invalid argument that does not follow the laws of thought.
+    - e.g., *All humans are mammals. Whales are mammals. Therefore, whales are humans.*
+ - A **no true Scotsman** (appeal to purity) fallacy takes a generalisation and doubles down to protect it by excluding counterexamples typically via emotionally charged language.
+    - e.g., *"Although your father is a Scotsman and dances, no* true *Scotsman would dance."*
+ - Arguments with **recency bias** put greater importance on recent data over historic data.
+    - e.g., *As GameStop's stock has risen over the past few days dramatically, it will continue to do so.*
+ - **Red herrings** change the issue of subject away from the original question.
+    - e.g., *You should support the new housing bill. We can't continue to see people living in the streets; we must have cheaper housing.*
+ - A **straw man** argument misrepresents the opposing position by making their arguments sound more extreme.
+    - e.g., *"We should relax laws on immigration." "The instant we let millions of people through our border is when our country falls."*
+
+## Causes of the Chinese Civil War
+
+### Decline of the Manchu Qing Dynasty
+**— Long-term structural political**
+
+In the Qing dynasty, from 1861 to 1908, Empress **Cixi** ruled China as an autocrat. Corruption was rampant in Beijing and officials could not control warlords in remote regions. Under Cixi, China became weak and was easily influenced by foreign powers.
+
+### Foreign involvement
+**— Long-term structural political**
+
+The influence of foreign powers increased outrage among citizens at the inability of the government to do things and led to greater internal dissent.
+
+The **opium wars** starting from 1839 were two armed conflicts in China between Western powers and the Qing dynasty. Both were won easily by the West due to their superior, more modern military technology. This resulted in China signing a series of what came to be known as "**unfair treaties**" starting with the Treaty of Nanjing in 1842, which ended the First Opium War and gave Hong Kong to Britain "in perpetuity" among other trading concessions.
+
+The **First Sino-Japanese War** (also known as the War or Jiawu) in 1894–1895 also resulted in Japan's easy victory due to obsolete Chinese military technology.
+
+!!! example
+    During the First Sino-Japanese War, Cixi took military money and spent it on palace renovations, demonstrating the corruption in and ineffectiveness of the regime.
+
+### Outdated agricultural practices and limited industrial development
+**— Long-term structural economic**
+
+!!! context
+    In 1900, Japan and the United States were major industrial powers and both were steadily modernising.
+
+China's rulers believed that Westerners were barbarians and that nothing could be learned from industrialisation. This contributed to their weak military strength as they fell further behind other countries.
+
+!!! example
+    By 1914, only ~6 000 km of rail was laid in China while the US had laid ~225 000 km.
+
+Additionally, the population boom meant that demand for food increased, but outdated agricultural practices and technologies could not keep up, resulting in famine.
+
+### Tradition and class structure
+**— Long-term structural social**
+
+China operated under a **patriarchy** and had traditions and practices which were thought to be even at the time to be cruel and outdated especially for women.
+
+!!! example
+    The life expectancy of a city in China (Shenyang, Daoyi) from 1792 to 1867 was less than 40 years for men and about 30 for women.
+
+**Submission to the Qing dynasty** was traditional as well — the Chinese could not marry Manchus nor live in Manchuria.
+
+!!! example
+    Men were required to wear a queue (a long pigtail behind a shaven forehead) to show subservience to the Manchu Empire.
+
+The **class structure** in China did not change for hundreds of years and old traditions and practices persisted into the 1900s.
+
+ - The land peasants worked on belonged to local landlords.
+ - 80% of the population remained peasants.
+ - Landlords took a large amount of crops as rent and the government also took a large portion as taxes.
+ - The burdens from landlords and the government were compounded with natural disasters such as floods and droughts, resulting in famine.
+ - Landlords could taken peasant women as they wished, force peasants to perform extra duties, and beat them if they were questioned.
+
+### Internal dissent
+**— Long-term structural political**
+
+The **Taiping Civil War** from 1850 to 1864 was one of the bloodiest wars ever and the largest conflict of the 19th century. Anti-Manchurian sentiment was high as the people did not like that the Qing dynasty was ruled not by themselves. At the time, they could not marry Manchus, could not settle in Manchuria, and men were required to wear queues as a reminder of submission to Qing rule.
+
+The **Boxer Rebellion** from 1899 to 1901 was caused by an organisation known as the Boxers (due to many of their members practising Chinese martial arts) desiring an end to foreign control in China. To do so, they violenced across northern China targeting foreign property, Christian missionaries, and Chinese Christians. Although initially supported by Cixi, support was split in the country and eventually she accepted help from foreign armies to end the rebellion.
+
+!!! example
+    The slogan of the Boxers was to "support the Qing government and exterminate foreigners".
+
+### Introduction of progressive ideas and rise of revolutionaries
+
+Sun Yixian
+
+### 1911 revolution
+
+failure of revolution
+aftermath of revolution
+
+### Warlords
+**— Short-term**
+
+warlords
+
+### Rise of revolutionaries 2: electric boogaloo
+**— Short-term**
+
+first united front
+
+screw this history too hard
+
+## Consolidation and maintenance of power in China
+
+### Use of legal methods
+
+In the aftermath of the Chinese Civil War, political pluralism was adopted for the purposes of stability and unification. At the **Chinese People's Political Consultative Conference (PCC)** on 21 September 1949:
+
+ - [Mao Zedong](https://en.wikipedia.org/wiki/Mao_Zedong) (Chinese: 毛泽东) was elected Chairman of the People's Republic of China,
+ - [Zhou Enlai](https://en.wikipedia.org/wiki/Zhou_Enlai) (Chinese: 周恩来) was appointed first premier (prime minister) by Mao,
+ - a de facto constitution in the form of the Organic Law was adopted, and
+ - China was divided into six regions with each region under two civilian and two military officers to maintain control
+
+<img src="/resources/images/ccp-hierarchy.png" width=700>(Source: Kognity)</img>
+
+#### Xinjiang
+
+The **reunification campaigns** from 1950 to 1953 consolidated Mao's power across a divided China and allowed him to gain control of its borders.
+
+!!! background
+    The region of [Xinjiang](https://en.wikipedia.org/wiki/Xinjiang) followed the Qing dynasty, but was semi-independent during the rule of the warlords and the Republic of China. With a population consisting largely of the [Uyghur](https://en.wikipedia.org/wiki/Uyghurs) minority ethnic group as opposed to the [Han](https://en.wikipedia.org/wiki/Han_Chinese) that made up 98% of the population in China, the province was ruled by a coalition government made of local leaders and **Guomindang (GMD)** members at the time of the establishment of the PRC.
+
+The **Chinese Communist Party (CCP)** was concerned that Xinjiang would start a separatist movement against them or become part of the USSR. Upon the CCP's army, the **People's Liberation Army (PLA)** moving into Xinjiang, provincial authorities pledged allegiance to the CCP and the province was fully under their control by 1951. It is referred to as the "Peaceful Liberation of Xinjiang" in Chinese historiography.
+
+In 1955, the region became an [autonomous region](https://en.wikipedia.org/wiki/Autonomous_regions_of_China) of China, granting it a local government and more legislative rights than other provinces.
+
+#### Controlling the population
+
+All citizens were required to belong to a self-sufficient entity known as a *danwei* (Chinese: 单位), or **work unit**. Permission from the work unit was needed to marry or have children. Under the threat of punishment if policy was not followed, everyone was assigned a home a food to eat along with others in their work unit. Regardless of their size, all work units were obligated to provide or share adequate facilities — schools, housing, health care, etc.
+
+A system of **household registration** to identify citizens known as *hukou* (Chinese: 户口) recorded the birth, death, and movement of people as well as their family members, connecting identifiable information to their location of permanent residence. Whether someone was given better benefits by the government was determined by their agricultural status (agricultural or non-agricultural — i.e. rural or urban, non-agricultural was better) and it was virtually impossible to switch to the other. During industrialisation from 1955 onward, Mao used this system to control rural-to-urban migration by way of a certificate on one's registration required to be able to move to urban area. 
+
+Public **records** per person known as *dang'an* (Chinese: 档案) contained personal information such as:
+
+ - employment records
+ - physical characteristics
+ - family background with photos
+ - transcripts and school records
+ - achievements, mistakes, and self-criticisms
+ - political activity, and more
+ 
+The government and work unit could access these files, instilling a culture of fear into the citizens as the most intrusive form of surveillance.
+
+#### 1954 Constitution
+
+The [constitution](https://en.wikipedia.org/wiki/1954_Constitution_of_the_People%27s_Republic_of_China) was based on the PCC and Organic Law. It established rights such as equality of citizens andbetween Han and minority groups, prohibiting racial discrimination and oppression. It also set up a legal system where all citizens had the right to a fair trial by judges appointed by the government, but this was not genuinely implemented until after Mao.
+
+### Use of force
+
+#### Hundred Flowers Campaign
+
+From 1956 to 1957, the CCP encouraged citizens to express their thoughts of the regime in the [Hundred Flowers Campaign](https://en.wikipedia.org/wiki/Hundred_Flowers_Campaign) (Chinese: 百花齐放), inviting intellectuals to criticise their policies.
+
+!!! quote "Mao Zedong, 1956:"
+    The policy of letting a hundred flowers bloom and a hundred schools of thought contend is designed to promote the flourishing of the arts and the progress of science.
+
+Initially, people were afraid of being arrested publicly and executed after the events of the Three- and Five- Antis campaigns, but eventually began to provide criticism. By May 1957, millions of letters were openly criticising the government. Rallies in the streets and posters and articles in magazines protested against corruption and censorship.
+
+!!! example
+    Students at Peking University created a "Democratic Wall" on which they criticised the CCP with posters and letters over their control over intellectuals, the harshness of campaigns against counter-revolutionaries, low living standards, economic corruption and privileges among members of the party.
+
+Mao abandoned the campaign in June 1957 and then began the [Anti-Rightist Campaign](https://en.wikipedia.org/wiki/Anti-Rightist_Campaign) in July where those who provided criticism were not persecuted. It is controversial whether Mao was genuinely surprised by the extent of the criticism or whether the campaign was to identify enemies of the CCP.
+
+!!! definition
+    In China, a **rightist** was someone who favoured capitalism over communism, but eventually became a label for anyone who disagreed with Mao.
+
+During the Anti-Rightist Campaign from 1957 to 1959, Mao attacked his critics in the Hundred Flowers Campaign, forcing them to take back what they said. Led by [Deng Xiaoping](https://en.wikipedia.org/wiki/Deng_Xiaoping) (Chinese: 邓小平), at least 550 000 citizens were declared rightists — most of them were sent to be re-educated in the countryside via labour reform. The result was that Mao accumulated more power over the party and over China.
+
+#### Force in the Cultural Revolution
+
+!!! background
+    The [**Cultural Revolution**](https://en.wikipedia.org/wiki/Cultural_Revolution) from 1966 to 1976 was declared by Mao to remove all capitalist and intellectual elements from China. In reality, it was Mao's attempt to remove his rivals in the government — [Liu Shaoqi](https://en.wikipedia.org/wiki/Liu_Shaoqi) (Chinese: 刘少奇) and Deng Xiaoping — believing their successful policies would damage his reputation and ideology, and to revive his cult of personality after the events of the Great Leap Forward.
+
+The "Four Olds" — old ideas, culture, customs, and habits — were denounced and books were burnt, porcelain destroyed, museums ransacked, and heritage sites destroyed.
+
+!!! example
+    More than 70% of Beijing's cultural artifacts were destroyed in August–September 1966.
+
+Because religion and intellectuals could change people's minds, Mao directed temples, shrines, and religious statues to be destroyed. Intellectuals such as teachers were tortured, beaten to death, and sent to prison. Priests and the clergy were imprisoned and denounced — religious worship was banned entirely. [Red Guards](https://en.wikipedia.org/wiki/Red_Guards) comprising students from elementary to high school carried out humiliations and attacks as per the social norm because Mao had ordered them to.
+
+!!! info
+    The **Little Red Book** was a pocket-sized collection of Mao's thoughts, assembled by his propaganda minister, [Lin Biao](https://en.wikipedia.org/wiki/Lin_Biao) (Chinese: 林彪). Published during the Cultural Revolution, it was viewed as the source of all truth and more than a billion copies circulated, second only to the Holy Bible. Members of the Red Guard were required to carry the book with them at all times, and its contents were viewed as the source of all truth.
+
+In addition, landlords, rich peasants, counter-revolutionaries, rightists, and capitalists were all targeted. Due to the vague nature of thse labels, anyone could accuse anyone of anything. An environment of fear formed as people reported suspected enemies, with some citizens reporting neighbours as revenge, and children even reported and condemned their parents. The population was effectively mobilised to act as a **secret police** without actual secret police. Casualties from the Cultural Revolution are difficult to measure and range from thousands to millions.
+
+!!! example
+    People were accused and persecuted for crimes such as forgetting a quote from the Little Red Book or owning a Western instrument.
+
+#### Tibet
+
+Declaring that Tibet was part of China, on 7 October 1950, the PLA invaded east Tibet under the banner of liberating them from Western imperialist powers, although the Tibet government recorded only ten foreigners in the country. As a poor country with little infrastructure and communications, the poorly equipped and trained Tibetan army of 8 000 was outmatched by the 40 000 sent by the PLA. On 23 May 1951, the CCP imposed on Tibet the [Seventeen Point Agreement for the Peaceful Liberation of Tibet](https://en.wikipedia.org/wiki/Seventeen_Point_Agreement) that confirmed Chinese soverignity over the province.
+
+In 1959, an [uprising in Tibet](https://en.wikipedia.org/wiki/1959_Tibetan_uprising) led to the mass arrests of Tibetans and increased social and religious control. The spiritual leader of Tibetan Buddhism, the [Dalai Lama](https://en.wikipedia.org/wiki/Dalai_Lama), fled to India and some Buddhist practices were forbidden. The Tenth [Panchen Lama](https://en.wikipedia.org/wiki/Panchen_Lama) under the Dalai Lama wrote the [70 000 Character Petition](https://en.wikipedia.org/wiki/70,000_Character_Petition) addressed to the Chinese government, denouncing the abusive policies such as mass imprisonment and the high prisoner death rate in Tibet under communist rule. Mao rejected the claims and arrested him.
+
+#### Guangdong purge
+
+Guangdong (Chinese: 广东), also known as Canton, is a province in southern China close to Hong Kong and Macau and had held an economically important port that was the headquarters of the GMD. During the **reunification campaigns**, an estimated 28 000 people were executed as it was purged of Nationalist forces.
+
+#### Antis campaigns
+
+The [**Three-Anti Campaign**](https://en.wikipedia.org/wiki/Three-anti_and_Five-anti_Campaigns) in 1951 denounced waste, corruption, and bureaucratic inefficiency. Its targets were CCP members, former GMD members, and bureaucratic officials not in the party. Confessions were required for the people denounced in public trials. An estimated 5% of government officials were purged.
+
+!!! example
+    One thousand officers were denounced in the first month — some for minor offenses like the use of an American car and some more serious such as the use of government money to purchase luxury furniture in offices.
+
+The **Five-Anti Campaign** in 1952 denounced bribery, theft of state property, tax evasion, state property theft, chating on government contracts, and stealing economic intelligence. Its targets were the bourgeoisie, merchants, industrialists, and the rest of the capitalist class. The campaign sent a wave of fear in the bourgoisie, not helped by the Chinese encouraged to support the campaign and denounce people they suspected to be guilty. Those convicted also had to confess their crimes in public trials, and an estimated 450 000 private businesses were convicted.
+
+The results of the campaigns were large-scale purges in the CCP and bourgeoisie, the ceoncept of class struggle devevloped in China, and it showed the population that it challenging the regime would be futile.
+
+### Charisma and propaganda
+
+Mao was seen as the figurehead of China and his image was proimently on display throughout China. His Little Red Book was distributed everywhere, and his charisma was a form of control over the population via his **cult of personality**. There was no distinction between Mao the person, the government, and China in the people's eyes. He received little backlash after the events of the Great Leap Forward and most of the population genuinely mourned his death.
+
+!!! example "Example: "Long Live the Chinese Communist Party that Chairman Mao Personally Founded", April 1973"
+    <img src="/resources/images/mao-propaganda.jpeg" width=500>(Source: Kognity)</img>
+
+**Propaganda** was under the control of the CCP's Central Propaganda Department up to the Cultural Revolution. Its aims were to spread Chinese ideology and the idea of Maoism against capitalism, indoctrinate the Chinese population, and reinforcce political messages to enhance Mao's cult of personality.
+
+!!! example
+    The **Combat Illiteracy Campaign** from 1950 to 1956 sent simple reading material all over China including remote places to increase literacy while spreading ideology and propaganda.
+
+From the 1940s to the 1950s, propaganda was generally colourfully visual because not many could read or write and it was cheaper and quicker to produce. Posters portrayed Mao as a god-like figure and all-powerful, symbolised as the father of the nation. They were put up everywhere in publicc spaces, poor citizens used them to decorate their homes, and Mao's picture was always in newspapers, stamps, and pins.
+
+After 1949, radios and loudspeakers were also extensively used, broadcasting government propaganda and national anthems constantly.
+
+**Role models** of real or fictitious people were used in propaganda to teach the population how to behave.
+
+!!! example
+    Lei Fang was a fictitious soldier who died at 22 and was depicted as a happy and positive person despite experiencing a difficult childhood because his immediate family died at an early age. Mao's propaganda minister, Lin Biao, promoted his image by pushing his diary with positive comments about Mao. In 1953, Mao encouraged everyone to learn the "Lei Feng spirit", and the "Learn from Lei Feng" campaign launched during the Cultural Revolution aimed at getting people to obey and be loyal to Mao.
+
+#### Thought Reform Movement
+
+At the same time of the Antis campaigns, the [**Thought Reform Movement**](https://en.wikipedia.org/wiki/Thought_reform_in_China) from 1951 to 1952 was aimed to get the citizens to accept Marxism-Leninism and Maoism. Teachers and college staff were ordered to become Marxist-Leninists, and intellectuals who studied abroad were forced to confess as "implementers of the imperialist cultural invasion". School curricula were restructured, and propaganda and indoctrination were heavily used to change the citizens.
+
+#### Cult of personality
+
+Mao's cult of personality reached its peak during the Cultural Revolution. The people were drawn in by his promises after a decade of wars and corruption after the fall of the Qing dynasty. As the son of a poor peasant who worked hard to improve his social status, he maintained his image of being connected to the land and the peasants.
+
+!!! example
+    As a publicity stunt, at the age of 72 years, he swam through the Yangtze River in July 1966 to prove that he was fit to rule. Propaganda at the time claimed he swam nearly 15 kilometres in just over an hour.
+
+#### Education
+
+To create citizens supportive of the state, the CCP controlled the curriculum, reading material, and other information that students were exposed to. Schools effectively became indoctrination centres.
+
+!!! example
+    - Chinese textbooks were censored.
+    - The Little Red Book was the primary literary text.
+    - Elementary education focused on rote (memorisation via repetition) over critical thinking.
+    - Secondary education focused on testing, exams, physical education, and in practice it was mostly children of high ranking party members and government officials who attended.
+
+**Language reforms** resulted in a new form of Mandarin that allowed for people averywhere in the country to communicate. This standardisation made Mandarin easier to learn but caused local languages to disappear.
+
+The **Socialist Education Movement** from 1964 to 1966 had the goal to cleanse politics, the economy, organisation, and ideology as the "Four Cleanups". Intellectuals were sent to the countryside to be re-educated by peasants. They still attended school, but also worked in factories and with the peasants.
+
+#### Propaganda in the Cultural Revolution
+
+During the Cultural Revolution, it was required that a portrait or sculpture of Mao was present in each home.
+
+Mao called for class struggle in all educational institutions, and education as an ideal was condemned. Peasants and industrial workers were made teachers and pupils, and they were encouraged to criticise their teachers, who were forced to wear dunce caps and were paraded around. Schools and universities closed down and were made into barracks for Red Guards.
+
+!!! info
+    130 million young people stopped attending school.
+
+The [**Down to the Countryside Movement**](https://en.wikipedia.org/wiki/Down_to_the_Countryside_Movement) beginning in 1968 when the Red Guard was becoming violent and difficult to control had Mao order them to return to schools, with more violent radical groups being forcibly suppressed by the PLA. Urban students were sent to the countryside to experience peasant life. This ripped millions of families apart and many did not go back to school or university when they were allowed to return home.
+
+!!! info
+    From 1968 to 1976, 17 million young people were sent to rural areas.
+
+#### Arts
+
+Mao's wife, [Jiang Qing](https://en.wikipedia.org/wiki/Jiang_Qing) (Chinese: 江青) was made the chief of new Chinese culture and the "cultural purifier". She imposed censorship on anything that did not meet the criteria of "revolutionary purity".
+
+**Statues** of Mao were erected in front of state offices, universities, and schools.
+
+Traditional Chinese **operas** were replaced by those focused on the proletariat overthrowing class enemies. The more famous ones have political overtones with communist and/or revolutionary themes.
+
+Folk **music** was made modern and an attempt was made to put traditional Chinese music on equal footing. Rousing songs that appealed to the masses were composed, and Western music was entirely banned — symphonic and classical music fell in this category as they were associated with elitism and the West. Mao's poems were put into choral and classical music — the Red Guard sang lyrics derived from the Little Red Book.
+
+**"The East is Red"** was a revolutionary song that was the *de facto* national anthem during the Cultural Revolution. It was played through loudspeakers everywhere at dawn and dusk, sung by students at the beginning of the first class of each day, and shows began with this song.
+
+!!! quote ""The East is Red""
+    The east is red, the sun is rising,  
+    China has brought forth a Mao Zedong.  
+    He works for the people's welfare.  
+    Hurrah, he is the people's great saviour.  
+    Chairman Mao loves the people,  
+    He is our guide,  
+    To build a new China,  
+    Hurrah, he leads us forward!  
+    The Communist Party is like the sun  
+    Wherever it shines, it is bright.  
+    Wherever there is a Communist Party,  
+    Hurrah, there the people are liberated!  
+
+#### Historians
+
+???+ quote "Yan Yen (poet):"
+    As a result of the Cultural Revolution, you could say the cultural trademark of my generation is that we had no culture.
+
+???+ quote "Michael Lynch:"
+    Cultural terrorism — Result of the Cultural Revolution: Cultural wasteland. Artists who resisted the revolution were sent to re-educational labour camps, where they were brutally treated. Pianists and string players were made to scratch at the ground so they would never be able to play well again.
+
+### Nature, extent, and treatment of opposition
+
+Mao never had any direct opposition.
+
+#### Struggle sessions
+
+Struggle sessions were a form of public humiliation where people **self-criticise** to find the mistakes they made and to free themselves from error. It was used in the USSR from the 1920s and by Mao during the Yan'an Rectification Movement in 1941, the Anti-Rightist Campaign, and the Cultural Revolution.
+
+#### Purges
+
+**Landlords** were publicly humiliated after land reform and tried then executed in their villages until there were no more landlords. Red Guards looted and destroyed homes, attacking landlords and local officials.
+
+!!! info
+    An estimated 2-3 million landlorrds were killed.
+
+After Mao's retreat from the Great Leap Forward, in 1966, he decided to reorganise the party by "eliminating members who had taken a capitalist road". In 1968, Liu Shaoqi and Deng Xiaoping were denounced and stripped of their positions. Liu was struggled against and humiliated while the propaganda campaign accused him of being a traitor. He died in November 1969 in prison. Deng was also struggled against and publicly shamed but put under house arrest with his wife, later sent to Jiangxi for re-education through labour.
+
+#### Laojiao and Laogai
+
+[**Laojiao**](https://en.wikipedia.org/wiki/Re-education_through_labor) (Chinese: 劳教)), or re-education through labour, involved sending prisoners to labour camps. Designed to re-educate intellectuals, the goal was for prisoners to live and work with farmers and workers. They were also required to attend political classes where they denounced themselves and criticised their own thinking to realign their thoughts with communism. Those who committed smaller crimes that did not warrant capital punishment were sent here.
+
+[**Laogai**](https://en.wikipedia.org/wiki/Laogai) (Chinese: 劳改), or reform through labour, is a lake in *Avatar: The Last Airbender*. Located in Ba Sing Se in the Earth Kingdom, it hid an underground prison where those interned were brainwashed. It is also compared to the USSR's [*gulag*](https://en.wikipedia.org/wiki/Gulag). As internment camps for criminals, they were built in areas with extreme wewather and interns were forced to perform hard labour such as digging dithes and building roads under extremely poor conditions.
+
+!!! quote
+    The Earth King has invited you to Lake Laogai.
+
+### Extent of authoritarian control
+
+As China is still under authoritarian control and many numbers and facts are inaccurate or unknown, it is difficult to obtain concrete information. However, it appears very **totalitarian**.
+
+The Chinese followed **Confucian** philosophy, so social harmony was very important to them. Individuals were expected to accept their position in society and respect authority and the hierarchy. This led to less opposition and greater conformity. Unlike in the West, where the state is viewed as the oppressor, the state was viewed in China as family and the protector of civilisation.
+
+**Maoism** was described as "Marxism adapted to Chinese conditions". It held the following differences compared to Marxism:
+
+ 1. Peasants are the **agents of change** as opposed to urban workers, and they are moldable via social engineering. Mao used this to set up his own cult that would keep others in line.
+ 3. All revolutions require constant renewal, and permanent, **constant revolution** is required to prevent counter-revolution. Mao related stability to dangerous bureaucracy and privileged classes such as imperial China and the USSR under Khrushchev. He used this to justify constant revolution such as the Hundred Flowers Campaign and Cultural Revolution, creating a culture where violent upheaval was a regular way of life.
+
+### Historians
+
+???+ quote "Liang Heng, age 12, recount of late 1967 after 11 million young people travelled to Beijing for Mao:"
+    If there was a single thing that meant ecstasy to everyone in those days, it was seeing Chairman Mao. Ever since I had been in Peking, the possibility had been in the back of my mind, and, like every other Red Guard, I would have laid down my life for the chance… On May 1st Peng Ming was planning to go with a small group to conduct performances of Revolutionary songs at the Summer Palace during the day…and I was sometimes asked to carry drums and other instruments, so I went with Peng Ming’s group to the park. We were completely unprepared for what happened. In the middle of singing a song that Peng Ming had composed himself, we heard the great news: Chairman Mao was in the park! Gathering our instruments together hastily, we ran gasping to the spot, but it was too late. He was gone. All that remained of him was the touch of his hand on the hands of a few who had been lucky enough to get close to him. But we didn’t leave in disappointment. That trace of precious warmth in the palms of others seemed to us a more than adequate substitute for the real thing. Those Chairman Mao had touched now became the focus of our fervor. Everyone surged toward them with outstretched arms in hopes of transferring the sacred touch to their own hands. If you couldn’t get close enough for that, then shaking the hand of someone who had shaken the hands with Our Great Saving Star would have to do.
+
+## Foreign policy of China
+
+!!! definition
+    **Sinocentrism** is the idea that China is the cultural, political, and/or economic centre of the world.
+
+Mao was concentrated on maintaining and consolidating his power in China, but he supported the idea of an **international communist revolution** and focused on re-establishing China's position as a great power in the world.
+
+After [Joseph Stalin](https://en.wikipedia.org/wiki/Joseph_Stalin)'s death, Mao was seen by many to be the leader of the communist world.
+
+!!! example
+    In 1964, China detonated its first nuclear bomb, claiming its necessity for defense and in opposing the "US imperialist policy of nuclear blackmail and nuclear threats". Originally advocating for the abolishment of nuclear weapons but blocked by the US, evidence seems to suggest that Mao was quite willing to use nuclear weapons, believing that China's population would recover quickly.
+
+### Sino-Soviet relations
+
+China's relationship with the USSR started off well enough but steadily deteriorated by the 1960s.
+
+#### Treaty of Friendship, Alliance, and Mutual Assistance
+
+On 16 December 1949, Mao took his first trip abroad to Moscow only a few months after the establishment of the PRC. He was not met with great enthusiasm — Soviet leaders would meet him but would not drink or eat lunch with him, and there were no celebrations upon his arrival. He was in essence treated like a minor politician from a small communist country.
+
+The visit lasted three months and culminated in the signing of the treaty on 14 February 1950. In the [Sino-Soviet Treaty of Friendship, Alliance, and Mutual Assistance](https://en.wikipedia.org/wiki/Sino-Soviet_Treaty_of_Friendship,_Alliance_and_Mutual_Assistance), the USSR:
+
+ - recognised the People's Republic of China as the legitimate government of China
+ - lent $300 million over five years to aid economic and logistic recovery from a decade of warfare
+ - sent assistance from 11 000+ Soviet consultants and experts to be paid for by the PRC
+
+#### Korean War
+
+!!! background
+    From 1910 to August 1945, Korea was occupied by imperial Japan. After the end of World War II, the USSR and the US agreed to temporarily divide Korea along the 38th parallel and established a communist government in the north and a democratic government in the south, respectively.
+    <center><img src="/resources/images/korea-map.png" width=350></img></center>
+    (Source: Kognity)
+
+Under a desire to unify Korea under communism, North Korean leader [Kim Il-Sung](https://en.wikipedia.org/wiki/Kim_Il-sung) asked Stalin for approval to attack South Korea but was denied due to the [Berlin Blockade](https://en.wikipedia.org/wiki/Berlin_Blockade) at the time. His approval was granted later in April 1950 and the surprise attack launched on 25 June 1950, capturing important cities such as Seoul.
+
+In response, the [United Nations Security Council](https://en.wikipedia.org/wiki/United_Nations_Security_Council) declared North Korea as the aggressor and sent troops from 15 countries led by the US to restore peace under American general [Douglas MacArthur](https://en.wikipedia.org/wiki/Douglas_MacArthur), successfully retaking the 38th parallel.
+
+!!! info
+    - The UN motion to send troops only succeeded because the Soviet delegate with [veto powers](https://en.wikipedia.org/wiki/United_Nations_Security_Council_veto_power) was absent as a protest against UN refusal to accept the PRC as the legitimate government of China.
+    - The United States, South Korea, and other nations sent 350 000, 400 000, and 50 000 troops, respectively.
+
+!!! background
+    - North Korea made significant contributions to the CCP during their liberation of mainland China.
+    - Sino-American relations during this time period were especially poor due to the Truman administration declaring their support for the Republic of China on Taiwan as the "main China".
+
+When UN forces crossed the Yalu River on the China-Korea border, Mao felt China's security was at stake and also recognised an opportunity to assert power. Additionally, concern over border security with a hostile east due to a revived Japan, a desire to replace the Soviet influence in North Korea with their own, and Stalin pressing Mao to assist in the war led China to intervene.
+
+In October 1950, the [Chinese People's Volunteers](https://en.wikipedia.org/wiki/People%27s_Volunteer_Army) — in actuality a group of forces from the main People's Liberation Army under a different name to avoid official war with the US — deployed 500 000 troops push the UN troops back to the 38th parallel, resulting in a stalemate back at status quo by 1951. Talks lasted two years with US President [Eisenhower](https://en.wikipedia.org/wiki/Dwight_D._Eisenhower) threatening the use of nuclear weapons should they drag on until an [**armistice**](https://en.wikipedia.org/wiki/Korean_Armistice_Agreement) was signed on 27 July 1953.
+
+???+ info
+    **Casualties in the Korean War**  
+    <img src="/resources/images/korean-war-casualties.png" width=500>(Source: Kognity)</img>
+
+From China's perspective, the war was both a success and a failure. Mao propagandised the war as a total success in their aim to "Resist America and Defend Korea".
+
+| Success | Failure |
+| --- | --- |
+| Mao gained considerable prestige for being able to fight the US to a standstill | Heavy casualties — Mao's eldest son was killed in an air raid |
+| North Korea remained communist | Sino-American relations deteriorated further, and China faced a total embargo from the US |
+| China preserved its Manchurian border where its heavy industry was concentrated | The USSR-lent military equipment had to be repaid |
+
+#### Sino-Soviet split
+
+Stalin and Mao's relationship was tense, and the relationship between the two countries deteriorated much faster under [Nikita Khrushchev](https://en.wikipedia.org/wiki/Nikita_Khrushchev).
+
+!!! background
+    The border between the Soviet Union and China was determined by many treaties signed by various officials over many years. This left many gray areas where both countries claimed soverignity.
+
+**Border conflicts** in 1969 between the two countries led to seven months of unofficial conflicts and border clashes over various islands and rivers. At this point in time, both countries had nuclear weapons. Tensions persisted until September 1969 when the Chinese Premier and Soviet Minister of Foreign Affairs met in Beijing in September 1969.
+
+**Personality and ideological conflicts** between the leaders and countries worsened relations. Mao acted like an obedient student and never openly contradicted Stalin while he was in power but was often annoyed by his level of control over the CCP. Stalin was annoyed by Mao's attitude but needed a strong ally.
+
+!!! example
+    - In 1921, Mao believed that the rural population would lead China to revolution while Stalin interpreted that the proletariat could only be urban workers.
+    - After the end of World War II, Stalin requested Mao to work with the GMD, but Mao decided to wipe out the party and take power instead.
+
+In 1956, after Stalin's death, Khrushchev gave a [secret speech](https://en.wikipedia.org/wiki/On_the_Cult_of_Personality_and_Its_Consequences) denouncing his rule. Mao was alarmed by the brutality of these attacks and interpreted the speech as criticism against him and his own leadership in China.
+
+!!! definition
+    - **Détente** between the US and Soviet Union was the relaxation of strained relations between the two countries.
+    - **Revisionism** in this context is the betrayal of original revolutionary ideas.
+    - **Peaceful coexistence** is the belief that both capitalist and communist nations can exist together without war.
+
+In 1957, Khrushchev organised a [conference in Moscow](https://en.wikipedia.org/wiki/1957_International_Meeting_of_Communist_and_Workers_Parties), inviting all communist states including China. Mao complained about Khrushchev's [revisionism](https://en.wikipedia.org/wiki/Revisionism_(Marxism)) and [peaceful coexistence](https://en.wikipedia.org/wiki/Peaceful_coexistence)/[detente](https://en.wikipedia.org/wiki/D%C3%A9tente) approaches with the United States. Mao believed that it was the duty of communists to conduct class warfare and that the Soviet Union was being too soft on the West by making concessions — they were not fit to lead the communist world. He thought that a final violent conflict was needed with capitalism.
+
+In 1958, Khrushchev was invited to visit China and Mao treated him with disdain, aiming to make his visit unpleasant: the Soviet delegation was placed in a hotel with no AC, Mao invited Khrushchev to swim in his private pool — aware of his inability to swim, and refused any proposals for military cooperation and defense initiatives. In response, Khrushchev pulled most advisors out from China and removed all of them by 1960.
+
+**Chinese meddling in Soviet international affairs** resulted in open defiance against the USSR. Unimpressed with de-Stalinisation, when Albania left the USSR in 1961, China supported them against the Soviet Union, pouring money into the country. In return, Albanian leader Enver Hoxha declared his support for Mao.
+
+Further examples of disagreements during the Sino-Soviet split include:
+
+ - 1958: Mao wanted the USSR to use their first satellite, Sputnik-1, to aid revolutionary efforts, but Khrushchev refused to risk nuclear conflict.
+ - 1959: [China invaded Tibet](https://en.wikipedia.org/wiki/Annexation_of_Tibet_by_the_People's_Republic_of_China), but the USSR refused support and withdrew its support from the Chinese nuclear program by refusing to give them a prototype weapon.
+ - 1962: China disagreed with the USSR in backing down and making in a deal in the [Cuban Missile Crisis](https://en.wikipedia.org/wiki/Cuban_Missile_Crisis), wanting them to support third world countries in their fight against communism.
+ - 1963: China opposed the USSR signing of the [Partial Nuclear Test Ban Treaty](https://en.wikipedia.org/wiki/Partial_Nuclear_Test_Ban_Treaty) against nuclear weapons, sparking a fierce propaganda war.
+
+### Cross-Strait relations
+
+Mao never recognised Taiwan as an independent state.
+
+In the [First Taiwan Strait Crisis](https://en.wikipedia.org/wiki/First_Taiwan_Strait_Crisis) from 1954 to 1955, the PLA bombed various islands near Taiwan and then seized the [Yijiangshan Islands](https://en.wikipedia.org/wiki/Yijiangshan_Islands) in a military conflict. This led to the [Formosa Resolution](https://en.wikipedia.org/wiki/Formosa_Resolution_of_1955) to be enacted by the US Congress — that US forces would defend Taiwan against any attack from the mainland.
+
+In 1958, Mao ordered the PLA to attack the Taiwan-surrounding and -controlled islands of Kinmen and Matsu without discussion with the USSR. As the US prepared for war because of the Formosa Resolution, Mao stood down because he did not have USSR backup. In the aftermath, Khrushchev accused Mao of being a [Trotskyist](https://en.wikipedia.org/wiki/Trotskyism) who had lost all sense of reality.
+
+### Sino-American relations
+
+The US viewed China as an aggressive country with the objective of threatening the security of the non-communist states surrounding it. China viewed the US as their enemy.
+
+!!! example
+    The [Red Scares](https://en.wikipedia.org/wiki/Red_Scare) in the US and the anti-American and anti-capitalist propaganda in China (e.g., "Death to the American imperialists") meant that the two countries could not easily communicate diplomatically in public.
+
+To "contain" communism in China, the US signed several treaties such as the [Southeast Asia Treaty Organization](https://en.wikipedia.org/wiki/Southeast_Asia_Treaty_Organization) and the [ANZUS Treaty](https://en.wikipedia.org/wiki/ANZUS) to ward off China. Additionally, they gave the Chinese seat at the UN to Taiwan, pushed allies to avoid entertaining diplomatic relations with Taiwan, supported countries that felt threatened by China, encouraged the split between the USSR and China, and implemented a trade embargo.
+
+From 1970 onward, the two countries began to grow closer — China wanted a new strong ally as Sino-Soviet relations were deteriorating and the US wanted a way out of the Vietnam War.
+
+Taking advantage of [ping-pong diplomacy](https://en.wikipedia.org/wiki/Ping-pong_diplomacy), the exchange of ping-pong players between the US and China, US Secretary of State Henry Kissinger went secretly to meet Chinese Premier Zhou Enlai.
+
+The [**Shanghai Communiqué**](https://en.wikipedia.org/wiki/Shanghai_Communiqu%C3%A9) signed between the two countries on 28 February 1972 was a statement issued by both countries during US President Richard Nixon's visit to China — the first visit to the PRC by any US President — and began the normalisation of relations between them. Nixon described the visit as a mission for peace with the goal to re-establish communications after a generation of hostility, and agreements were reached to expand cultural, educational, and journalistic contracts.
+
+### Historians
+
+???+ quote "*Mao: a Biography* - Ross Terill (revisionist and somewhat sympathetic to Mao), 1995:"
+    Mao knew little of the world outside China, and nothing of the capitalist world… Yet Mao took a lively interest in the world beyond the Soviet Bloc during the early 1960s… It was as if the split with Russia in 1960 took a burden off the back of Chinese diplomacy. Instead of being junior partner in someone else’s show, Mao made China its own one-man show on the broadening stage of the Third World.
+
+???+ quote "*Mao: Profiles in Power* - Shaun Breslin, 1998:"
+    Mao’s main objective in all of his foreign policy initiative from 1949 to 1976 was to safeguard China’s borders and restore China to its rightful position on the world stage. Mao had a traditional Sinocentrism: the notion that China is the central place in the world, and that only those who recognise and accept Chinese superiority can be considered to be civilised.
+
+
+???+ quote "*Mao Zedong* - Maurice Meisner (sympathetic to socialist ideology and goals), 2007:"
+    Mao’s foreign policy clothed itself in revolutionary rhetoric, but was conservatively cautious in substance, based on narrow calculation of China’s national self-interest… In Mao’s view the Soviet Union posed a greater danger to China than did the United States.
+
+## Resources
+
+ - [IB History Syllabus](/resources/g11/ib-history-syllabus.pdf)
+ - [Textbook: Origins and Development of Authoritarian and Single-Party States](/resources/g11/textbook-authoritarian-states.pdf)
+ - [Textbook: Causes and Effects of 20th-Century Wars, Second Edition ](/resources/g11/textbook-cause-20-century-wars.pdf)
+ - [Textbook: The Move to Global War](/resources/g11/textbook-move-global-war.pdf)

docs/chy4uz.md

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+# HL History - 2
+
+The course code for this page is **CHY4UZ**.
+
+o7

docs/eng3uz.md

@@ -1,8 +1,8 @@
-# HL English - A
+# HL English - 1
 
 The course code for this page is **ENG3UZ**.
 
-## Literary Techniques/Devices
+## Literary techniques/devices
 
 ### Description
 
@@ -33,6 +33,7 @@ The course code for this page is **ENG3UZ**.
 	- e.g., *The cat in the hat knows a lot about that!*
  - Cacophony: The use of words and phrases that are harsh to the ear by using consonants that require pressure to to say (e.g., p, b, d, g, k, ch-, sh-).
 	- e.g., *How much wood could a woodchuck chuck if a woodchuck could chuck wood?*
+ - Catalexis: A line missing a syllable at the end or beginning.
  - Consonance: A number of words with the same consonant sound, not at the beginning, that appear close together.
 	- e.g., *Shelley sells shells by the seashore.*
  - Dialect: A regional variety of language with spelling, grammar, and pronunciation that differentiates a population from others around them.
@@ -134,11 +135,69 @@ The course code for this page is **ENG3UZ**.
  - Theme: The "main idea" or underlying meaning of a literary work, which can be given directly or indirectly.
 	- e.g., *"Never forget that* you are royalty, *and that hundreds of thousands of souls have suffered and perished so you could become what you are. By their sacrifices, you have been given the comforts you take for granted. Always remember them, so that their sacrifices shall never be without meaning."* (*Eon Fable*, ScytheRider)
 
+## General writing
+
+### Active and passive voice
+
+Active writing involves having the subject of a sentence perform the action, while passive writing involves the subject receiving the action.
+
+!!! example
+    Active: *Joey mangled the teacher.*<br>
+    Passive: *The teacher was mangled by Joey.*
+
+### Redundancy
+
+When possible, any words that do not add meaning should not be present.
+
+## Essay writing
+
+An essay is a relatively brief non-fiction piece of writing (can be read in one sitting) that is focused on one subject.
+
+### Types of essays
+
+**Compare and contrast** essays look for similarities and differences between two concepts, objects, or ideas. Arguments are either structured **subject by subject** or **point by point**.
+
+**Cause and effect** essays attempt to establish a causal connection between ideas or events, in essence explaining why something happens/ed.
+
+**Definitional** essays focus on defining a term, idea, or concept.
+
+**Narrative** essays make a point by telling a non-fictional event in the structure of a short story in first person.
+
+ - They are usually told chronologically.
+ - They usually have a purpose/thesis that is stated in the opening sentence.
+ - Dialogue is permitted in narrative essays.
+ - They are written with vivid imagery and descriptions to involve the reader with the goal of relating in some way to the thesis.
+ - Much like fictional stories, they should have conflicts and events.
+ - They are usually written in first person.
+
 ## Essay analysis
 
+### Thesis statement
 
+The thesis statement of an essay is effectively its central assertion, and may appear in different places:
+
+ - An **initial** thesis appears within the first paragraph or so.
+ - A **delayed** thesis appears anywhere else, but often appears at the end of the essay.
+ - An **inferred** thesis is one that does not appear in the essay at all. It is instead up to the reader to glean an inferred thesis via inference.
+
+### Purpose
+
+Essays may be classified under two general categories — persuasive/argumentative or expository/informative. Persuasive essays argue to convince a reader to take their position, while expository essays aim to explain a topic without bias.
+
+### Audience
+
+The target audience of an essay and those that it would appeal to are useful in determining the point of an essay. Some factors that may indicate the audience include,
+
+ - the topic of the essay
+ - bias of the author
+ - diction/language used in the essay
+ - the use of jargon or slang
+ - the formality of the essay — formal essays are typically more organised and appeal to logic more than informal essays
+ - literary techniques
+ - the tone/attitude of the author of the essay via the use of emotionally charged words
 
 ## Resources
 
- - [Analysis of a Poem](/resources/g11/central-asserion-1.pdf)
+ - [Analysis of a Poem](/resources/g11/central-assertion-1.pdf)
  - [Essay Analysis](/resources/g11/essay-analysis.pdf)
+ - [Literary Criticism Overview](/resources/g11/literary-criticism.pdf)

docs/eng4uz.md

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+# HL English - 2
+
+The course code for this page is **ENG4UZ**.

docs/fsf3u7.md

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+# SL French - 1
+
+The course code for this page is **FSF3U7**.
+
+Ahaha good luck with this I'm outta here
+
+## Resources
+
+ - [Textbook: Oxford IB French B Course Companion](/resources/g11/textbook-french-b-second-edition.pdf) ([Answers](/resources/g11/textbook-french-b-second-edition-answers.pdf))

docs/fsf3uz.md

@@ -0,0 +1,9 @@
+# SL French - 2
+
+The course code for this page is **FSF3UZ**.
+
+o7
+
+## Resources
+
+ - [Textbook: Oxford IB French B Course Companion](/resources/g11/textbook-french-b-second-edition.pdf) ([Answers](/resources/g11/textbook-french-b-second-edition-answers.pdf))

docs/g12/snc4mz.md

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+---
+sidebarDepth: 5
+---
+
+# HL Chemistry 3
+
+The course code for this page is **SNC4MZ**.
+
+## Organic chemistry
+
+::: info DEFINITION
+
+- An **organic molecule** is one with at least one carbon atom covalently bonded to another carbon or hydrogen atom (i.e., at least one C-H or C to C bond)
+
+:::
+
+Carbon is unique in organic chemistry as it is the only element with the following properties:
+
+- It is in the second row of the periodic table, meaning it has less electron shells, thus forming stronger bonds
+- It can covalently bond to up to 4 other atoms
+- Because each of its valence electrons is involved in bonding, it can form single through triple bonds
+- The molecular geometry can be anything from tetrahedral to linear depending on its bonding
+
+Carbon is also able to bond to itself in the following ways:
+
+- long straight chains
+- long straight chains with branches
+- rings
+
+<img src="/resources/images/cool-carbon.png" width=700 />
+
+### Simple hydrocarbons
+
+::: info DEFINITION
+
+- A **branched hydrocarbon** is one with at least one "side group" extending from the main hydrocarbon chain.
+- A **functional group** is a group of atoms responsible for the characteristic properties of a molecule (e.g. C=C)
+- A **homologous series** is a family of organic compounds with the same functional group but the hydrocarbon chain length changes by 1 $\ce{CH2}$ group.
+
+:::
+
+These only contain carbon and hydrogen.
+
+**Alkanes** are a homologous series that only contain single bonds between carbons, and are named with the number of carbons with the suffix "-ane".
+
+<img src="/resources/images/alkanes.png" width=700 />
+
+| Carbon atoms | Prefix |
+| ------------ | ------ |
+| 1            | Meth   |
+| 2            | Eth    |
+| 3            | Prop   |
+| 4            | But    |
+| 5            | Pent   |
+| 6            | Hex    |
+| 7            | Hept   |
+| 8            | Oct    |
+| 9            | Non    |
+| 10           | Dec    |
+
+::: warning EXAMPLE
+A molecule with only hydrogen and three carbon atoms all held together with single covalent bonds is called "propane".
+:::
+
+**Alkenes** contain **at least** one carbon-carbon double bond and are named with a prefix with the total number of carbon atoms and "-ene".
+
+**Alkynes** contain **at least** one carbon-carbon triple bond and are named with a prefix with the total number of carbon atoms and "-ene".
+
+::: danger WARNING
+The lack of standardisation prior to IUPAC means that some IUPAC names have common names that are still widely used today.
+
+- acetylene: **ethyne**
+- vinyl: **propene**
+- ethylene: **ethene**
+
+:::
+
+The general formula for an **acyclic** hydrocarbon with no rings is as follows, where $n$ is the number of carbon atoms, $x$ is the number of double bonds, and $y$ is the number of triple bonds.
+$$\ce{C_nH_{2n+2-2x-4y}}$$
+
+### Representing organic compounds
+
+A simple **molecular formula** is the least useful as it provides no information on structure and bonding.
+$$\ce{C6H14}$$
+
+A **complete structural diagram** shows all atoms by their chemical symbols and uses lines like a Lewis Dot diagram to represent bonds. VSEPR shapes do not need to be taken into account because these are 2D representations of the molecule.
+
+A **condensed structural diagram** is a complete structural diagram but C-H bonds are aggregated into a formula.
+$$\ce{CH3 - CH2 - CH2 - CH2 - CH2 - CH3}$$
+
+A **structural formula** or **expanded molecular formula** is a condensed structural diagram but there are no bond lines. The bond organisation is inferred based on the number of hydrogens on each carbon. Carbon chain side groups (branches) are shown with parentheses.
+$$\ce{CH3CH2CH2CH2CH2CH3}$$
+
+A **condensed structural formula** is a structural formula but any consecutive repeated $\ce{CH2}$ groups are factored with parentheses.
+$$\ce{CH3(CH2)_4CH3}$$
+
+A **line diagram** or **skeletal structural formula** removes carbons and hydrogens and replaces all carbon-carbon bonds with lines, where the number of lines represents the type of bond. Each line is bent where a carbon atom would be, except for triple bonds as those are linear. Non-carbon groups such as $\ce{OH}$ can be shown in collapsed form.
+
+::: warning EXAMPLE
+These are the ways to represent pentane, $\ce{C5H12}$. The structural formula is mislabeled as a condensed structural diagram.
+<img src="/resources/images/pentane.png" width=700 />
+:::
+
+### General nomenclature
+
+To name an organic compound:
+
+1. Find the **longest acyclic chain** of carbon atoms as the parent chain.
+2. Assign numbers from 1 to $n$ for each carbon atom in the parent chain.
+   - The numbers should be arranged in a way that the highest priority functional group in the chain is assigned the lowest number possible.
+   - Apply the **first branch rule** only if there is a tie: If there are side chains, the parent chain should be numbered such that the location of any side chains have the lowest number possible.
+
+- If there is a tie, the location with the most branches wins.
+- If there is a tie, the rest of the chain is compared in sequence applying the first branch rule.
+- If there is a tie, the first location with the side chain group name that is alphabetically greater wins.
+- If there is a tie, it doesn't matter which side is picked as the whole thing is symmetrical.
+
+3. Name the main chain based on the name of the functional group and location number for the functional group in the format "number-name".
+4. Name the side groups.
+   - If the group is not carbon, name it by its identity.
+   - Otherwise, name the hydrocarbon based on the number of carbons in the side group with the ending "yl".
+   - If there is more than one identical side group in the **whole chain**, combine their numbers and names with a Greek prefix.
+   - Assign a number representing the carbon atom of the parent chain that the side group is attached to in the form "numbers-name".
+5. Arrange the name with each side group with their numbers in alphabetical order, discounting any prefixes due to duplicates, followed by the parent chain.
+6. Join everything together:
+   - Drop the ending vowel from the prefix if there is a double vowel unless it is "i".
+   - Separate numbers from words with dashes.
+   - Separate numbers from numbers with commas.
+   - Do not separate words from words.
+
+::: tip
+In hydrocarbons:
+
+- Atoms with double or triple bonds share equal priority as the highest functional group.
+- The main chain will be named as an alkane if there are only single bonds.
+- If there is exactly one double or triple bond, it will be named as an alkene or alkyne with its position inserted between the prefix and ending.
+  - e.g., "pentane", "pent-2-ene"
+- If there are multiple double or triple bonds, their numbers are also included, but an "a" is appended to the prefix and a Greek prefix added to the suffix.
+  - e.g., "penta-1,3-diene", "hexa-1,3,5-triyne"
+- If there are both double and triple bonds, the "-ene" becomes "-en" and is always before "-yne".
+  - e.g., "pent-4-en-2-yne"
+
+:::
+
+::: warning EXAMPLE
+tf
+:::
+
+Other **side chains** with equal priority as double or triple bonds _in side chains_ include:
+
+- halogens, which have their "-ine" suffix replaced with "o" (e.g., "chloro")
+- $\ce{NO2}$: "nitro-"
+- benzene (as a side chain): "phenyl"
+
+If there is no other option and there is a **branched side chain**, name it based on the total number of carbon atoms in the side chain.
+
+::: warning EXAMPLE
+tf
+:::
+
+### Cyclic aliphatic hydrocarbons
+
+These contain rings that **are not** benzene rings.
+
+$$\ce{C_nH_{2n-2x}}$$
+
+::: danger WARNING
+Cyclic hydrocarbons **do not** contain any triple bonds as it would force the carbon ring to widen too much.
+:::
+
+Cyclic aliphatic hydrocarbons are named the same way as acyclic hydrocarbons except they have a "**cyclo-**" at the start of the name of their parent chain.
+
+::: warning EXAMPLE
+cyclohexa-1,3-diene
+:::
+
+The initial double bond should be numbered such that the lowest number is assigned to both sides of the bond (numbers 1 and 2 should be to either side of the double bond). If there is more than one double bond, the ring should be numbered such that the lowest number is assigned to both.
+
+The **first branch rule** still applies. (See [HL Chemistry 3#General nomenclature](/snc4mz/#general-nomenclature).)
+
+::: warning EXAMPLE
+tf
+:::
+
+::: danger WARNING
+Rings can be side chains, and are named accordingly (e.g., "cyclopropyl"). The "cyclo-" prefix is counted when sorting names alphabetically as it describes the group.
+:::
+
+### Cyclic aromatic hydrocarbons
+
+These contain benzene rings, which do not actually have single/double bonds as they actually have delocalised pi bonds.
+
+<img src="/resources/images/benzene.png" width=700 />
+
+As benzene rings do not have double bonds, they are named according to the **first branch rule**.
+
+### Isomers
+
+**Structural isomers** are two chemicals that have the same chemical formulas but have different structural formulas, resulting in different chemical properties.
+
+**Hydrocarbon chain isomers** are two chemicals with the same chemical formulas but have different carbon/hydrogen arrangements.
+
+::: warning EXAMPLE
+The following are two hydrocarbon chain isomers (and, by extension, structural isomers) of $\ce{C5H12}$.
+<img src="/resources/images/structural-isomer-g5h12.png" width=700 />
+:::
+
+**Positional isomers** are two chemicals with the same chemical formulas **and functional groups** but have different structural formulas.
+
+::: warning EXAMPLE
+The following are positional isomers (and, by extension, structural isomers) of $\ce{C4H8}$.
+<img src="/resources/images/positional-isomers.png" width=700 />
+:::
+
+**Functional group isomers** are chemicals with the same chemical formulas but **different functional groups**.
+
+::: warning EXAMPLE
+The following are functional group isomers (and, by extension, structural isomers) of $\ce{C3H6O2}$.
+<img src="/resources/images/functional-group-isomers.png" width=700 />
+:::
+
+**Geometric** or **cis/trans isomers** are two chemicals have the same chemical formulas and atom arrangements but are positioned differently, thus having ambiguous names.
+
+In order for this to occur, there must be two different atoms or groups of atoms bonded to each carbon atom in the double bond.
+
+- A **cis** hydrocarbon isomer will have its main chain enter and exit the double bond on the **same side**.
+- A **trans** hydrocarbon isomer will have its main chain enter and exit the double bond on **opposite sides**.
+
+Unlike the examples below, these should be named with "cis" or "trans" at the beginning as a **separate word without a hyphen**.
+
+::: warning EXAMPLE
+The following are two geometric isomers of but-2-ene:
+<img src="/resources/images/cis-trans-but-2-ene.png" width=700 />
+:::
+
+- In acyclic compounds, this is because the double bond prevents simply rotating one side but not the other as it would force breaking the pi bond.
+- In cyclic compounds, this is because the ring's other side is similar to a double bond, preventing rotation around the axis.
+
+::: warning EXAMPLE
+The following are cis-trans isomers of dichlorocyclobutane (notice the chlorine):
+<img src="/resources/images/cis-trans-ring.png" width=700 />
+:::
+
+Isomers may have different physical properties in:
+
+- **polarity**: a cis isomer may cause a molecule to be polar as opposed to its trans variant
+- **packing efficiency**: a non-branching hydrocarbon chain will pack better than a branching one, and a continuously trans chain will pack better than a cis one
+
+These change the strength and type of intermolecular forces involved so affect their melting/boiling points.
+
+Isomers may also have different chemical properties as cis isomers are more likely to bump into themselves to make some reactions more viable
+
+### Benzene reactions
+
+::: info DEFINITION
+
+- An **electrophile** is any species that is or would be electron deficient (+) in the presence of a pi bond.
+  ::
+
+In reactions involving a benzene ring, the ring itself is **stable** and will not break apart because of the strength of delocalised pi bonds.
+
+Therefore, only the hydrogens can be swapped out via **electrophilic substitution**, where an hydrogen atom is substituted with an electrophile. The concentration of electrons in the delocalised pi area attracts electrophiles to initiate the bond.
+
+In the mechanism diagram below, $\ce{E+}$ represents the electrophile. Curly arrows are used to show the movement of electrons from the **delocalised area to the electrophile** and **hydrogen atom to the delocalised area**.
+
+<img src="/resources/images/benzene-substitution-mechanism.png" width=900 />
+
+The **first step** (the change from the first to the second diagram) is the **slow step** due to the highest activation energy due to the requirement to break a bond.
+
+<img src="/resources/images/benzene-substitution-mechanism-graph.png" width=900 />
+
+#### Benzene nitration
+
+::: info DEFINITION
+
+- A **nitrating mixture** is a mixture of concentrated sulfuric and nitric acids.
+
+:::
+
+In a **nitrating mixture**, benzene will react with positive nitronium ions at **~50°C** to form nitrobenzene, outlined in the reaction mechanism diagrams below.
+
+$$\ce{C6H6 + HNO3_{(aq)} ->[conc H2SO4][50^\circ C] C6H5NO2 + H2O_{(l)}}$$
+
+<img src="/resources/images/benzene-nitration-mechanism.png" width=900 />
+
+The first step is to **form the nitronium ion** through a Bronsted-Lowry acid-base reaction between the acids.
+
+$$\ce{HNO3_{(aq)} + H2SO4_{(aq)} <=> H2NO3+_{(aq)} + HSO4-_{(aq)}}$$
+
+The lone pair on the oxygen of the nitric acid attracts a hydrogen atom, which becomes an $\ce{H+}$ ion as sulfuric acid's oxygen takes its electrons. The hydrogen ion bonds to the nitric acid.
+
+$$\ce{H2NO3+_{(aq)} <=> H2O_{(l)} + NO2+_{(aq)}}$$
+
+The oxygen-hydrogen group is conveniently able to form water by taking both electrons it was sharing with the nitrogen. The other single-bonded oxygen compensates with a dative covalent bond with the nitrogen to form the nitronium ion.
+
+The second step is to **react with benzene** through electrophilic substitution, with electrons moving back from the dative oxygen-nitrogen bond back to the oxygen.
+
+### Alkane reactions
+
+::: info DEFINITION
+
+- **Halogenation** is the introduction of a halogen into a compound.
+
+:::
+
+#### Substitution halogenation
+
+Because a sigma bond must be broken, alkanes are not very reactive. In the presence of light, alkanes will react with halogens in their standard state through halogenation, replacing one of their hydrogens. **Fluorine** is an exception that does not require light because it is highly reactive.
+
+If the halogen is in excess and the reaction continues, more of the halogen (**not the hydrogen-halogen product**) will react with the alkane until all hydrogens have been substituted.
+
+::: warning EXAMPLE
+$$\ce{CH3CH3 + Cl2_{(g)} ->[light] CH3CH2Cl + HCl_{(g)}}$$
+
+If $\ce{Cl2}$ is in excess:
+
+$$
+\ce{
+CH3CH2Cl + Cl2_{(g)} ->[light] CH3CHCl2 + HCl_{(g)} \\
+... \\
+CCl3CHCl + Cl2_{(g)} ->[light] CCl3CCl3 + HCl_{(g)}
+}
+$$
+
+:::
+
+The order that hydrogens are substituted in is **random**. If there is more than one possibility, all of them are written as products, ignoring balancing.
+
+::: warning EXAMPLE
+Propane reacts with chlorine gas to form either 1-chloropropane or 2-chloropropane.
+$$\ce{CH3CH2CH3 + Cl2 ->[hf] CH3CH2CH2Cl + CH3CHClCH3 + HCl}$$
+:::
+
+::: warning EXAMPLE
+1-bromoethane reacts with chlorine gas to form either 1,1-dibromoethane (40% chance) or 1,2-dibromoethane (60% chance) because each hydrogen is equally likely to be substituted, and there are 2 and 3 that would form them, respectively.
+$$\ce{CH2ClCH3 + Cl2 ->[hf] CHCl2CH3 + CH2ClCH2Cl + HCl}$$
+:::
+
+#### Free radical substitution
+
+::: info DEFINITION
+
+- A **free radical** is a species with a lone unpaired electron.
+- **Homolytic fission** is the dissociation of a chemical bond in a neutral molecule where each product takes one electron, generating two free radicals.
+- **Heterolytic fission** is the dissociation of a chemical bond in a neutral molecule where one product takes both electrons.
+
+:::
+
+The free radicals are first produced with the help of light energy.
+
+$$\ce{Br2 ->[hf] Br. + Br.}$$
+
+They are then spread to organic compounds and reformed.
+
+$$
+\ce{
+Br. + CH4 -> .CH3 + HBr \\
+Br2 + .CH3 -> CH3Br + Br.
+}
+$$
+
+This cycle only ends when all radicals are used up, through reactions that end up with a net loss in radicals, such as:
+
+- $\ce{Br. + Br. -> Br2}$ (unlikely, contributes a little)
+- $\ce{.CH3 + Br. -> CH3Br}$ (likely)
+- $\ce{.CH3 + .CH3 -> CH3CH3}$ (likely)
+
+::: danger WARNING
+The free radical is on the carbon atom, not the hydrogen atoms, so the marker goes at the beginning.
+:::
+
+### Alkene/yne addition reactions
+
+::: info DEFINITION
+
+- A **carbocation** is a compound with a $\ce{C+}$ atom.
+- The **primary** (1°), **secondary** (2°), and **tertiary** (3°) carbocations are carbocations bonded to one, two, and three other carbon atoms, respectively.
+  :::
+
+The presence of double/triple bonds make alkenes and alkynes more reactive and also allow the **addition** of species as pi bonds are easier to break. Addition always takes precedence over substitution when possible.
+
+These **spontaneous** reactions break the double/triple bond down a level and slot themselves in (i.e., alkynes form alkenes, alkenes form alkanes).
+
+$$
+\ce{alkene + Br2 -> alkaneBr2} \\
+\ce{alkyne + Br2 -> alkeneBr2}
+$$
+
+<img src="/resources/images/alkene-addition.png" width=900 />
+
+1. If the non-alkene/yne reactant does not have a dipole moment, the electrons concentrated in the double/triple bond of the alkene/yne induce a dipole by repelling the electrons closest to it.
+2. The positive dipole (such as H in HBr) is attracted to the double bond, and **two electrons** in the bond are used to form a **dative** bond with the positive dipole.
+3. No longer needing its old bond, the previously positive dipole loses **both electrons** in its old bond to the negative dipole.
+4. The now positive carbon atom attracts the now negative ion.
+5. The negative ion forms a **dative** bond with the positive carbon atom.
+
+::: danger WARNING
+
+- If an **alkene is formed**, the same randomness of where the atoms attach applies, so it is possible that a cis/trans isomer is formed.
+- If an **asymmetrical alkane** is formed, the same randomness of where the atoms attach applies after applying Markovnikov's rule, so it is possible that positional isomers are formed.
+  :::
+
+**Markovnikov's rule** states that in Soviet Russia, the rich get richer. Hydrogens preferentially bond to the carbon with the **most hydrogens** if there is one — otherwise it randomly chooses one available.
+
+This is because carbocations with that are _more highly substituted_ (are bonded to more carbon atoms) are more stable, so they last longer and are more likely to form a bond with the negative dipole.
+
+The preferred product is the **major product** while the other is the **minor product**. Some minor product will still be produced if the negative dipole is speedy enough, although it will be vastly outnumbered by the major product.
+
+#### Halogenation
+
+Unlike alkane substitution, addition halogenation is spontaneous.
+
+$$\ce{alkene + Br2 -> alkaneBr2}$$
+
+::: warning EXAMPLE
+This process is used to test for alkenes/alkynes in a solution. As bromine water is red-brown, if alkenes/alkynes are present, the water will be **decolourised** from red-brown to become more colourless.
+:::
+
+::: warning EXAMPLE
+<img src="/resources/images/halogenation.jpeg" width=700 />
+:::
+
+#### Hydrogenation
+
+The addition of hydrogen follows the same principle as that of halogenation.
+
+$$\ce{alkene + H2 ->[\text{heat, high pressure, Ni/Pt/Pd}] alkane}$$
+
+::: warning EXAMPLE
+<img src="/resources/images/hydrogenation.png" width=700 />
+:::
+
+#### Hydrohalogenation
+
+The addition of both a hydrogen and halogen follows similar principles.
+
+$$\ce{alkene + HBr -> alkaneBr}$$
+
+#### Hydration
+
+Hydration is the addition of an $\ce{H-OH}$ group (colloquially known as water) onto an alkene/yne within 6 mol/L $\ce{H+}$ to produce an alcohol.
+
+$$\ce{alkene + H2O ->[6 mol/L H+] alkaneOH}$$
+
+::: warning EXAMPLE
+<img src="/resources/images/hydration.jpeg" width=700 />
+:::
+
+### Nucleophilic substitution
+
+::: info DEFINITION
+
+- A **nucleophile** is a species with a lone pair or a negative charge.
+  :::
+
+Nucleophilic substitution replaces a group of atoms attached to a C with a nucleophile. Both processes involve the **leaving group** taking both electrons, becoming negative in the process, and forming a carbocation as the other product, which attracts and bonds with the nucleophile.
+
+Effectively all reactions here involve the formation or stealing of dative covalent bonds.
+
+Where $\ce{X}$ is a halogen:
+$$\ce{R-X_{(l)} + OH-_{(aq)} -> R-OH_{(aq)} + X-_{(aq)}}$$
+
+If substituting with hydroxide, it must be **warm** and **aqueous** (dilute).
+
+Generally:
+
+| Carbocation type | Substitution type |
+| ---------------- | ----------------- |
+| Primary          | S<sub>N</sub>2    |
+| Secondary        | Both/either       |
+| Tertiary         | S<sub>N</sub>1    |
+
+#### S<sub>N</sub>1
+
+This **two-step** reaction involves the heterolytic fission of the C-X bond to form a carbocation + halide ion (slow), followed by the nucleophile's lone pairs/negative charge attracting it to the carbocation.
+
+The "1" refers to the order of the rate-limiting step being a **unimolecular** collision.
+
+<img src="/resources/images/sn1-1.png" width=700 />
+<img src="/resources/images/sn1-2.png" width=700 />
+
+::: danger WARNING
+Be sure to draw VSEPR, unlike in the diagrams above.
+:::
+
+#### S<sub>N</sub>2
+
+This **single-step** reaction has the nucleophile forming a bond with the central atom **opposite the leaving group** in a "back-side attack". The oncoming nucleophile repels the other groups, causing them to move away, effectively **reflecting** ("inverting") the remaining groups across the vertical axis.
+
+The "2" refers to the order of the rate-limiting step being a **bimolecular** collision.
+
+<img src="/resources/images/sn2-substitution.png" width=900 />
+
+::: danger WARNING
+Dashes must be drawn for the transition state for bonds breaking/forming. In this case, drawing the front/back lines for the bottom two atoms may be ignored in favour of regular lines instead to avoid the ambiguity of forming bonds.
+:::
+
+#### Factors affecting substitution type
+
+**Steric hindrance** is the effect of other parts of a molecule getting in the way to the central atom, preventing a reaction. If there is not enough space for a backside attack, S<sub>N</sub>2 cannot happen. Therefore, this makes 3° S<sub>N</sub>2 substitution not viable.
+
+**Steric stress reduction** is the resistance of groups against being forced together. In a 3° carbocation, pushing the groups together for a backside attack increases steric stress. This encourages S<sub>N</sub>1 substitution **only for 3°** to maintain a tetrahedral geometry.
+
+The **positive inductive effect** is the effect that causes more highly substituted carbons to be more stable. Electrons on neighbouring carbon atoms can move closer to the carbon ion, creating an electron-donating effect that slightly balances its charge, increasing its stability and thus window of opportunity for a **S<sub>N</sub>1** substitution.
+
+### Alcohols
+
+An **alcohol** is an organic compound with a $\ce{-OH}$ (hydroxyl) functional group.
+
+It has a **higher priority** than double and triple bonds, and alcohol names are suffixed with **-ol**.
+
+::: danger WARNING
+The -ol suffix is a standard suffix following the same numbering rules as -en and -yne. As functional groups are ordered from lowest to highest priority in their name, similar to how a -yne can have an -en, an -ol can also have an -en and **-yn** before it.
+:::
+
+- Therefore, $\ce{CH3OH}$ is methanol, _not_ methol.
+
+::: warning EXAMPLE
+Some alcohols and their common names:
+
+- **Glycerol**: propan-1,2,3-triol
+- **Ethyl alcohol** or drinking alcohol: ethanol
+- **Isopropanol** or rubbing alcohol: propan-2-ol
+  :::
+
+The **type** of an alcohol (primary/secondary/tertiary) is that of the would-be carbocation it is attached to.
+
+#### Alcohol combustion
+
+Alcohols are combustible, and can undergo complete and incomplete combustion.
+
+$$
+\ce{alcohol + O2 -> CO2 + H2O (complete) \\
+alcohol + O2 -> CO2 + H2O + CO + C (incomplete)}
+$$
+
+#### Alcohol elimination
+
+Under significantly more acidic conditions than hydration, the opposite process can be used to revert an alcohol into its base components.
+
+$$\ce{alcohol ->[12 mol/L H2SO4] H2O + alkene}$$
+
+::: danger WARNING
+When choosing a new double bond to form in the alkene, it must bond to the carbon the OH group was attached to. In elimination, **Markovnikov's rule does not apply**.
+:::
+
+### Aldehydes
+
+::: info DEFINITION
+
+- A **carbonyl** is $\ce{C=O}$.
+- A **hydroxyl** is $\ce{-OH}$. In a side group, it is named **hydroxy**.
+  :::
+
+In the presence of an oxidising agent that is **limited** and acid, **primary** alcohols will oxidise to form aldehydes, where a hydroxyl group becomes a carbonyl group and the hydrogen migrates to the carbon.
+
+- $\ce{K2Cr2O7}$
+- $\ce{Cr2O7^2-}$
+- $\ce{KMnO4}$
+- $\ce{MnO4-}$
+
+An aldehyde is named like an alcohol but has a higher naming priority, with a suffix of **-al**. As aldehydes must be at the end of a chain, numbering their position is not required.
+
+::: warning EXAMPLE
+
+- butanal ($\ce{CH3CH2CH2COH}$)
+- The common name of **methanal** is **formaldehyde**.
+  :::
+
+<img src="/resources/images/alcohol-aldehyde.png" width=900 />
+
+Aldehydes will continue to react to ketones if the oxidising agent is not limited. To prevent this, the aldehyde is separated and removed from the mixture through distillation.
+
+<img src="/resources/images/aldehyde-distillation.png" width=900 />
+
+The mixture is heated to a temperature greater than the aldehyde's boiling point but less than the alcohol's, such that the gaseous aldehyde enters the condenser and is cooled by the water jacket.
+
+An aldehyde can also be reduced in a process similar to **hydrogenation** to reverse the reaction.
+
+$$\ce{aldehyde + H2 ->[\text{high temp, high pressure, Pt/Pd/Ni}] alcohol}$$
+
+### Ketones
+
+In the presence of an oxidising agent and acid, **secondary** alcohols will oxidise to form ketones, where the hydrogen plops off completely.
+
+<img src="/resources/images/alcohol-ketone.png" width=900 />
+
+Because there is no possible reaction afterward (no more hydrogens), distillation is not required.
+
+Ketones have equal priority to aldehydes and are named the same but with a suffix of **-one**. A position number _is_ required because ketones can be located anywhere on the chain.
+
+::: warning EXAMPLE
+
+- 3-ethyl-4,4-difluoro-5-hydroxylhexan-2-one
+- 1,1-dibromo-4-cyclopropylhex-5-en-2-one
+  :::
+
+### Carboxylic acids
+
+Aldehydes will react again if there is excess oxidising agent to form a carboxylic acid.
+
+<img src="/resources/images/alcohol-acid.png" width=900 />
+
+Instead of distillation, **reflux** is used to keep the aldehyde in the mixture. The vaporised aldehyde condenses and returns to the mixture.
+
+<img src="/resources/images/alcohol-reflux.png" width=900 />
+
+Carboxylic acids have higher priority than aldehydes/ketones and are named the same but with a suffix of **-oic acid**. Similar to aldehydes, because the $\ce{COOH}$ can only exist on the end of a chain, position numbers are omitted.
+
+::: warning EXAMPLE
+
+- **Benzoic acid**: $\ce{benzene-COOH}$
+- 3,3-difluoropent-4-enoic acid
+- 3-ethylhexanedioic acid
+- The common name of **ethanoic acid** is **acetic acid**.
+- The common name of **ethanedioic acid** is **oxalic acid**.
+- The common name of **methanoic acid** is **formic acid**.
+- Look up citric acid because I'm not writing that down.
+  :::
+
+#### Carboxylic acid salts
+
+If the ionising hydrogen is removed ($\ce{COOH -> COO-}$), a carboxylic acid can form a salt by reacting with a metal to form an **ionic compound**. Salts are named as an ionic compound would be, with the acid component resuffixed to **-oate**.
+
+$$\ce{R-COOH + NaOH -> R-COONa + H2O}$$
+
+::: warning EXAMPLE
+
+- sodium ethanoate
+- lithium benzoate
+  :::
+
+#### Identifying alcohols
+
+The **Lucas test** is used to in part determine the type of alcohol (primary/secondary/tertiary) through the **nucleophilic substitution** of OH with Cl. To perform this substitution, **anhydrous** zinc chloride and **concentrated** HCl must be present.
+
+$$\ce{R-OH + HCl ->[ZnCl2] R-Cl + H2O}$$
+
+This test is only valid on **small** alcohols because (<6 carbons) as longer ones are insoluble.
+
+The insoluble halogenoalkane becomes visible, making the solution **cloudy**. Because the reaction is an S<sub>N</sub>1 reaction:
+
+- Primary alcohols will **not** react
+- Secondary alcohols react slowly
+- Tertiary alcohols react rapidly
+
+Alternatively, **oxidising** alcohols to aldehydes/ketones through S<sub>N</sub>2 by reducing $\ce{Cr2O7^2-}$ (orange) to $\ce{Cr^3+}$ (green) will identify the alcohol.
+
+- Primary alcohols will react quickly
+- Secondary alcohols will react slowly
+- Tertiary alcohols will **not** react
+
+### Ethers
+
+::: info DEFINITION
+
+- A **condensation reaction** or **dehydration synthesis** involves two small molecules reacting to form water and another molecule.
+  :::
+
+Ethers are formed by reacting two alcohols through dehydration synthesis in sulfuric acid.
+
+$$\ce{R-OH + HO-R ->[H2SO4] R-O-R + H2O}$$
+
+To name ethers, the shorter alkyl group is named as a side chain while the longer is as the main chain, separated by "oxy".
+$$\ce{short + oxy + long}$$
+
+Usually, if the "side chain" is at position 1, the position number is omitted.
+
+::: warning EXAMPLE
+
+- pentoxypentane (pentan-1-ol + pentan-1-ol)
+- 2-ethoxybutane (ethan-2-ol + butan-1-ol)
+- 2-chloro-3-methoxypentane (chloro is at position 2, methoxy is at position 3 on the pentane)
+- The common name of **ethoxyethane** is **diethyl ether**.
+  :::
+
+### Esters
+
+When an alcohol and carboxylic acid react in sulfuric acid **and heat**, the only the $\ce{O}$ from the alcohol remains in the ester while that in the acid forms a water. The formed $\ce{COO}$ is known as the **ester linkage**.
+
+<img src="/resources/images/ester-formation.png" width=900 />
+
+Esters are named with the alcohol as the side group and the acid as its salt variant with a space in between. If the side chain looks like an alkane, its position number and -ane suffix can be dropped.
+$$\text{alcohol-yl acid-oate}$$
+
+::: danger WARNING
+The carbon in the ester linkage is included as a carbon of the main chain of the ester.
+:::
+
+::: warning EXAMPLE
+
+- Propyl pentanoate or propan-1-yl pentanoate is formed from propan-1-ol and pentanoic acid.
+- Propyl 2-chloroethanoate
+- Hexan-3-yl propanoate
+  :::
+
+Esters hydrolyse to their original components if catalysed by an acid or base.
+$$\ce{ester + H2O ->[H2SO4] alcohol + carboxylic acid}$$
+$$\ce{ester + H2O ->[NaOH] alcohol + RCOONa ->[react with acid] alcohol + carboxylic acid}$$
+
+### Amines
+
+Amines are $\ce{NR3}$ derived from ammonia ($\ce{NH3}$), where R is either H or a carbon group. Similar to alcohols, they can be primary/secondary/tertiary depending on the number of carbon groups attached. The **main chain** is the longest carbon chain.
+
+Amines have a priority between double/triple bonds and alcohols, and are named like alcohols but with a suffix of **-amine**.
+
+If there are any side groups attached to the nitrogen, they are named as if they were side groups on the main chain with a **number of $N$**.
+
+::: warning EXAMPLE
+<img src="/resources/images/amine-name-simple.png" width=700 />
+<img src="/resources/images/amine-name-mid.png" width=700 />
+<img src="/resources/images/amine-name-hard.png" width=700 />
+:::
+
+#### Amine synthesis
+
+Amines can be formed through **halogenoalkane substitution**, where ammonia or another amine is alkylated in an S<sub>N</sub>2 reaction.
+$$\ce{NH3 + CH3Cl -> CH3NH4Cl ->[OH-] CH3NH2}$$
+
+::: warning EXAMPLE
+$\ce{CH3NH2 + CH3Cl -> CH3NH2CH3Cl ->[OH-] CH3NH2CH3}$
+:::
+
+### Amides
+
+Amides are formed from a reaction between an amine and a carboxylic acid through dehydration synthesis, similar to the formation of an ester. The $\ce{N-C=O}$ link is known as the **amide link**.
+
+$$\ce{R-COOH + N-R -> R-CON-R}$$
+
+Amides carry the suffix **-amide** and are otherwise named equivalently to esters, but _without_ spaces.
+
+::: warning EXAMPLE
+<img src="/resources/images/amide-names.png" width=700 />
+:::
+
+### Nitriles
+
+Nitriles consist of a cyanide(s) attached at the end of a carbon chain.
+
+$$\ce{R-C#N}$$
+
+As they can only be placed at the end of a carbon chain, a positional number is not used. These have the highest priority of all organic compounds and use the suffix **-nitrile** and the prefix **cyano-**.
+
+::: warning EXAMPLE
+
+- methanenitrile
+- methanedinitrile
+  :::
+
+Nitriles are synthesised through the nucleophilic substitution of halogenoalkanes, **extending their carbon chain**.
+$$\ce{R-X + C#N- -> R-C#N + X-}$$
+
+### Reduction reactions
+
+**Hydride reagents** include $\ce{LiAlH4}$ and $\ce{NaBH4}$, the former of which requires ether because it reacts violently with water. Always use $\ce{LiAlH4}$ unless specified otherwise.
+
+**Aldehydes** can be reduced to **primary alcohols**.
+
+$$\ce{aldehyde ->[LiAlH4, ether, then acid] 1^\circ alcohol}$$
+
+**Amides** can be reduced to their **amines**, reacting twice such that the O pops off. The name is a simple `amide.replace("amide", "amine")`.
+
+$$\ce{amide ->[LiAlH4, ether, then acid] amine}$$
+
+::: danger WARNING
+$\ce{LiAlH4}$ is required for this reaction.
+:::
+
+**Carboxylic acids** can be reduced to **primary alcohols** with the $\ce{C=O}$ plopping off.
+
+$$\ce{carboxylic acid ->[LiAlH4, ether, then acid] 1^\circ alcohol}$$
+
+**Esters** can be reduced to **two primary alcohols** with each alcohol keeping an O and gaining an H to make OH.
+
+$$\ce{ester ->[LiAlH4, ether, then acid] 1^\circ alcohol + 1^\circ alcohol}$$
+
+**Nitriles** can be double reduced to **amines**.
+
+$$\ce{nitrile ->[LiAlH4, ether, then acid] amine}$$
+
+### Retro-synthesis
+
+Retro-synthesis is basically a language of math but for chem, with products on the left and reactants on the right. The bottom right contains initial reactant(s) and the top left contains the product(s).
+
+"A is made from B which is made from C":
+
+$$
+\ce{
+A => B react with alcohol using H2SO4 in reflux \\
+B => C
+}
+$$
+
+::: warning EXAMPLE
+
+$$
+\ce{
+ethanoic acid => ethanol (react w/K2Cr2O7 in H+) \\
+ethanol => chloroethane (react w/warm dilute hydroxide)
+}
+$$
+
+:::
+
+### Simple polymers
+
+::: info DEFINITION
+
+- **Polymers** are large molecules made from many monomers in long chains.
+- **Plastics** are polymers formed through addition.
+- A **homopolymer** has identical monomers.
+- A **heteropolymer** has multiple distinct monomers.
+- A **monomer** is the repeating segment in a polymer.
+  :::
+
+Polymer properties change based on the type of linkages, the presence of side chains, and the extent of crosslinking between other chains.
+
+The **addition formation** of an **addition polymer** opens up pi bonds which are used to bond to other monomers. Monomers are continuously added until the process ends with hydrogen atoms capping the ends.
+
+<img src="/resources/images/addition-polymer.png" width=700 />
+
+Only the two carbons directly involved in the double bond go in the main chain of the polymer, with all others expressed as side groups.
+
+<img src="/resources/images/addition-polymer-notation.png" width=900 />
+
+**Polymer notation** is the formula/condensed formula/structural diagram of the **repeating unit only** with crossed out brackets and the number of repetitions at the bottom right (or $n$ if unknown). Side groups should be clearly expressed as side groups. Polymers are named with the prefix **poly-** on the repeating unit.
+
+::: warning EXAMPLE
+$$\ce{-(-CH2-CH2-) -_3}$$
+:::
+
+#### Crosslinking
+
+::: info DEFINITION
+
+- **Crosslinking** is the bond between side chains of separate polymers, connecting them.
+  :::
+
+The crosslinking between polymers depends on the side chains. If there are multiple double bonds in monomers, those can be used in different chains which can attach them together.
+
+::: warning EXAMPLE
+divinylbenzene
+:::
+
+::: warning EXAMPLE
+If an OH side group meets another OH side group, they may react to form $\ce{O=O}$ and connect the two polymers.
+:::
+
+### Polyesters and polyamides
+
+::: info DEFINITION
+
+- **Condensation polymers** are polymers formed via dehydration synthesis and produce water.
+  :::
+
+A **polyester** has monomers connected via an ester linkage on both ends. **Unlike addition polymers**, any carbons between the functional groups are included in the parent polymer chain.
+
+<img src="/resources/images/polyester-formation.png" width=700 />
+
+The repeating unit should be **copy-pastable** — it should not end with oxygen on both ends. The link is broken where it would normally break — between the C-O of the ester linkage, such that the O goes to the side of the alcohol.
+
+<img src="/resources/images/polyester-notation.png" width=700 />
+
+A **polyamide** has monomers connected via an amide linkage on both ends.
+
+<img src="/resources/images/polyamide-formation.png" width=700 />
+
+::: danger WARNING
+There should be a hydrogen attached to the nitrogen at the end of the amine.
+:::
+
+### E/Z isomers
+
+E/Z isomers are a generalised form of cis-trans isomers, where priority is determined by atomic number. If both sides with the higher atomic number are on the **same** side, the isomer is a Z-isomer (German: _ze zame zide_). E/Z isomers are placed at the beginning surrounded by parentheses.
+
+::: warning EXAMPLE
+(Z)-2-bromo-1-chloro-1-fluoroethene:
+<img src="/resources/images/ez-example.png" width=700 />
+:::
+
+If the atoms are of equal priority, the sum of atomic numbers that they are directly connected to are compared (double bonds count twice), repeating as necessary.
+
+::: warning EXAMPLE
+(Z)-1-chloro-1-fluoro-2-methyl-1-butene (left) and (E)-1-chloro-1-fluoro-2-methyl-1-butene (right).
+<img src="/resources/images/special-ez-isomer.png" width=700 />
+:::
+
+If there are multiple E/Z isomers, they are separated by commas and numbered according to their earliest position on the main chain.
+
+::: warning EXAMPLE
+(2Z, 3E)-R
+:::
+
+### Optical isomers
+
+::: info DEFINITION
+
+- An **enantiomer** is an optical isomer.
+- A **chiral centre** is a carbon atom with four different groups attached to it.
+- The **chirality** of a carbon atom represents its ability to form an enantiomer.
+- A **racemic mixture** is a mixture of exactly one half of each enantiomer of a species such that it is not optically active.
+- A **dextrorotary** enantiomer rotates rightward (+).
+- A **levrorotary** enantiomer rotates leftward (-).
+  :::
+
+Optical isomers are mirrored across the y-axis with the same compounds put on the same bonds. **Four distinct groups** must be attached to the central carbon atom to have optical isomers.
+
+In the data booklet, all amino acids are chiral except for glyine and proline.
+
+::: warning EXAMPLE
+<img src="/resources/images/enantiomer.ex.png" width=700 />
+:::
+
+An **optically active** species is one that can rotate the plane of polarised light. Please see [SL Physics 1#Polarisation](/sph3u7/#polarisation) for more information.
+
+A species that rotates the plane clockwise is positive, while counter-clockwise is negative. Both enantiomers have the same magnitude of polarisation except for the direction. If there is a mixture of both enantiomers, the angle changes depending on the proportion of each isomer.
+
+Enantiomers have the same physical properties except for the direction of polarised light. They also have mostly the same chemical properties except for chemical reactions with other enantiomers of different compounds.
+
+### Properties of organic compounds
+
+**Alcohols** are able to form hydrogen bonds, so are soluble in water. Increasing the length of the main chain decreases solubility as the rest of the molecule is non-polar, but this can be compensated by adding more hydroxyls too.
+
+In general:
+
+- m/ethanols are miscible
+- butanols are 10-15% v/v miscible
+- alcohols longer than octanols are effectively insoluble
+
+Although the boiling point of an alcohol will always be higher than its corresponding alkane, the difference between the two will decrease as chain length increases as the proportion of force the alcohol provides decreases relative to the larger contributor in the LDF from the main chain.
+
+Low mass **esters** smell good, and large mass esters are oily/waxy.
+
+**Amines** smell bad and are all Bronsted-Lowry weak bases because they can accept protons and form dative bonds.
+
+The solubility of compounds is directly related to their melting/boiling point — compounds that cannot hydrogen bond with themselves but can with water have an advantage.
+
+From greatest to lowest melting point:
+
+**Hydrogen bonding**
+
+- Water is able to hydrogen bond with two other molecules per molecule, efficiently using all its lone pairs and Hs.
+- Carboxylic acids are less efficient than water but more than alcohols as an the OH can attract to an O on a different molecule.
+- Alcohols
+- Primary/secondary amines can hydrogen bond but the N-H bond is less polar than O-H, decreasing its strength.
+
+**Dipole-dipole interaction**
+
+- Aldehydes and ketones
+- Esters are less polar than aldehydes because the single bond O attracts electrons from the C=O.
+- Ethers have horizontal components to their dipole vectors that cancel out, so they are least polar.
+
+**London dispersion forces**
+
+- Alkynes' triple bonds means that packing is easier, increasing LDFs.
+- Alkanes
+- Alkenes' double bonds means that there are less electrons than their alkane counter parts, reducing LDFs.
+
+$$
+\ce{
+water >> \\
+carboxylic acids > alcohols > amines >> \\
+ethers > aldehydes/ketones >> \\
+alkynes > alkanes > alkenes
+}
+$$
+
+## Resources
+
+- [IB Chemistry Data Booklet](/resources/g11/ib-chemistry-data-booklet.pdf)
+- [IB HL Chemistry Syllabus](/resources/g11/ib-chemistry-syllabus.pdf)
+- [Significant Figures/Digits](/resources/g11/chemistry-sig-figs.pdf)
+- [Error Analysis and Significant Figures (long)](/resources/g11/error-analysis-sig-figs.pdf)
+- [General Guidelines for Writing a Formal Laboratory Report](/resources/g11/lab-report-guidelines.pdf)
+- [Designing an IB Investigation](/resources/g11/designing-investigation.pdf)
+- [Textbook: Pearson Higher Level Chemistry](/resources/g12/textbook-hl-chem.pdf) ([Answers](/resources/g12/textbook-hl-chem-answers.pdf)) - [mini Eifueo](/resources/g12/textbook-hl-chem-eifueo.pdf)

docs/index.md

@@ -1,21 +1,24 @@
+---
+sidebarDepth: 2
+footer: Thanks to James Su, Lakshy Gupta, and Vincent Guo for providing supplementary data for accuracy and conciseness.
+---
+
 # Home
 
-This website is intended to be primarily for personal use, but is available publically online.
+This website is intended to be primarily for personal use, but is available publicly online.
 
 Testing math rendering: $a^2+b^2=c^2$
 
 | Tables | too!    |
-| --- | --- |
+| ------ | ------- |
 | yeet   | no yeet |
 
 ## Contact
 
-If you would like to contribute by submitting fixes, requesting pages, and/or complaining about issues, feel free to open an issue on the [issue tracker](https://git.eggworld.tk/eggy/eifueo/issues) or submit a [pull request](https://git.eggworld.tk/eggy/eifueo/pulls), or contact the site administrator at [341213551@gapps.yrdsb.ca](mailto:341213551@gapps.yrdsb.ca).
+If you would like to contribute by submitting fixes, requesting pages, and/or complaining about issues, feel free to open an issue on the [issue tracker](https://git.eggworld.tk/eggy/eifueo/issues) or submit a [pull request](https://git.eggworld.tk/eggy/eifueo/pulls), or contact the site administrator at 341213551 at gapps.yrdsb.ca
 
 ## Source
 
 The source for Eifueo is available [here](https://git.eggworld.tk/eggy/eifueo).
 
 ## Acknowledgements
-
-Thanks to James Su, Lakshy Gupta, and Vincent Guo for providing supplementary data for accuracy and conciseness.

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+# SL Math - Analysis and Approaches - 2
+
+The course code for this page is **MCV4U7**.
+
+## Integration
+
+Integration is an operation that finds the **net** area under a curve, and is the opposite operation of differentiation. As such, it is also known as **anti-differentiation**.
+
+The area under a curve between the interval of x-values $[a,b]$ is:
+$$A=\lim_{x\to\infty}\sum^n_{i=1}f(x_i)\Delta x$$
+
+which can be simplified to, where $dx$ indicates that integration should be performed with respect to $x$:
+$$A=\int^b_a f(x)dx$$
+
+While $\Sigma$ refers to a finite sum, $\int$ refers to the sum of a limit.
+
+As integration is the opposite operation of differentiation, they can cancel each other out.
+$$\frac{d}{dx}\int f(x)dx=f(x)$$
+
+The **integral** or **anti-derivative** of a function is capitalised by convention. Where $C$ is an unknown constant:
+$$\int f(x)dx=F(x)+C$$
+
+When integrating, there is always an unknown constant $C$ as there are infinitely many possible functions that have the same rate of change but have different vertical translations.
+
+!!! definition
+    - $C$ is known as the **constant of integration**.
+    - $f(x)$ is the **integrand**.
+
+### Integration rules
+
+$$
+\begin{align*}
+&\int 1dx &= &&x+C \\
+&\int (ax^n)dx, n≠-1 &=&&\frac{a}{n+1}x^{n+1} + C \\
+&\int (x^{-1})dx&=&&\ln|x|+C \\
+&\int (ax+b)^{-1}dx&=&&\frac{\ln|ax+b|}{a}+C \\
+&\int (ae^{kx})dx &= &&\frac{a}{k}e^{kx} + C \\
+&\int (\sin kx)dx &= &&\frac{-\cos kx}{k}+C \\
+&\int (\cos kx)dx &= &&\frac{\sin kx}{k}+C \\
+\end{align*}
+$$
+
+Similar to differentiation, integration allows for constant multiples to be brought out and terms to be considered individually.
+
+$$
+\begin{align*}
+&\int k\cdot f(x)dx&=&&k\int f(x)dx \\
+&\int[f(x)\pm g(x)]dx&=&&\int f(x)dx \pm \int g(x)dx
+\end{align*}
+$$
+
+### Indefinite integration
+
+The indefinite integral of a function contains every possible anti-derivative — that is, it contains the constant of integration $C$.
+$$\int f(x)dx=F(x)+C$$
+
+### Substitution rule
+
+Similar to limit evaluation, the substitution of complex expressions involving $x$ and $dx$ with $u$ and $du$ is generally used to work with the chain rule.
+$$
+u=g(x) \\
+\int f(g(x))\cdot g´(x)\cdot dx = \int f(u)\cdot du
+$$
+
+??? example
+    To solve $\int (x\sqrt{x-1})dx$:
+    $$
+    let\ u=x-1 \\
+    ∴ \frac{du}{dx}=1 \\
+    ∴ du=dx \\
+    \begin{align*}
+    \int (x\sqrt{x-1})dx &\to \int(u+1)(u^\frac{1}{2})du \\
+    &= \int(u^\frac{3}{2}+u^\frac{1}{2})du \\
+    &= \frac{2}{5}u^\frac{5}{2}+\frac{2}{3}u^\frac{3}{2}+C \\
+    &= \frac{2}{5}(x-1)^\frac{5}{2} + \frac{2}{3}(x-1)^\frac{3}{2} + C
+    \end{align*}
+    $$
+
+### Definite integration
+
+To find a numerical value of the area under the curve in the bounded interval $[a,b]$, the **definite** integral can be taken.
+$$\int^b_a f(x)dx$$
+
+$a$ and $b$ are known as the lower and upper **limits of integration**, respectively.
+
+<img src="/resources/images/integration.png" width=700>(Source; Kognity)</img>
+
+Regions **under** the x-axis are treated as negative while those above are positive, cancelling each other out, so the definite integral finds something like the net area over an interval.
+
+If $f(x)$ is continuous at $[a,b]$ and $F(x)$ is the anti-derivative, the definite integral is equal to:
+$$\int^b_a f(x)dx=F(x)\biggr]^b_a=F(b)-F(a)$$
+
+As such, it can be evaluated manually by integrating the function and subtracting the two anti-derivatives.
+
+!!! warning
+    If $u$-substitution is used, the limits of integration must be adjusted accordingly.
+
+To find the total **area** enclosed between the x-axis, $x=a$, $x=b$, and $f(x)$, the function needs to be split at each x-intercept and the absolute value of each definite integral in those intervals summed.
+$$A=\int^b_a \big|f(x)\big| dx$$
+
+### Properties of definite integration
+
+The following rules only apply while $f(x)$ and $g(x)$ are continuous in the interval $[a,b]$ and $c$ is a constant.
+
+$$
+\begin{align*}
+&\int^a_a f(x)dx&=&&0 \\
+&\int^b_a c\cdot dx&=&&c(b-a) \\
+&\int^a_b f(x)dx&=&&-\int^b_a f(x)dx \\
+&\int^c_a f(x)dx&=&&\int^b_a f(x)dx + \int^c_b f(x)dx
+\end{align*}
+$$
+
+The **constant multiple** and **sum** rules still apply.
+
+### Area between two curves
+
+To find the area enclosed between two curves, the graph should be sketched if possible and their points of intersection determined to identify which parts of each function are on top of the other at any given time. An interval chart may be helpful. For each section, where $f(x)$ is always greater than $g(x)$ in the interval $[a,b]$:
+$$A=\int^b_a [f(x)-g(x)]dx, f(x)\geq g(x)\text{ in } [a,b]$$
+
+If the limits of integration are not given, they are the outermost points of intersection of the two curves.
+
+### Volumes of solids of revolution
+
+Shapes formed by rotating a line or curve about a fixed axis, such as cones, spheres, and cylinders are all known as **solids of revolution**. By splicing each shape into infinitely small disks, the cylinder volume formula can be used to find the volume of the solid.
+$$
+\begin{align*}
+V&=\lim_{x\to 0}\sum^b_{x=a}\pi y^2 dx \\
+&=\pi\int^b_a y^2 dx
+\end{align*}
+$$
+
+The area between two curves can also be rotated to form a solid, in which case its formula is:
+$$V=\pi\int^b_a \big[g(x)^2-f(x)^2\big]dx, g(x)>f(x)$$
+
+## Probability
+
+!!! definition
+    - $\cap$ is the **intersection sign** and means "and".
+    - $\cup$ is the **union sign** and means "or".
+    - $\subset$ is the **subset sign** and indicates that the value on the left is a subset of the value on the right.
+    - The **sample space** of an experiment is a list/set of all of the possible outcomes.
+    - An **event** is a subset of a sample space that contains outcomes that meet a particular requirement.
+
+### Sets
+
+A **set** is a collection of things represented with curly brackets that can be assigned to a variable.
+
+!!! example
+    $A=\{0,1,2\}$ assigns the variable $A$ to a collection of numbers $0, 1, 2$.
+
+The variable $U$ is usually reserved for the **universal set**: a set that contains all of the elements under discussion for a particular situation.
+
+Where both $A$ and $B$ are sets:
+
+ - $A\cap B$ returns a new set with only objects that belong to both $A$ **and** $B$.
+ - $A\cup B$ returns a new set with only objects that are inclusively in either $A$ **or** $B$.
+ - $A\subset B$ is true only if all of the elements in $A$ are also in $B$.
+ - $A'$ or $A^c$ return the **complement** of a set: they return all elements in the universal set that are **not** in $A$.
+ - $n(A)$ returns the number of elements in set $A$.
+
+An empty/**null** set contains no objects and is represented either as $\{\}$ or $\emptyset$.
+
+Two sets are **disjoint** or **distinct** if they have no common elements between them.
+
+!!! warning
+    Generally, unless specified otherwise, "between" should be inferred to mean "inclusively between".
+
+### Probability rules
+
+The probability of an event is represented by $P(A)$, where $A$ is the event.
+$$P(A)=\frac{n(A)}{n(U)}$$
+
+As event $A$ must be a subset of all possible outcomes $U$, where $1$ indicates that the event always happens and $0$ the opposite:
+$$0\leq P(A)\leq 1$$
+
+The **complement** of event $A$ is the probability that it does not happen. It is written as $A^c$, $A'$, or $\pu{not } A$.
+$$P(A')=1-P(A)$$
+
+Events $A$ and $B$ are **disjoint** or mutually exclusive if no outcomes between them are common and can never happen simultaneously. As such the probability of one of the events happening is equal to their sum.
+$$
+P(A\cup B)=P(A)+P(B) \\
+P(A\cap B)=0
+$$
+
+Events $A$ and $B$ are **exhaustive** if their union includes all possible outcomes in the sample space: $A\cup B=U$.
+$$P(A\cup B)=1$$
+
+The **principle of inclusion and exclusion** forms a general rule for the union between two *independent* events:
+$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$
+
+For independent events:
+$$P(A\cap B)=P(A)\times P(B)$$
+
+### Conditional probability
+
+A vertical bar is used between two events to denote that the event on the left occurs knowing that the right has already occurred.
+$$P(A|B) = \frac{P(A\cap B)}{P(B)}$$
+
+!!! example
+    $P(A|B)$ is the probability of event $A$ given $B$ has occurred.
+
+### Probability distributions and discrete random variables
+
+The **discrete random variable**, $X$, represents a **quantifiable**, measurable, discrete quantity. The lowercase $x$ represents a possible value of $X$.
+
+The probability that $X$ takes on any one of the specific possible outcomes is written as $P(X=x)$. The sum of the probability all possible outcomes must still remain $1$:
+$$\Sigma P(X=x)=1$$
+
+!!! example
+    In an experiment of tossing a coin twice, possible values of $X$ include $0,1,2$ so $x\in\{0, 1, 2\}$.
+
+A **probability distribution** is a distribution of outcomes and their probabilities. Events/outcomes are placed on the top row while probability is provided on the bottom in the form of a fraction. Probability distributions can also be graphed with the outcomes on the x-axis and their probabilities on the y-axis with lines similar to a bar graph sitting on the grid lines to represent a probability..
+
+!!! example
+    For the coin ross experiment in the previous example, where $X$ is the number of tails when tossing a coin twice:
+    
+    | $x$ | $0$ | $1$ | $2$ |
+    | --- | --- | --- | --- |
+    | $P(X=x)$ | $\frac{1}{4}$ | $\frac{1}{2}$ | $\frac{1}{4}$ |
+
+The **expected value** of an experiment or the "expectation of $X$" is the mean value of $X$ that is expected to be obtained over many trials. It is equal to the sum of the value of all outcomes multiplied by their probability.
+$$
+\begin{align*}
+E(X)&=\Sigma P(X=x)x \\
+&=\mu=x_1p_1+x_2p_2+...+x_kp_k
+\end{align*}
+$$
+
+!!! warning
+    It is possible that the expected value will not be a value in the set, and the expected value should **not be mistaken** with the outcome with the highest probability.
+
+### Binomial distribution
+
+**Bernoulli trials** have a fixed number of trials that are independent of each other and identical with only two possible outcomes — a success or failure.
+
+Where $r$ is the number of successes in a Bernoulli trial:
+$$P(X=r)={n\choose r}p^rq^{n-r}$$
+
+where ${n\choose r}=\frac{n!}{r!(n-r)!}$
+
+A binomial distribution is a probability distribution of two possible events, a success or a failure. The distribution is defined by the number of trials, $n$, and the probability of a success, $p$. The probability of failure is defined as $q=1-p$.
+
+$X\sim$ denotes that the random variable $X$ is distributed in a certain way. Therefore, the binomial distribution of $X$ is expressed as:
+$$X\sim B(n, p)$$
+
+In a binomial distribution, the expected value and **variance** are as follows:
+$$
+E(X)=np \\
+Var(X)=npq
+$$
+
+On a graphing display calculator, where $r$ is the number of successes:
+$$
+\begin{align*}
+P(X=r)&=\text{binompdf}(n,p,r) \\
+P(X<r)&=\text{binomcdf}(n,p,r-1) \\
+P(X\leq r)&=\text{binomcdf}(n,p,r) \\
+P(a\leq X\leq b)&=\text{binomcdf}(n,p,b) - \text{binomcdf}(n,p,a-1)
+\end{align*}
+$$
+
+### Normal distribution
+
+Also known as **Gaussian distribution** or in its graphical form, a normal or bell curve, the normal distribution is a **continuous** probability distribution for the random variable $x$.
+
+<img src="/resources/images/gaussian-distribution.png" width=700>(Source: Kognity)</img>
+
+In a normal distribution:
+
+ - The mean, median, and mode are all equal.
+ - The normal curve is bell-shaped and symmetric about the mean.
+ - The area under the curve is equal to one.
+ - The normal curve approaches but does not touch the x-axis as it approaches $\pm \infty$.
+
+From $\mu-\sigma$ to $\mu+\sigma$, the curve curves downward. $\mu\pm\sigma$ are the **inflection points** of the graph. It is expressed graphically as:
+$$X\sim N(\mu,\sigma^2)$$
+
+where
+
+$$f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^\frac{-(x-\mu)^2}{2\sigma^2}$$
+
+~68%, ~95%, and ~99.7% of the data is found within one, two, and three standard deviations of the mean, respectively.
+
+### Standard normal distribution
+
+The **standard normal distribution** has a mean of 0 and standard deviation of 1. The horizontal scale of the standard normal curve corresponds to **$z$-scores** that represent the number of standard deviations away from the mean. To convert an $x$-score to a $z$-score:
+$$z=\frac{x-\mu}{\sigma}$$
+
+A **Standard Normal Table** can be used to determine the cumulative area under the standard normal curve to the left of scores -3.49 to 3.49. The area to the *right* of the score is equal to $1-z_\text{left}$. The area *between* two z-scores is the difference in between the area of the two z-scores.
+
+To standardise a normal random variable, it should be converted from the form $X\sim N(\mu,\sigma^2)$ to $Z\sim N(0,1)$ via the formula to convert between x- and z-scores.
+
+The probability of a z-score being less than a value can be rewritten as phi.
+$$P(z<a)=\phi(a)$$
+
+Some z-score rules partially taken from probability rules:
+
+$$
+\begin{align*}
+P(z>-a)&=P(z<a) \\
+1-P(z>a)&=P(z<a)
+\end{align*}
+$$
+
+On a graphing display calculator:
+
+The `normalcdf` command can be used to find the cumulative probabilty in a normal distribution in the format $\text{normalcdf}(a,b,\mu,\sigma)$, which will solve for $P(a<x<b)$. $-1000$ is generally a sufficiently low value to solve for just $P(x<b)$. 
+
+## Vectors
+
+Please see [SL Physics 1#1.3 - Vectors and Scalars](/sph3u7/#13-vectors-and-scalars) for more information.
+
+One vector can be represented in a variety of methods. The algebraic form $(1, 2)$ can also be represented in the alternate algebraic forms $[1, 2]$ and $1\choose 2$.
+
+Where $v$ is the vector, $A$ is the initial and $B$ is the terminal point of the vector, a vector can be identified by any of the following symbols:
+
+ - $\vec{AB}$
+ - $\vec{v}$
+ - $\boldsymbol{v}$ (bolded)
+
+The special **zero vector** $\vec{0}$ is a vector of undefined direction and zero magnitude.
+
+Vectors with the same magnitude but opposite directions are equal to one another except one is the negative of the other.
+
+**Colinear** vectors are those that parallel with one another — that is, with identical or opposite directions. Vectors that are colinear must also be **scalar multiples** of each other:
+$$\vec{u}=k\vec{v}$$
+
+**Position** vectors are vectors where the initial point is at the origin — where the terminal point is $A$, a position vector can be written as $\vec{OA}$.
+
+**Colinear points** are points that lie on the same straight line. If two colinear vectors that share a common point can be formed between three points, those points are colinear.
+
+### Unit vector
+
+The **unit vector** of a vector is a vector of the same direction as the original with a magnitude of $1$. It is denoted via a caret/hat.
+$$\hat{v}$$
+
+From the original vector $\vec{u}$, the unit vector $\hat{u}$ can be found by dividing by the magnitude of the vector.
+$$\hat{u}=\frac{\vec{u}}{|\vec{u}|}$$
+
+The **standard unit vectors** $\hat{i}$ and $\hat{j}$ are unit vectors designated to point in the directions of the positive x- and y-axes.
+$$
+\hat{i}=(1,0) \\
+\hat{j}=(0,1)
+$$
+
+Any vector in two dimensions can be expressed as a sum of scalar multiples of the vectors.
+$$
+\begin{align*}
+\vec{u}&=\vec{OP} \\
+&=(a,b) \\
+&=a\hat{i}+b\hat{j} \\
+&={a\choose b} \\
+|\vec{u}|&=\sqrt{a^2+b^2}
+\end{align*}
+$$
+
+The angle between two vectors is the smaller angle formed when the vectors are placed **tail to tail**.
+
+### Three-dimensional vectors
+
+The additional standard unit vector $\hat{k}$ is used for the z-dimension.
+$$
+\begin{align*}
+\vec{u}&=\vec{OP} \\
+&=(a,b,c) \\
+&=a\hat{i}+b\hat{j}+c\hat{k}
+\end{align*}
+$$
+
+In general, the x-plane is the one in and out of the page, the y-plane left and right, and the z-plane up and down.
+
+### Vector operations
+
+Please see [SL Physics 1#Adding/subtracting vectors diagrammatically](/sph3u7/#addingsubtracting-vectors-diagrammatically) for more details. The sum of two vectors is known as the **resultant** while the negative (opposite direction) version of that vector is known as the **equilibrant**.
+
+The sum of two vectors can also be solved diagrammatically by envisioning the head-to-tail as a parallelogram.
+
+<img src="/resources/images/vector-parallelogram.png" width=700>(Source: Kognity)</img>
+
+### Dot product
+
+Also known as the scalar product, the dot product between two vectors returns a **scalar** value representing the horizontal displacement after multiplication. Wheree $\theta$ is the angle contained between the vectors $\vec{u}$ and $\vec{v}$ when arranged tail-to-tail:
+$$\vec{u}\bullet\vec{v}=|\vec{u}||\vec{v}|\cos\theta$$
+
+!!! note
+    This implies that vectors perpendicular to one another must have a dot product of zero.
+
+Much like regular multiplication, dot products are:
+
+ - communtative: $\vec{u}\bullet\vec{v}=\vec{v}\bullet\vec{u}$
+ - distributive over vectors: $\vec{u}\bullet(\vec{v}+\vec{w})=\vec{u}\bullet\vec{v}+\vec{u}\bullet\vec{w}$
+ - associative over scalars: $(m\vec{u})\bullet(n\vec{v})=mn(\vec{u}\bullet\vec{v})$
+ 	- $m(\vec{u}\bullet\vec{v})=(m\vec{u})\bullet\vec{v}=(mv)\bullet\vec{u}$
+
+When working with algebraic vectors, their dot products are equal to the products of their components.
+$$\vec{u}\bullet\vec{v}=u_xv_x+u_yv_y$$
+
+### Vector line equations in two dimensions
+
+!!! definition
+    The **Cartesian** or **scalar** form of a line is of the form $Ax+By+C$.
+
+The vector equation for a straight line solves for an unknown position vector $\vec{r}$ on the line using a known position vector $\vec{r_0}$ on the line, a direction vector parallel to the line $\vec{m}$, and the variable **parameter** $t$. It is roughly similar to $y=b+xm$.
+$$\vec{r}=\vec{r_0}+t\vec{m},t\in\mathbb{R}$$
+
+The equation can be rewritten in the algebraic form to be
+$$[x,y]=[x_0,y_0]+t[m_1,m_2], t\in\mathbb{R}$$
+
+The direction vector is effectively the slope of a line.
+$$\vec{m}=[\Delta x, \Delta y]$$
+
+For a line in scalar form:
+$$\vec{m}=[B, -A]$$
+
+To determine if a point lies along a line defined by a vector equation, the parameter $t$ should be checked to be the same for the $x$ and $y$ coordinates of the point.
+
+!!! warning
+    Vector equations are **not unique** — there can be different position vectors and direction vectors that return the same line.
+
+The **parametric** form of a line breaks the vector form into components.
+$$
+\begin{align*}
+x&=x_0+tm_1 \\
+y&=y_0+tm_2,t\in\mathbb{R}
+\end{align*}
+$$
+
+The **symmetric** form of the equation takes the parametric form and equates the two equations to each other using $t$.
+$$\frac{x-x_0}{m_1}=\frac{y-y_0}{m_2},m_1,m_2\neq 0$$
+
+If one of the **direction numbers** $m_1$ or $m_2$ is zero, the equation is rearranged such that only one position component is on one side.
+
+!!! example
+    Where $m_2=0$:
+    $$\frac{x-x_0}{m_1},y=y_0$$
+
+### Vector line equations in three dimensions
+
+There is little difference between vector equations in two or three dimensions. An additional variable is added for the third dimension.
+
+The vector form:
+$$\vec{r}=\vec{r_0}+t\vec{m},t\in\mathbb{R}$$
+
+The parametric form:
+$$[x,y,z]=[x_0,y_0,z_0]+t[m_1,m_2,m_3],t\in\mathbb{R}$$
+
+The symmetric form:
+$$\frac{x-x_0}{m_1}=\frac{y-y_0}{m_2}=\frac{z-z_0}{m_3}$$
+
+### Intersections of vector equation lines
+
+Two lines are parallel if their direction vectors are scalar multiples of each other.
+$$\vec{m_1}=k\vec{m_2},k\in\mathbb{R}$$
+
+Two lines are coincident if they are parallel and share at least one point. Otherwise, they are distinct.
+
+If two lines are not parallel and in two dimensions, they intersect. To solve for the point of intersection, the x and y variables in the parametric form can be equated and the parameter $t$ solved.
+
+In three dimensions, there is a final possibility should the lines not be parallel: the lines may be *skew*. To determine if the lines are skew, the x, y, and z variables of **two** parametric equations should be equated to their counterparts in the other vector as if they intersect. The resulting $t$ and $s$ from the first and second line respectively should be substituted into the third equation and an equality check performed.  Should there not be a solution that fulfills the third equation, the lines are skew. Otherwise, they intersect.
+
+### Vector projections
+
+If two vectors $\vec{a}$ and $\vec{b}$ are placed tail-to-tail, the **component** of $\vec{a}$ in the direction of $\vec{b}$ is known as the **vector projection of $\vec{a}$ onto $\vec{b}$**. Represented by $Projection$, its magnitude is called the **scalar projection**.
+$$Proj_\vec{b}\vec{a}=\biggr(\frac{\vec{a}\bullet\vec{b}}{|\vec{b}|^2}\biggr)\vec{b}$$
+
+$$
+\begin{align*}
+|Proj_\vec{b}\vec{a}|&=\frac{\vec{a}\bullet\vec{b}}{|\vec{b}|} \\
+&=|\vec{a}|\cos\theta
+\end{align*}
+$$
+
+!!! warning
+    The magnitude of any projection is always **positive**. If $\cos\theta$ returns a negative value, it needs to be absed again.
+
+Vector projections are applied in work equations — see [SL Physics 1](/sph3u7/#work) for more information.
+
+### Cross product
+
+The cross product or **vector product** is a vector that is perpendicular of two vectors that are not colinear. Where $\vec{u}_1,\vec{u}_2,\vec{3}$ represent the x, y, and z coordinates of the position vector $\vec{u}$, respectively:
+$$
+\begin{align*}
+\vec{u}\times\vec{v}&=
+\begin{vmatrix}
+\hat{i} & \hat{j} & \hat{k} \\
+\vec{u}_1 & \vec{u}_2 & \vec{u}_3 \\
+\vec{v}_1 & \vec{v}_2 & \vec{v}_3
+\end{vmatrix} \\
+\\
+&=-\hat{j}\begin{vmatrix}
+\vec{u}_1 & \vec{u}_3 \\
+\vec{v}_1 & \vec{v}_3
+\end{vmatrix}
++\hat{i}\begin{vmatrix}
+\vec{u}_2 & \vec{u}_3 \\
+\vec{v}_2 & \vec{v}_3
+\end{vmatrix}
++\hat{k}\begin{vmatrix}
+\vec{u}_1 & \vec{u}_2 \\
+\vec{v}_1 & \vec{v}_2
+\end{vmatrix} \\
+\\
+&=[\vec{u}_2\vec{v}_3-\vec{u}_3\vec{v}_2,\vec{u}_3\vec{v}_1-\vec{u}_1\vec{v}_3,\vec{u}_1\vec{v}_2-\vec{u}_2\vec{v}_1]
+\end{align*}
+$$
+
+Cross products are:
+
+ - anti-communtative: $\vec{u}\times\vec{v}=-(\vec{u}\times\vec{v})$
+ - distributive: $\vec{u}\times(\vec{u}+\vec{w})=\vec{u}\times\vec{v}+\vec{u}\times\vec{w}$
+ - associative over scalars: $m(\vec{u}\times\vec{v})=(m\vec{u})\times\vec{v}=(m\vec{v})\times\vec{u}$
+
+The **magnitude** of a cross product is opposite that of the dot product. Where $\theta$ is the smaller angle between the two vectors ($0\leq\theta\leq180^\circ$):
+$$|\vec{u}\times\vec{v}|=|\vec{u}||\vec{v}|\sin\theta$$
+
+This is also equal to the area of a parallelogram enclosed by the vectors — where one is the base and the other is the adjacent side.
+
+To determine the **direction** of a cross product, the right-hand rule can be used. Spreading the fingers out:
+
+ - the thumb is the direction of the first vector
+ - the index finger is the direction of the second vector
+ - the palm faces the direction of the cross product
+
+### Applications of vector operations
+
+A **triple scalar product** is the result of a cross product performed first then put in a dot product.
+$$|\vec{c}\bullet(\vec{a}\times\vec{b})|$$
+
+In a **parallelpiped**, or a three-dimensional shape with six faces each a parallelogram with an identical one opposite it, the volume is the triple scalar product of the distinct three vectors that make up its side lengths:
+$$A=|\vec{c}\bullet(\vec{a}\times\vec{b})|$$
+
+For an object moving at **constant velocity in 2D space**, where $\vec{s}$ is its displacement, $\vec{s}_0$ is its initial displacement at $t=0$, $t$ is the time elapsed, and $\vec{v}$ is its velocity:
+$$\vec{s}=\vec{s}_0+t\vec{v}$$
+
+**Torque** ($\vec{\tau}$ or $\vec{M}$) is the ability to rotate an object — effectively angular/rotational force — and is the cross product of the **outward-pointing radius vector** ($\vec{r}$) and the **force** vector ($\vec{F}$).
+$$
+\begin{align*}
+\vec{\tau}&=\vec{r}\times\vec{F} \\
+&=|\vec{r}||\vec{F}|\sin\theta
+\end{align*}
+$$
+
+<img src="/resources/images/torque.svg" width=700>(Source: Wikimedia)</img>
+
+The direction of the torque can be found using the **right-hand rule**.
+
+**Force** and **velocity** are vectors with magnitude and direction. See [SL Physics 1#Force diagrams](/sph3u7/#force-diagrams) and [SL Physics 1#Velocity](/sph3u7/#velocity) for more information.
+
+### Operations with vector components
+
+If **Cartesian vectors** (see [SL Physics 1#Adding/subtracting vectors algebraically](/sph3u7/#addingsubtracting-vectors-algebraically) for more details) cannot be used, the **sine and cosine laws** can be used, which are, respectively:
+
+Where $a$, $b$, and $c$ are the lengths of a triangle, and $A$, $B$, and $C$ are their angles opposite to them:
+$$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$
+
+$$c^2=a^2+b^2-2ab\cos C$$
+
+### Vector planes
+
+A **plane** ($\pi$) is a flat surface extending infinitely in all directions and can be represented by a parallelogram.
+
+So long as any of the following are known, their plane can be constructed:
+
+ - Two intersecting or parallel lines
+ - A line and a point not on the line
+ - 3 non-colinear points
+
+In each scenario, an initial point $r_0$ and two direction vectors $\vec{u},\vec{v}$ can be derived to form the equation for a plane:
+$$\vec r = \vec r_0 + s\vec u + t\vec v,s,t\in\mathbb R$$
+
+This can be expanded to form the parametric form of the equation:
+$$
+\begin{align*}
+x&=x_0+su_1+tv_1 \\
+y&=y_0+su_2+tv_2 \\
+z&=z_0+su_3+tv_3,s,t\in\mathbb R
+\end{align*}
+$$
+
+Where $A,B.C.D$ are all integers, the **scalar** or Cartesian equation of a plane in three dimensions can be expressed as follows:
+$$Ax+By+Cz+D=0$$
+
+!!! info
+    $[A,B,C]$ is the **normal direction vector** of a plane.
+
+### Interactions of planes
+
+A line intersects a plane if the dot product between the two is not zero, and the resulting scalar multiple found can be used to find the point of intersection. Otherwise, once the equations are substituted into each other, if the statement is true, the line and plane are **parallel and coincident**. Otherwise, they are parallel.
+
+The shortest distance between two **skew lines** $L_1$ and $L_2$ is equal to:
+$$
+\begin{align*}
+d&=|Proj_\vec{n}\vec{P_1P_2}| \\
+&=\frac{|\vec{P_1P_2}\bullet(\vec m_1\times\vec m_2)|}{|\vec m_1\times\vec m_2|}
+\end{align*}
+$$
+
+The shortest distance between a point $P(x_1,y_1,z_1)$ and plane $\pi: Ax+By+Cz+D=0$ is equal to:
+$$d=\frac{|Ax_1+By_1+Cz_1+D|}{\sqrt{A^2+B^2+C^2}}$$
+
+The shortest distance between two parallel planes is equal to:
+$$d=\frac{|D_1-D_2|}{\sqrt{A^2+B^2+C^2}}$$
+
+Two planes are parallel if their direction vectors are scalar multiples of each other:
+$$\vec n_1 = k\vec n_2$$
+
+If they are also coincident, the D-values will also be identical or equal to the same scalar multiple as the normal:
+$$D_1=D_2$$
+
+Otherwise, the planes intersect, the line along which is equal to the cross product between the two direction vectors.
+$$\vec m=\vec n_1\times\vec n_2$$
+
+An initial point vector can be solved by setting any of the variables ($x,y,z$) to zero and solving for the others. Alternatively, the parameter $t$ can be set equal to one of the variables instead and the parametric equation derived that way.
+
+The **angle between two planes** is equal to the angle between their normal direction vectors, which can be determined using the dot product formula.
+
+When looking at three planes:
+
+If all three normals are scalar multiples:
+
+ - If all three $D$-values are those same scalar multiples, the planes are parallel and coincident and they have infinite points of intersection along the plane equation.
+ - Otherwise, there are no solutions and the planes are parallel and distinct and/or parallel and coincident for two.
+
+If two normals are scalar multiples:
+
+ - If the two parallel planes are coincident with the same $D$-values, there will be a line of intersection much like solving for intersection between two planes.
+ - Otherwise, the two parallel planes are distinct, forming a Z-pattern with the third plane and so there is no solution.
+
+If no normals are scalar multiples:
+
+ - If the triple scalar product of the three planes is equal to zero, the normal vectors are not coplanar and so there will be a point of intersection.
+ - Alternatively, by solving the scalar equations for the planes, if:
+	- the result is a contradiction (e.g., $0 = 3$), there is no solution
+	- the result is true with no variable (e.g., $0 = 0$), there are is an infinite number of solutions along a line
+	- the result contains a variable (e.g., $t = 4$), there is a single point of intersection at the parameter $t$.
+
+## Matrices
+
+A **matrix** is a two-dimensional array with rows and columns, represented by a capital letter and a grid denoted by square brackets.
+$$
+A=
+\begin{bmatrix}
+1 & 2 & 3 \\
+4 & 5 & 6
+\end{bmatrix}
+$$
+
+$A_{ij}$ represents the element in the $i$th row and the $j$th column.
+
+A **coefficient matrix** contains coefficients of variables.
+$$
+A=
+\begin{bmatrix}
+1 & 2 & 3 \\
+4 & 5 & 6
+\end{bmatrix}
+$$
+
+An **augmented matrix** also contains constants, separated by a vertical line.
+$$
+A=
+\left[\begin{array}{rrr|r}
+1 & 2 & 3 & 5 \\
+4 & 5 & 6 & 10
+\end{array}\right]
+$$
+
+!!! example
+    The equation system
+    $$
+    x+2y-4z=3 \\
+    -2x+y+3z=4 \\
+    4x-3y-z=-2
+    $$
+    can be written as the matrix
+    $$
+    A=
+    \left[\begin{array}{rrr|r}
+    1 & 2 & -4 & 3 \\
+    -2 & 1 & 3 & 4 \\
+    4 & -3 & -1 & -2
+    \end{array}\right]
+    $$
+
+### Gaussian elimination
+
+Gaussian elimination is used to solve a system of linear relations, such as that of plane equations. It aims to reduce a matrix into its **row echelon form** shown below to solve for each variable.
+$$
+A=
+\left[\begin{array}{rrr|r}
+a & b & c & d \\
+0 & e & f & g \\
+0 & 0 & h & i
+\end{array}\right]
+$$
+
+The following **row operations** can be performed on the matrix to achieve this state:
+
+ - swapping (interchanging) the position of two rows
+	- $R_a \leftrightarrow R_b$
+ - multiplying a row by a non-zero constant
+	- $AR_a \to R_a$
+ - adding/subtracting rows, overwriting the destination row
+	- $R_a\pm R_b\to R_b$
+ - multiplying a row by a non-zero constant and then adding/subtracting it to another row
+	- $AR_a + R_b \to R_b$
+
+!!! example
+    In the matrix from the previous example, by performing $R_1\leftrightarrow R_2$:
+    $$
+    A=
+    \left[\begin{array}{rrr|r}
+    -2 & 1 & 3 & 4 \\
+    1 & 2 & -4 & 3 \\
+    4 & -3 & -1 & -2
+    \end{array}\right]
+    $$
+    $5R_1\to R_1$:
+    $$
+    A=
+    \left[\begin{array}{rrr|r}
+    -10 & 5 & 15 & 20 \\
+    1 & 2 & -4 & 3 \\
+    4 & -3 & -1 & -2
+    \end{array}\right]
+    $$
+    $10R_2+R_1\to R_1$:
+    $$
+    A=
+    \left[\begin{array}{rrr|r}
+    0 & 25 & -25 & 50 \\
+    1 & 2 & -4 & 3 \\
+    4 & -3 & -1 & -2
+    \end{array}\right]
+    $$
+
+## Resources
+
+ - [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)
+ - [IB Math Analysis and Approaches Formula Booklet](/resources/g11/ib-math-data-booklet.pdf)
+ - [Textbook: Calculus and Vectors 12](/resources/g11/calculus-vectors-textbook.pdf)
+ - [Textbook: Oxford SL Mathematics](/resources/g11/textbook-oxford-math.pdf)
+ - [Course Pack Unit 1: Integration](/resources/g11/s2cp1.pdf) ([Annotated](/resources/g11/s2cp1-anno.pdf))
+ - [Course Pack Unit 2: Probability](/resources/g11/s2cp2.pdf)([Annotated](/resources/g11/s2cp2-anno.pdf))
+ - [Course Pack Unit 3: Vectors](/resources/g11/s2cp3.pdf) ([Annotated](/resources/g11/s2cp3-anno.pdf))
+ - [Course Pack Unit 4: Vector Applications](/resources/g11/s2cp4.pdf) ([Annotated](/resources/g11/s2cp4-anno.pdf))
+ - [Course Pack Unit 5: Planes](/resources/g11/s2cp5.pdf) ([Annotated](/resources/g11/s2cp5-anno.pdf))
+ - [TI-84 Plus Calculator Guide](/resources/g11/ti-84-plus.pdf)

docs/mhf4u7.md

@@ -1,7 +1,65 @@
-# SL Math - Analysis and Approaches - A
+# SL Math - Analysis and Approaches - 1
 
 The course code for this page is **MHF4U7**.
 
+## Review
+
+### Logarithm rules
+
+The logarithm of a product can be rewritten as the sum of two logarithms.
+$$\log_c(ab)=\log_c(a)+\log_c(b)$$
+
+The logarithm of a quotient can be rewritten as the difference of two logarithms.
+$$\log_c\biggr(\frac{a}{b}\biggr)=\log_c(a)-\log_c(b)$$
+
+The exponentials of a logarithm can be brought down to be coefficients.
+$$\log_c(a^n)=n\log_c(a)$$
+
+Some simple values can be easily found.
+
+$$
+a^{\log_a(x)}=x \\
+\log_a(a)=1 \\
+\log_a(1)=0
+$$
+
+## 3 - Geometry and trigonometry
+
+To find the result of a primary trig ratio, the related acute angle (RAA) should first be found before referring to the CAST rule to determine quadrants before identifying all correct answers in the domain.
+
+### Circles
+
+The equation below is true for every point on a circle with radius $r$.
+$$x^2+y^2=r^2$$
+
+The area of a **sector** requires knowledge of the radius and angle in **radians** that the sector encompasses.
+$$A=\frac{r^2\theta}{2}$$
+<img src="/resources/images/sector.png" width=500>(Source: Kognity)</img>
+
+### Trigonometric identities
+
+The **Pythagorean identity** relates the radius of a circle to its x and y components.
+$$\sin^2\theta+\cos^2\theta=1$$
+
+The **quotient identity** relates the side lengths of a right-angled triangle.
+$$\tan\theta=\frac{\sin\theta}{\cos\theta}$$
+
+The **double angle identities** can be used to convert one trig ratio to another.
+$$
+\sin 2\theta = 2\sin\theta\cos\theta \\
+\cos 2\theta = 2\cos^2\theta-1 \\
+\cos 2\theta = \cos^2\theta-\sin^2\theta \\
+\cos 2\theta = 1-2\sin^2\theta \\
+\tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta}
+$$
+
+### Euler's number
+
+Euler's number $e$ is a constant irrational number represented as a special limit in calculus.
+$$e=\lim_{x\to ∞}\biggr(1+\frac{1}{x}\biggr)^x$$
+
+The inverse of $e^x$ is $\log_e(x)$, which is known as the **natural logarithm** and can be rewritten as $\ln(x)$ ("lawn x").
+
 ## 4 - Statistics and probability
 
 !!! note "Definition"
@@ -54,7 +112,7 @@ A **frequency distribution** is a table that lists categories/ranges and the num
 A frequency distribution table includes:
 
  - A number of classes, all of the same width.
-	- This number is arbitrarily chosen, but a commonly used formula is $\lceil\sqrt{\text{# of elements}}\rceil$.
+	- This number is arbitrarily chosen, but a commonly used formula is $\lceil1+3.3\log({\text{# of elements})}\rceil$.
 	- The width (size) of each class is $\lceil\frac{\text{max} - \text{min}}{\text{# of classes}}\rceil$.
 	- Each class includes its lower bound and excludes its upper bound ($\text{lower} ≤ x < \text{upper}$)
 	- The **relative frequency** of a data set is the percentage of the whole data set present in that class in decimal form.
@@ -253,6 +311,9 @@ This is known as the **difference quotient**.
 
 The **instantaneous rate of change (IRoC)** at point $P(a, f(a))$ is represented by the slope of the **tangent line ($m_T$)**. The slope of the tangent line can be found by finding the difference quotient with $h$ as a very small value, e.g., $0.001$.
 
+!!! warning
+    The above method of finding the IRoC should be disregarded in favour of finding the derivative.
+
 ### Sequences
 
 A sequence is a **function** with a domain of all positive integers in sequence, but uses subscript notation ($t_n$) instead of function notation ($f(x)$).
@@ -264,7 +325,7 @@ A sequence is a **function** with a domain of all positive integers in sequence,
 If the sequence is infinite, as $n$ becomes very large:
 
  - If the sequence continuously grows, it **tends to infinity**. (E.g., $a_n = n^2, n ≥ 1$)
- - If the sequence gets closer to a real number and converges on it, it **converges to a real limit**, or is convergent**. (E.g., $a_n = \frac{1}{n}, n ≥ 1$)
+ - If the sequence gets closer to a real number and converges on it, it **converges to a real limit**, or is **convergent**. (E.g., $a_n = \frac{1}{n}, n ≥ 1$)
  - If the sequence never approaches a number, it **does not tend to a limit**, or is **divergent**. (E.g., $a_n = \sin(n \pi)$)
 
 ### Limits
@@ -367,6 +428,9 @@ Substituting a variable in for the variable to be solved and then solving in ter
     = \frac{1}{16}
     $$
 
+!!! note
+    If $\lim_{x \to a} \frac{f(x)}{g(x)}$ exists and direct substitution is not possible, $x - a$ *must* be a factor of both $f(x)$ and $g(x)$ so that the discontinuity can be removed. Therefore, $f(a) = 0$ and $g(a) = 0$.
+
 ### Limits and continuity
 
 If a function has holes or gaps or jumps (i.e., if it cannot be drawn with a writing utensil held down all the time), it is **discontinuous**. Otherwise, it is a **continuous** function. A function discontinuous at $x=a$ is "discontinuous at $a$", where $a$ is the "point of discontinuity".
@@ -430,27 +494,254 @@ results in the equation of the derivative function. Direct substitution of $h$ w
     f´(x)=4x
     $$
 
-### Derivative rules
-
-The degree of a derivative is always the degree of the original function$-1$.
-
-The power rule applies to all functions of the form $f(x)=x^n,x \in \mathbb{R}$, such that:
-$$f´(x) = nx^{n-1}$$
-
 ### Drawing derivative functions
 
 If the slope of a tangent is:
 
  - positive/negative, that value on the derivative graph is also positive/negative, respectively
- - zero (e.g., linear equations), that value on the derivative graph is on the x-axis 
+ - zero, that value on the derivative graph is on the x-axis 
 
 Points of inflection on the original function become maximum/minimum points on the derivative graph.
 
+The derivative of a linear equation is always constant, and the derivative of a constant value is $0$..
+
+### Derivative rules
+
+These rules can be used in place of/to supplement finding derivative functions using first principles and are usually much faster to calculate. These rules assume that all of the functions involved are differentiable.
+
+The degree of a derivative is always the degree of the original function$-1$.
+
+The **power rule** applies to all functions of the form $f(x)=x^n,x \in \mathbb{R}$, such that:
+$$f´(x) = nx^{n-1}$$
+
+??? example
+    $$f(x) = x^5$$
+    $$f´(x) = 5x^4$$
+
+The **constant multiple rule** applies to all functions of the form $f(x) = k·g(x)$, where $k$ is any real number, such that:
+$$f´(x) = k·g´(x)$$
+
+??? example
+    $$f(x) = 2x^2$$
+    $$f´(x) = 2·2x$$
+    $$f´(x) = 4x$$
+
+The **sum rule** applies to all functions of the form $f(x) = g(x) + h(x)$ such that:
+$$f´(x) = g´(x) + h´(x)$$
+
+??? example
+    $$f(x) = 2x^2 + 3x$$
+    $$f´(x) = 4x + 3$$
+
+The **product rule** applies to all functions of the form $f(x) = g(x)h(x)$ such that:
+$$f´(x) = g´(x)h(x) + g(x)h´(x)$$
+
+??? example
+    $$f(x) = (2x+5)(x-1)$$
+    $$f´(x) = 2(x-1) + (2x+5)·1$$
+    $$f´(x) = 4x + 1$$
+
+The **extended product rule** applies to all functions of the form $f(x) = g(x)h(x)j(x)$ such that:
+$$f´(x) = g´(x)h(x)j(x) + g(x)h´(x)j(x) + g(x)h(x)j´(x)$$
+
+The **quotient rule** applies to all functions of the form $f(x) = \frac{g(x)}{h(x)}$ such that:
+$$f´(x) = \frac{g´(x)h(x)-g(x)h´(x)}{[h(x)]^2}, h(x) ≠ 0$$
+
+??? example
+    $$f(x) = \frac{2x+5}{x-1}$$
+    $$f´(x) = \frac{2(x-1) - (2x+5)·1}{(x-1)^2}$$
+    $$f´(x) = -\frac{7}{(x-1)^2}$$
+
+The **chain rule** applies to all functions of the form $f(x) = g(h(x))$ such that:
+$$f´(x) = g´(h(x)) · h´(x)$$
+
+??? example
+    $$f(x) = (4x^2-3x+1)^7$$
+    $$f´(x) = 7(4x^2-3x+1)^6 (8x-3)$$
+
+### Trigonometric derivative rules
+
+$$
+\frac{d}{dx}\sin x = \cos x \\
+\frac{d}{dx}\cos x = -\sin x \\
+$$
+
+These primary derived rules can be used to further derive the derivatives of the other trignometric ratios:
+$$
+\frac{d}{dx}\tan x = \sec^2 x \\
+\frac{d}{dx}\csc x = -\csc x\cdot\cot x \\
+\frac{d}{dx}\sec x = \sec x\cdot\tan x
+$$
+
+The **chain rule** applies to trigonometric functions and will be applied recursively if needed.
+
+!!! example
+    $$\frac{d}{dx}[\sin g(x)]^n = n[\sin g(x)]^{n-1}\cdot\cos x\cdot g´(x)$$
+
+Trigonometric identities are not polynomial so values on an interval need to be determined by substituting values between vertical asymptotes and critical points.
+
+### Extended derivative rules
+
+For an **exponential function** where $f(x)=b^x,b≠0$ or $f(x)=b^{g(x)}$, respectively:
+
+$$
+f´(x)=b^x\cdot\ln(b) \\
+f´(g(x))=b^{g(x)}\cdot\ln(b)\cdot g´(x)
+$$
+
+For a **logarithmic function** where $f(x)=\log_b(x)$ or $f(x)=\log_b(g(x))$, respectively:
+
+$$
+f´(x)=\frac{1}{\ln(b)\cdot x} \\
+f´(x)=\frac{g´(x)}{\ln(b)\cdot g(x)}
+$$
+
+From the above base derivatives the derivatives for functions involving $e$ and the **natural logarithm** can be found:
+
+$$
+\frac{d}{dx}e^x=e^x \\
+\frac{d}{dx}e^{g(x)}=e^{g(x)}\cdot g´(x) \\
+\frac{d}{dx}\ln(x)=\frac{1}{x} \\
+\frac{d}{dx}\ln(g(x))=\frac{g´(x)}{g(x)}
+$$
+
+This opens up the possibility of **logarithmic differentiation**, which is required for exponential or logarithmic functions with a variable base. The **natural logarithm** of both sides should be taken prior to differentiation and logarithmic rules applied to simplify the equation.
+
+### Higher order derivatives
+
+The **second derivative** of $f(x)$ is the derivative of the first derivative of $f(x)$, that is, $f´´(x)$.
+
+The $n$th derivative of $f(x)$ is $f^{(n)}(x)$, and is the derivative of the $n-1$th derivative. It is written as $\frac{d^ny}{dx^n}$ in Leibniz notation.
+
+!!! example
+    The second derivative of an object's position with respect to time is its acceleration. See [SL Physics A#Displaying motion](/sph3u7/#displaying-motion) for more information.
+
+### Interval charts
+
+To identify the positive or negative regions of an equation, an interval line or chart can be used. To do so:
+
+1. Factor the equation as much as possible and identify the x-intercepts.
+2. Place the x-intercepts on a line.
+3. Find the sign of the end behaviour by taking the sign of the leading coefficient.
+4. When crossing an x-intercept, if the degree of that factor is even, the sign stays the same; otherwise, it alternates.
+5. Repeat for every other region.
+
+### Implicit differentiation
+
+Implicit differentiation differentiates both sides of an equation with respect to $x$ and solves for $\frac{dy}{dx}$ ($y´$). Note that if $y$ is isolated, this is effectively the same as explicit differentiation. When differentiating implicitly, it must be shown that the derivative of both sides with respect to x ($\frac{d}{dx}$) is being taken.
+
+!!! warning
+    The **chain rule** must be applied when differentiating terms that contain $y$.
+
+!!! example
+    $$
+    \frac{x^2}{4} + y^2 = 1 \\
+    \frac{2x}{4} + 2y · \frac{dy}{dx} = 0 \\
+    \frac{dy}{dx} = -\frac{\frac{x}{2}}{2y} \\
+    \frac{dy}{dx} = -\frac{x}{4y}
+    $$
+
+### Related rates
+
+When solving for questions that ask for rate of change related to other rates of change, ensure that:
+
+ - variables are defined
+ - equations are written in terms of derivates
+ - the equations are differentiated **with respect to time**
+ - apply derivative rules (especially the chain rule) to every variable that is not a constant (i.e., that changes with respect to time)
+ - substitute values only at the end
+
+## 5.2 - Increasing and decreasing functions
+
+ - If $f´(x) > 0$ in the interval $[a,b]$, $f$ is **increasing** on $[a,b]$.
+ - If $f´(x) < 0$ in the interval $[a,b]$, $f$ is **decreasing** on $[a,b]$.
+ - If $f´(x) = 0$ in the interval $[a,b]$, $f$ is **constant** on $[a,b]$.
+ - The points where $f´(x)=0$ are the **critical**/maximum/minimum points.
+
+Functions only change whether they are increasing/decreasing/constant at the **critical points**/"relative extrema".
+
+These points and whether the intervals between them increase/decrease can be found by using an **interval chart/line** using the first derivative.
+
+!!! example
+    If $f(x)=\frac{2x-3}{x^2+2x-3}$:
+    
+    - $f$ is decreasing on $(-∞, -3) \cup (-3, 0) \cup (3, ∞)$.
+    - $f$ is increasing on $(0, 1) \cup (1, 3)$.
+
+### Extrema
+
+Extrema are the maximum and minimum points in a function or an interval of a function. They must be at **critical points**—where $f(x)=0$ or $f(x)=\text{DNE}$, and may include the **boundary points** if looking for extrema in a given interval.
+
+The highest and lowest point(s) of $f(x)$ are known as the **absolute** maximum/minimum of $f(x)$.
+
+Any other **relative/local** maxima or minima are such that all of the points around that point are higher or lower.
+
+**Fermat's theorem** states that if $f(c)$ is a local extremum, $c$ must be a critical number of $f$. Therefore, if $f$ is continuous in the closed interval $[a,b]$, the absolute extrema of $f$ must occur at $a$, $b$, or a critical number.
+
+To find the extrema of a **continuous** function $f(x)$, where $x=a$ is a critical value, the **first derivative test** may be used with the assistance of an interval chart/line. If only an interval of a function is under consideration, the boundary points must be taken under consideration as well.
+
+ - If $f´(a)$ changes from positive to negative, there is a relative/local minimum at $x=a$.
+ - If $f´(a)$ changes from negative to positive, there is a relative/local maximum at $x=a$.
+ - If the sign is the same on both sides, there is no extrema at $x=a$.
+ - The greatest/least relative/local maximum/minimum is the absolute maximum/minimum.
+
+Alternatively, the second derivative test may be used instead. At the critical point where $x=a$, a positive $f´´(a)$ indicates a **local minimum** while a negative $f´´(a)$ indicates a **local maximum**. If $f´´(x)=0$, the test is inconclusive and the first derivative test must be used.
+
+!!! example
+    The absolute minimum of $f(x)=x^2$ is at $(0,0)$. There is no absolute maximum nor are there any other relative/local maximum/minimum points.
+
+!!! warning
+    - There can be multiple absolute maxima/minima if there are multiple points that are both highest/lowest.
+    - If a function is a horizontal line, the absolute maximum and minimum for $x \in \text{domain}$ is $y$.
+
+### Concavity
+
+!!! definition
+    A **point of inflection** on a curve is such that $f´´(x)=0 \text{ or DNE}$ and the signs of $f´´(x)$ around the point change (e.g., positive to negative).
+
+ - An interval is **concave up** if it increases from left to right and tangent lines are drawn below the curve, so $f´´(x)>0$. It is shaped like a smile.
+ - An interval is **concave down** if it increases from left to right and tangent lines are drawn **above** the curve, and $f´´(x)<0$. It is shaped like a frown.
+
+Changes in concavity only occur at points of inflection.
+
+### Cusps
+
+A cusp is a point on a continuous function that is not differentiable as the slopes on both sides approach -∞ and ∞. Concavity does not change at a cusp, but they may be considered for local extrema.
+
+### Optimisation
+
+To optimise for a maximum or minimum of a variable:
+
+ - Identify an equation with only one variable dependent on another
+ - Find the first derivative and identify critical points
+ - Use the second derivative test to identify if the critical point is a maximum or minimum
+ - Check constraints and throw away any inadmissible results
+
+Diagrams with labelled variables may be helpful.
+
+### Asymptote behaviour
+
+The vertical asymptotes of a function are at values of $x$ that make the denominator of the simplified function $0$. The behaviour near them can be found using limits as $x$ approaches those points.
+
+The horizontal asymptotes of a function can be found as $x$ approaches positive and negative infinity. To determine behaviour near them, the sign of $\lim_{xs \to ±∞} f(x) - L$, where $L$ is the y-coordinate of the asymptote. A positive limit indicates that $f(x)$ is above the asymptote while a negative limit indicates that $f(x)$ is below the asymptote.
+
+### Curve sketching
+
+ - Determine the domain of the function and consider discontinuities (holes and asymptotes)
+ - Determine the y-intercept and if easy, x-intercepts
+ - Determine the behaviour near vertical and horizontal asymptotes
+ - Identify critical points by solving $f´(x)=0$ or $f´(x)=\text{DNE}$
+ - Use the first or second derivative tests to test critical points
+ - Identify points of inflection by solving $f´´(x)=0$ or $f´´(x)=\text{DNE}$ and test concavity on both sides of possible points
+
 ## Resources
 
  - [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)
  - [IB Math Analysis and Approaches Formula Booklet](/resources/g11/ib-math-data-booklet.pdf)
  - [Calculus and Vectors 12 Textbook](/resources/g11/calculus-vectors-textbook.pdf)
- - [Course Pack Unit 1: Descriptive Statistics](/resources/g11/s1cp1.pdf)
- - [Course Pack Unit 2: Limits and Rate of Change](/resources/g11/s1cp2.pdf)
- - [TI-84 Plus Basic Calculator Functions](/resources/g11/ti-84-plus.pdf)
+ - [Course Pack Unit 1: Descriptive Statistics](/resources/g11/s1cp1.pdf) ([Annotated](/resources/g11/s1cp1-anno.pdf))
+ - [Course Pack Unit 2: Limits and Rate of Change](/resources/g11/s1cp2.pdf) ([Annotated](/resources/g11/s1cp2-anno.pdf))
+ - [Course Pack Unit 3: Derivatives and Applications](/resources/g11/s1cp3.pdf) ([Annotated](/resources/g11/s1cp3-anno.pdf))
+ - [Course Pack Unit 4: Curve Sketching and Optimisation](/resources/g11/s1cp4.pdf) ([Annotated](/resources/g11/s1cp4-anno.pdf))
+ - [Course Pack Unit 5: Trigonometric, Exponential, and Logarithmic Functions](/resources/g11/s1cp5.pdf) ([Annotated](/resources/g11/s1cp5-anno.pdf))
+ - [TI-84 Plus Guide](/resources/g11/ti-84-plus.pdf)

docs/resources.md

@@ -1,3 +0,0 @@
-# IB Resources
-
-

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+# HL Chemistry - 2
+
+The course code for this page is **SCH4UZ**.
+
+## Thermal concepts
+
+!!! definition
+    - A **system** consists of reactants and products being studied, often represented as a chemical equation.
+    - The **surroundings**/**environment** are all matter outside of the system capable of absorbing or releasing energy.
+
+ - **Open** systems allow **energy and matter** to move in and out of the system.
+ - **Closed** systems allow only **energy** to move in and out of the system.
+ - **Isolated** systems do not allow energy or matter to move in and out of the system. This is an ideal but unrealistic scenario.
+
+### Changes
+
+As **breaking bonds requires energy** and **forming bonds releases energy**:
+
+!!! definition
+    - An **endothermic** reaction overall requires energy.
+    - An **exothermic** reaction overall releases energy.
+
+**Physical** changes such as state changes or dissolving substances may release or require energy depending on the energy of intermolecular bonds being broken and formed.
+
+!!! example
+    - Ice melting requires energy to break the stronger bonds in a solid.
+    - Dissolving salt in water breaks the intermolecular bonds holding the salt together but regains it all by forming new bonds with the water.
+
+**Chemical** changes all involve breaking old bonds to form new bonds. Depending on the energy required/released in breaking/forming those bonds, the reaction may end up endothermic or exothermic. Regardless, all reactions need a small initial **activation energy** to begin.
+
+!!! info
+    Acid-base reactions are always exothermic.
+
+### Specific heat capacity
+
+Please see [SL Physics 1#3.1 - Thermal concepts](/sph3u7/#31-thermal-concepts) for more information.
+
+## Enthalpy
+
+Represented as $H$ in joules, enthalpy represents the total energy in a system. Absolute enthalpy is not measurable, so change in enthalpy ($\Delta H$) is often used instead. The magnitude of enthalpy change is dependent on the type of change and quantity of substance that is changing.
+
+A **negative** $\Delta H$ indicates that energy has left the system and so is an **exothermic** reaction.
+
+In a balanced chemical equation, change in enthalpy is written to the right after the product.
+$$
+a + b \to c\ \ \Delta H = x\text{ kJ}
+$$
+
+!!! example
+    Energy is required for the decomposition of water so its enthalpy is positive.
+    $$\ce{H2O_{(l)} -> H2_{(g)} + 1/2 O2_{(g)}\ \ \Delta H = +280 kJ}$$
+
+$\Delta H$ can also be included in a balanced thermochemical equation as a reactant or product instead of listed at the end. In this case, it is always positive and its sign determines whether it is a reactant or product.
+$$
+a + b + x\text{ kJ} \to c
+$$
+
+!!! example
+    Using the same formula as in the previous example:
+    $$\ce{H2O_{(l)} + 285.5 kJ -> H2_{(g)} + 1/2 O2_{(g)}}$$
+
+### (Standard) Molar enthalpy of reaction
+
+The **molar enthalpy of reaction** $\Delta H_x$ expresses the change in enthalpy when exactly one mole of the substance is involved in the reaction.
+
+!!! example
+    The molar enthalpy of combustion (also known as the **heat of combustion**) of ethanol is $\pu{-1367 kJ/mol}$, indicating that every one mole of ethanol combusted releases 1367 kilojoules of energy.  
+    $\Delta H_\text{combustion} = \ce{-1367 kJ/mol C2H6O}$
+
+The **standard molar enthalpy of reaction** $\Delta H^\theta_x$ is the molar enthalpy of reaction when initial and final conditions of the reaction are at standard atmospheric temperature and pressure (SATP, 25°C @ 100 kPA). Therefore, the activation energy, energy released/required during the reaction, and energy released/required following the reaction to return to SATP are all included.
+
+!!! warning
+    This includes energy required for some substances to change state, such as water vapour from combustion cooling to 25°C.
+
+### Energy profiles
+
+Also known as **reaction profiles**, energy profiles are a visual representation of the change in chemical potential energy of the system.
+
+ - Absolute enthalpy ($H$) is placed on the y-axis while the reaction progress (time, sort of) is placed on the x-axis.
+ - A horizontal line representing the enthalpy before the change is placed at the beginning labelled with the reactants.
+ - A horizontal line representing the enthalpy after the change is placed at the end labelled with the products.
+ - The change in enthalpy is labelled with an arrow in the direction of the change with its value if known.
+ - A hump shows the reaction in progress (even exothermic reactions require some activation energy).
+
+### Bond enthalpies
+
+$\Delta H_B$, also known as **bond association energies**, the enthalpy of a bond type (e.g., $\ce{C-H}$) is the energy required to break **1 mol** of that bond type when the reactants and products are **gaseous** so energy is not lost from state changes. Compared of other methods of determining reaction enthalpy, this method is less accurate due to the other compounds affecting bond strength and thus enthalpy on a per-molecule basis.
+
+The change in enthalpy of a reaction can be approximated by considering the bonds broken and formed:
+$$\Delta H = \sum n\Delta H_B\text{reactants} - \sum n\Delta H_B\text{products}$$
+
+## Calorimetry
+
+!!! definition
+    - A **calorimeter** measures changes in energy.
+
+A basic calorimeter uses a lid and insulation to keep matter in and minimise energy changes with its surroundings. A thermometer is used to measure the temperature change of the water, and a stirrer is common to ensure accurate thermometer readings. The reactants are placed in water to react.
+
+<img src="/resources/images/basic-calorimeter.png" width=700>(Source: Kognity)</img>
+
+It is assumed that all the heat lost/gained by the reaction is gained/lost from the water.
+
+$$-q_\ce{H2O}=\Delta H_\text{reaction}$$
+
+In the event that reactants cannot be placed in water to react (e.g., combustion), a **bomb calorimeter** is used, which contains a metal sealed box submerged in the waterfilled with reactant and oxygen. A circuit leads into the box to start the reaction with a spark.
+
+!!! warning
+    Assumptions in calorimetry:
+    
+    - All energy released/absorbed from the system goes to/from the surroundings of the calorimeter (water). This usually needs to be corrected for in bomb calorimeters by measuring the heat capacity and mass of the metal box inside the calorimeter as well.
+    - No energy is transferred outside the calorimeter — the insulation should work properly.
+    - The calorimeter itself does not absorb or release energy — this is not a good assumption but can be compensated for.
+    - A dilute aqueous solution is assumed to have the same density and specific heat capacity as water — this assumption is best when the solute is diluted close to 1 mol/L.
+
+### Measuring calorimeters
+
+Instead of recording the temperature of the calorimeter at any one point, a range of temperatures over time per trial should be plotted to obtain a curve. As calorimeters are not perfect and absorb/release energy, it will generate a graph that peaks and slowly returns to ambient temperature. To remedy this, the line returning the temperature to normal should be **linearly regressed** and extrapolated to the reaction start time to obtain a more accurate peak temperature.
+
+## Hess's law
+
+Hess's law asserts that the change in enthalpy works like displacement - so long as the products and reactants are the same, any reaction with any number of intermediate steps will result in the same change in enthalpy.
+$$\Delta H = \sum \Delta H \text{ of intermediate reactions}$$
+
+### Formation equations
+
+A **formation equation** is a balanced chemical equation where exactly one mole of product and its reactants in **elemental form** are in their standard state — -gens are diatomic, phosphorus is $\ce{P4}$, sulfur is $\ce{S8}$, and at SATP (25°C, 100 kPa).
+
+!!! info
+    Fractions are permitted as coefficients on the reactant side to get exactly one mole of product.
+
+!!! example
+    $$\ce{6C_{(s)} + 6H2_{(g)} + 3O2_{(g)} -> C6H12O6}$$
+    $$\ce{2C_{(s)} + 3/2 H2_{(g)} + 1/2 Cl2_{(g)} -> C2H3Cl_{(g)}}$$
+
+The **standard enthalpy of formation** $\Delta H^\theta_f$ is the energy change from the formation of one mole of its substance from its elements in their standard states. It can be determined by subtracting the sum of the enthalpy of each element/compound on the reactant side and adding those on the product side.
+$$\Delta H = \sum n\Delta H\text{ products} - \sum n\Delta H\text{ reactants}$$
+
+!!! warning
+    It is assumed that there is no state change that would affect enthalpy when calculating *standard* enthalpy of formation.
+
+### Enthalpy cycles
+
+Enthalpy cycles are a visual representation of Hess's law. It is used to show that the energy is the same from initial reactants to a product regardless of any intermediate steps.
+
+!!! example
+    $\Delta H_1 = \Delta H_2 + \Delta H_3$. Note that both arrows point to the intermediate product.
+    <img src="/resources/images/enthalpy-cycles.png" width=700>(Source: Kognity)</img>
+
+## Born-Haber cycles
+
+!!! definition
+    - The **standard enthalpy of atomisation** $\Delta H^\theta_{atm}$ is the energy required to change 1 mol of an element at SATP in its standard state to 1 mol of atoms of that element in its gaseous state.
+
+To form an ionic compound from elements in their standard states:
+
+ - the elements must be converted into gaseous atoms, (enthalpy of atomisation)
+ - the atoms must lose or gain electrons to form ions, (electron affinity/ionisation energy)
+ - and then the gaseous ions must bond to form an ionic compound.
+
+The products should be listed on each level of a Born-Haber cycle, and relatively to-scale arrows should point in the direction of enthalpy change, where upwards increases enthalpy.
+
+<img src="/resources/images/born-haber-simple.png" width=700>(Source: Kognity)</img>
+
+Second ionisation energy may increase the peak enthalpy after it has lowered from first ionisation energy. In this case, unlike the below figure, the first and second ionisation energies can be combined into a single arrow representing the sum of both.
+
+<img src="/resources/images/born-haber-ionisation.png" width=700>(Source: Kognity)</img>
+
+### Lattice enthalpy
+
+The lattice enthalpy of an ionic compound is the energy required to dissociate 1 mol of an ionic solid to its gaseous ions.
+
+ - It decreases as ionic radius increases due to greater distance and charge separation
+ - It increases as difference in charge increases because the greater charges are more strongly attracted
+ - The above only apply if the other (ionic radius/charge) is the same or similar
+ - Difference in charge has a much greater effect than ionic radius as it is multiplicative while the effect of increasing radius is additive
+
+### Enthalpy of solution and hydration
+
+The enthalpy of hydration is the enthalpy change when 1 mol of a gaseous ion is dissolved in water to make an infinitely dilute solution such that it is unaffected by attraction or repulsion from other ions.
+
+!!! example
+    The enthalpy of $\ce{Na+_{(g)} -> Na+_{(aq)}}$ is the enthalpy of hydration of $\ce{Na+}$.
+
+The enthalpy of solution is the enthalpy change when 1 mol of a substance dissolves in water. It is equal to the sum of the enthalpy of hydration and lattice enthalpy.
+
+$$\Delta H_{sol}=\Delta H_{hy} + \Delta H_{latt}$$
+
+!!! example
+    The enthalpy change of $\ce{NaCl_{(s)} -> Na+_{(aq)} + Cl-_{(aq)}}$ is the enthalpy of solution of $\ce{NaCl}$.
+
+## Entropy
+
+**Entropy**, $S$, is a measure of structural disorder in a system in $\pu{J/K/mol}$. Absolute enthalpy is always positive, similar to enthalpy. An increase in disorder results in more entropy which results in a greater chance that a system will be in a certain state.
+
+A reaction that increases entropy can continue even in the absence of extra energy, which results in endothermic reactions.
+
+Reactions that would increase entropy are **entropically favoured**, so entropy will work to make it happen.
+
+The following changes increase entropy:
+
+ - changes in state of one substance to a more disordered state, i.e., solid → liquid → gas,
+ - mixing particles of different types, e.g., solid to aqueous,
+ - increasing the number of moles of total gas or decreasing the number of moles of a solid,
+ - and increasing the number of moles of gas on the product side compared to the reactant side, which has the greatest effect.
+
+### Spontaneity
+
+The **spontaneity** of a reaction is its tendency to continue without extra energy input after its initial activation energy.
+
+Gibbs free energy or **standard free energy** ($\Delta G$/$\Delta G^\theta$, $\pu{kJ}$ or $\pu{kJ/mol}$) is a measure of the sponetaneity of a chemical change. Spontaneous reactions must have a negative $\Delta G$, while those that are positive will require more energy to continue.
+$$\Delta G^\theta = \Delta H^\theta - T\Delta S^\theta$$
+
+## Chemical kinetics
+
+The **rate of a reaction** is the change of reactant to product per unit of time. The following are all viable methods of measuring rate of reaction:
+
+ - change in gas volume via gas collection,
+ - change in mass,
+ - change in light absorption,
+ - titration,
+ - and change in conductivity.
+
+In an ideal gas, the kinetic energy of particles is spread in a **Maxwell-Boltzmann distribution**, where the total area under the curve is equal to the total number of particles in the sample.
+
+<img src="/resources/images/maxwell-boltzmann.png" width=700>(Source: Kognity)</img>
+
+As temperature increases, the distribution's total area *does not change* but the overall spread moves to the right as more particles have higher kinetic energies.
+
+<img src="/resources/images/mbdist-temperature.png" width=700>(Source: Kognity)</img>
+
+### Collision theory
+
+Collision theory states that for a chemical reaction to take place between two particles:
+
+ - they must collide,
+ - they must have proper **collision geometry** or **collision orientation** — similar to viruses bumping into cells, the "keys" must hit "locks" — in this case usually they must strike the bond,
+ - they must collide with enough energy to break the initial bond.
+
+If all of these conditions are met, the collision is an **effective collision** — a collision that results in a chemical reaction.
+
+The rate of a reaction increases with:
+
+ - the frequency of collisions,
+ - and the proportion of collisions that are effective collisions
+
+Over time, the rate generally decreases because initially the highest concentration of reactants results in the highest collision frequency, which goes down as reactants are consumed. The proportion of effective collisions will also decrease as reactants also collide with product. Eventually, the reaction will stop or be so slow it appears to have stopped.
+
+<img src="/resources/images/change-of-rate-over-time.jpg" width=700>(Source: Kognity)</img>
+
+The following factors affect the rate of reaction:
+
+ - **Surface area/particle size of a solid:** as only particles on the surface of a solid can be collided with, smaller solid particles have greater surface area where more collisions can happen, leading to greater collision frequency.
+ - **Concentration/pressure of reactant**: A greater concentration leads to more reactant particles to collide in a given volume, increasing collision frequency.
+ - **Temperature**:
+    - Increasing temperature increases reactant particles' kinetic energy, increasing collision frequency,
+    - however it primarily increases the chance of particles having sufficient activation when they do collide, changing the proportion of effective collisions.
+
+### Activation energy
+
+Because electron clouds repel reach other, without extra energy, particles would not get close enough to break bonds. This energy required for particles to become closer is known as the **activation energy** of a reaction. All chemical reactions have an activation energy requirement.
+
+### Catalysts
+
+A catalyst is a substance that increases the rate of a reaction without being consumed. Not all reactions have catalysts, and increasing catalyst quantity does not necessarily always increase the rate of reaction.
+
+Catalysts operate by reducing the activation energy needed by creating an **alternative reaction pathway** with a lower activation energy, so a larger proportion of particles are able to reach that lower energy requirement.
+
+<img src="/resources/images/catalyst-energy.png" width=700>(Source: Kognity)</img>
+
+Visualised with a Maxwell-Boltzmann distribution:
+
+<img src="/resources/images/mbdist-catalyst.png" width=700>(Source: Kognity)</img>
+
+Catalysts can also improve the chances of correct collision geometry by encouraging certain orientations.
+
+## Rates of reaction
+
+The **law of mass action** states that the rate of any reaction is directly proportional to the product of each reactant **concentration**. For a reaction of the form $\ce{aA + bB -> products}$, the rate law holds that:
+
+$$r=k[A]^a [B]^b$$
+
+where $k$ is the **rate constant**, an empirically determined value that is only valid for one reaction at one temperature. Its units are equal to whatever balances out the equation — where $n$ is the order of reaction, it is equal to $\ce{dm^{3(n-1)}} / \pu{mol}^{n-1} / \pu{s}$.
+
+!!! warning
+    Solids and liquids have constant concentrations, so their factor is incorporated as part of $k$ and **not included** as a separate factor (e.g., not as $[C]^c$).
+
+The **individual order of reaction** is the value of the exponent of a specific reactant in the rate law. It must be a real positive number.
+
+!!! example
+    The individual order of the reaction with respect to $A$ is $a$, and the order of reaction is $a+b$.
+
+To determine the individual order of reaction of a reactant, two identical experiments with equal quantities of the **other** reactants are needed. Where $c$ is the concentration of the reactant between the two trials, $r$ is the rate, and $n$ is the individual order of that reactant:
+
+$$\biggr(\frac{c_2}{c_1}\biggr)^n = \frac{r_2}{r_1}$$
+
+!!! example
+    For the following data, changing the concentration of $\ce{OCl-}$ by a factor of 3 causes a rate change by a factor of 9, therefore the individual order of $\ce{OCl-}$ is 2.
+    
+    | Initial $\ce{[OCl-]}$ | Initial $\ce{[I-]}$ | Initial rate |
+    | --- | --- | --- |
+    | $1.0\times10^{-3}$ | $4.0\times10^{-3}$ | $1.0\times10^{-3}$ |
+    | $3.0\times10^{-3}$ | $4.0\times10^{-3}$ | $9.0\times10^{-3}$ |
+
+### Integrated rate laws
+
+Throughout the course of one trial of one reaction, a **concentration-time graph** can be used to find details about its rate. Where concentration is the concentration of the reactant in question over time:
+
+<img src="/resources/images/concentration-time.png" width=700>(Source: Kognity)</img>
+
+A reactant with an individual order of
+
+ - **zero** shows a negative linear line, and $k=-\text{slope}$.
+ - **one** shows exponential decay, and $k=-\text{slope}$ of a graph of $\ln(\text{concentration})$ against time, which should be linear.
+ - **two** shows a *deeper* exponential decay, and $k=\text{slope}$ of a graph of $\frac{1}{\text{concentration}}$ against time, which should be linear.
+
+Additionally, a **concentration-rate graph** can be used.
+
+<img src="/resources/images/concentration-rate.png" width=700>(Source: Kognity)</img>
+
+A reactant with an individual order of
+
+ - **zero** shows a horizontal line.
+ - **one** shows a positive linear line that passes through the origin.
+ - **two** shows the right side of a positive quadratic that passes through the origin.
+
+### Half-life
+
+The half-life ($t_{1/2}$) of a reaction represents the time required for half of the sample to be used.
+
+In the context of radiation, it is the time for half of the nuclei in a radioactive sample to decay.
+
+In a **zero-order** reaction, each half-life is half of the previous.
+
+In a **first-order** reaction, it is constant regardless of concentration, and can the concentration can be expressed with an equation, where $[A]$ is the concentration of a wanted substance, $k$ is the rate constant, and $[A_0]$ is the initial concentration.
+
+$$\ln[A]=\ln[A_0]-kt_{1/2}$$
+
+In a **second-order** reaction, each half-life is double the previous.
+
+### Reaction mechanisms
+
+!!! definition
+    - A **multi-step reaction** consists of more than one reaction as intermediate steps.
+    - An **elementary step** is the basic step of a multi-step reaction, usually involving one or two molecules but never more than three.
+    - A **reactant** is present initially but not at the end of a reaction unless in excess.
+    - A **product** is not present initially but appears at the end of a reaction.
+    - A **catalyst** is present both at the start and end of a reaction. It may be consumed and regenerated in intermediate steps.
+    - A **reaction intermediate** is not present at the start or end of a reaction as it is generated and consumed in the intermediate steps.
+    - The **molecularity** of a reaction represents the number of molecules that react in an elementary reaction from uni- to termolecular.
+    - An **activated complex** or **transition state** is the point where new bonds are being formed at the same time bonds are being broken.
+
+A reaction involving any more than three particles will always take place under **multiple steps** because of the near-impossibility of such a perfect collision. Even reactions with three particles are often **multi-step**.
+
+The **reaction mechanism** is the step-by-step sequence of all elementary steps of a reaction. An elementary step that is repeated consecutively should be surrounded with square brackets and a coefficient.
+
+!!! example
+    $$\ce{2\times\big[HOBr + HBr -> Br2 + H2O\big]}$$
+
+!!! example
+    The reaction $\ce{NO2_{(g)} + CO_{(g)} -> NO_{(g)} + CO2_{(g)}}$ has a theoretical reaction mechanism of:
+    $$
+    \begin{align*}
+    \ce{
+    NO2_{(g)} + NO2_{(g)} &-> NO3_{(g)} + NO_{(g)} \\
+    NO3_{(g)} + CO_{(g)} &-> NO2_{(g)} + CO2_{(g)}
+    }
+    \end{align*}
+    $$
+    $\ce{NO3_{(g)}}$ is a reaction intermediate.
+
+Multi-step reactions will have a **rate-determining step**, which is the slowest step and so is responsible for the rate law of the reaction, acting as a bottleneck. If reaction intermediates are present, the **original** reactants or catalysts that form that intermediate are still used in the rate law.
+
+!!! example
+    The reaction $\ce{H2_{(g)} + Q2_{(g)} + 2Z2_{(g)} -> 2HZ_{(g)} + 2QZ_{(g)}}$ has the following reaction mechanism:
+    $$
+    \begin{align*}
+    \ce{
+    H2_{(g)} + Q2_{(g)} &-> 2HQ_{(g)} \\
+    2\times\big[HQ_{(g)} + Z2_{(g)} &-> HZ_{(g)} + QZ_{(g)}\big]\ \text{ (slow)}
+    }
+    \end{align*}
+    $$
+    As normally for this reaction $\ce{r=k[HQ][Z2]}$, because $\ce{HQ}$ is a reaction intermediate, it is instead $\ce{r=k[H2][Q2][Z2]}$ after substituting in the first step, **ignoring product coefficients**.
+
+Often, the step with the highest activation energy is the slowest because of collision theory. Alternatively, the one with the least favourable collision geometry, such as if there are more particles that have to collide, may be the slowest.
+
+If a reactant doesn't appear in the rate-limiting step (including via intermediates), changing its concentration will not affect the rate of reaction and so it will have an individual order of 0 in the final rate law.
+
+A reaction mechanism is only plausible if:
+
+ - each elementary reaction has **three** or less reactant particles,
+ - the rate-determining step is consistent with the rate law provided, and
+ - the elementary steps add up to the overall equation.
+
+### Arrhenius equation
+
+The Arrhenius equation relates the temperature to the rate of a reaction.
+
+Where:
+
+ - $k$ is the rate constant,
+ - $R$ is the ideal gas constant,
+ - $E_a$ is the activation energy for the reaction,
+ - $A$ is the proportionality/Arrhenius constant for the reaction,
+ - and $e$ is Euler's number
+
+$$k=Ae^\frac{-E_a}{RT}$$
+
+Graphing $\ln k$ against $\frac{1}{T}$ forms the linear relation:
+$$\ln k = \frac{-E_a}{R}\frac{1}{T}+\ln A$$
+
+where the slope of the graph is $\frac{-E_a}{R}$ and the y-intercept is $\ln A$.
+
+The number of moles of gas particles that are above the activation energy threshold is expressed in the second term of the equation: $e^\frac{-E_a}{RT}$.
+
+## Equilibrium
+
+!!! definition
+    - A reaction is at **dynamic equilibrium** if both the forward and reverse reaction continue at **equal and constant** rates, and there are no **macroscopic** changes such as temperature, colour, mass, or concentration.
+
+A chemical equation at equilibrium is represented with two single-headed arrows, indicating that a reaction has proceeded to the point that concentrations are constant, and rates are equal and constant.
+$$\ce{A + B <=> C}$$
+
+<img src="/resources/images/equilibrium-rate.png" width=700>(Source: Kognity)</img>
+
+In order for a system to eventually tend to equilibrium, the system must:
+
+ - be closed, with constant concentrations of reactant and product,
+ - maintain a constant temperature, and
+ - maintain a constant pressure if the reactant or product is a gas.
+
+For a given reaction, as long as the reactants and products are stoichiometrically matched, any combination will tend to the same equilibrium.
+
+!!! example
+    The following initital concentrations for the reaction $\ce{C + O2 -> CO2}$ will all tend to the same equilibrium.
+    
+    - 2 mol $\ce{C}$ and 2 mol $\ce{O2}$
+    - 2 mol $\ce{CO2}$
+    - 1 mol $\ce{C}$, 1 mol $\ce{O2}$, and 1 mol $\ce{CO2}$
+
+At equilibrium, the concentrations of the reactants and products must end up constant (but **not necessarily equal**).
+
+!!! example
+    <img src="/resources/images/equilibrium-concentration.jpeg" width=700>(Source: Kognity)</img>
+
+**Phase equilibrium** is when two or more states of exactly one pure substance are in dynamic equilibrium.
+
+!!! warning
+    A solution or an aqueous compound cannot be in phase equilibrium because it is not a pure substance.
+
+!!! example
+    Water constantly evaporates and condenses. Because the rate of evaporation is only dependent on the surface area of the water, the rate of condensation increases until the two are equal and constant at phase equilibrium.
+
+A **solubility equilibrium** requires at least two substances — a solute and a solvent.
+
+### Equilibrium constant
+
+!!! definition
+    - The **position of equilibrium** is the concentration of reactants and products at dynamic equilibrium.
+
+The equilibrium constant $K_c$ or $K_eq$ is related to the concentration of reactants and products in a given system at equilibrium at a given temperature. It is equal to the product of all products divided by the product of all reactants.
+$$
+\ce{aA + bB + cC <=> fF + gG + hH} \\
+\begin{align*}
+K_c &= \ce{\frac{[F]^f [G]^g [H]^h}{[A]^a [B]^b [C]^c}} \\
+&= \frac{\Pi[\text{products}]^p}{\Pi[\text{reactants}]^r}
+\end{align*}
+$$
+
+The units of $K_c$ varies similar to the rate constant so they are often omitted.
+
+!!! warning
+    Only concentrations that change during the course of the reaction should appear in $K_c$, so solids and liquid water should not be included. 
+
+If $K_c$ is greater than 1000, the reaction is **product-favoured**, meaning that there will be a greater concentration of products at equilibrium. If $K_c$ is less than 0.001, the reaction is **reactant-favoured**.
+
+Contrary to the house of cards of lies told to you in lower grades, all reactions are equilibrium reactions, but some have $K_c$s that are so large or small that they effectively occur to completion or don't occur at all.
+
+#### ICE tables
+
+An initial-change-equilibrium (ICE) table is used to work with equilibrium concentrations and **only contains concentrations**.
+
+It consists of:
+
+ - the original concentrations of each compound in the "initial" row,
+ - the change in concentration in the form of a variable of each compound after one "iteration" of the reaction in the "change" row, and
+ - the end equilibrium concentration of each compound in the "equilibrium" row. The "initial" and "change" rows should sum to the "equilibrium" row.
+
+!!! example
+    An ICE table with 1 mole each of $\ce{H2}$ and $\ce{I2}$ in $\pu{2.00 dm3}$ of water that eventually ends up with an equilibrium concentration of $\ce{[H2]}=\pu{0.11 mol/dm3}$ will form the following ICE table.
+    
+    | | $\ce{H2_{(g)}}$ | $\ce{I2_{(g)}}$ | $\ce{2HI_{(g)}}$ |
+    | --- | --- | --- | --- |
+    | Initial | 0.50 | 0.50 | 0 |
+    | Change | $-y$ | $-y$ | $+2y$ |
+    | Equilibrium | 0.11 | 0.50$-y$ | $+2y$ |
+
+When working with values involving $K_c$, if the initial concentration of a chemical is much bigger than $K_c$ ($[A]/K_c > 500$), it is possible to assume that it will not change at all.
+
+This assumption is valid if the impact of the calculated shift is less than 5%.
+
+!!! example
+    If the equilibrium concenration is equal to $0.250-2y$, and the initial concentration is very big, assume that the equilibrium concentration is $0.250$, removing the $-2y$ from the equation.
+    
+    As long as $2y$ is less than 5% of 0.250, the assumption is valid.
+
+!!! info
+    In this course, when working with $K_c$ and ICE tables, only three things should be possible when solving for concentrations: you can get a perfect square, you can use the quadratic equation, or you can use the approximation rule.
+
+### Le Chatelier's principle
+
+Le Chatelier's principle states that: If there is a change in a system at equilibrium, the position of equilibrium will readjust to minimise the effect of the change.
+
+The changes that this principle affects — and therefore affect equilibrium — include changes in temperature, concentration, and pressure. These changes are assumed to occur instantaneously, which may result in sudden theoretical spikes in concentration-time graphs.
+
+The initial rate of the change will start **fast** and then slow down, appearing as a sharp change instead into a curve in a concentration-time/reaction progress graph that **never return to its original value**.
+
+!!! tip
+    Drawing horizontal dotted lines that represent the original position of equilibrium and vertical lines to represent the moment of system change makes it clearer to read.
+
+Increasing the **temperature** of a system causes it to shift in favour of the **endothermic** side, and vice versa. 
+
+Of the three changes, this is the only one that would change $K_c$ as it changes the rate constants, which are temperature-specific ($K_c\propto\frac{r_\text{reverse}}{r_\text{forward}}$). Therefore, as temperature **increases**, $K_c$ also **increases**, and vice versa.
+
+!!! example
+    If heat is added to a solution of KCl, more KCl will dissolve to minimise the change in temperature as it is an endothermic process.
+
+Increasing the **concentration** of a reactant or product will cause the position to shift **away** from the increased side, and vice versa.
+
+??? example
+    If there is an **instantaneous** spike of $\ce{N2}$ to a system at equilibrium, it will be consumed along with $\ce{H2}$ to form $\ce{NH3}$, **but not enough to return to its original value**.
+    
+    <img src="/resources/images/equilibrium-concentration.png" width=500>(Source: Kognity)</img>
+    
+    The same applies if instead $\ce{NH3}$ is reduced.
+    
+    <img src="/resources/images/equilibrium-concentration-2.png" width=500>(Source: Kognity)</img>
+
+Increasing the **pressure** of a gas will cause the position to shift in whatever direction would **decrease** the total moles of gas.
+
+!!! warning
+    Inert (uninvolved in a reaction) gases such as catalysts will not affect the position of equilibrium as it does not affect the **partial pressure** of the gas. In a similar vein, adding water to an aqueous solution will not cause any changes in equilibrium position.
+
+!!! warning
+    If given a system not at equilibrium, if a change is made that would change the prior equilibrium, it should be assumed that the system reaches equilibrium before the change is made, regardless if it is specified.
+
+### Gibbs free energy 2
+
+The value of Gibbs free energy changes as the reaction progresses, similar to enthalpy. At equilibrium, $\Delta G=0$, so a reaction is a result of a system attempting to minimise Gibbs free energy.
+
+**Standard Gibbs free energy** represents the Gibbs free energy of a chemical at standard state (1 mol/L for solutions, 100 kPa partial pressure for gases).
+$$\Delta G^\circ = \sum n\Delta G^\circ_\text{f products} - \sum n\Delta G^\circ_\text{f reactants}$$
+
+A negative $\Delta G^\circ$ indicates that the reaction will shift right to reach equilibrium as $\Delta G^\circ$ always decreases in **magnitude** as the reaction proceeds. It also means that the forward reaction is **spontaneous** while the backwards is not.
+
+### Reaction quotient
+
+The reaction quotient ($Q$) is a tool to compare the current state of a system to its equilibrium state.
+$$Q=\frac{\Pi[\text{products currently}]^p}{\Pi[\text{reactants currently}]^r}$$
+
+At equilibrium, $Q=K_c$ as they are the same equation, so the equilibrium will shift in whatever direction that would bring $Q$ closer to $K_c$
+
+!!! example
+    If $Q > K_c$, there are more products than reactants than at equilibrium, so the reaction will shift to make more **reactants**.
+
+### Dynamic equilibrium
+
+When $\Delta G$ is at a minimum, both sides of the reaction are equally spontaneous. Realistically, $\Delta G$ never reaches zero because entropy. TODO: wtf
+
+<img src="/resources/images/product-favoured-gibbs.png" width=700>(Source: Kognity)</img>
+
+Where $\Delta G$ is the Gibbs free energy at a given point of the reaction, $R$ is the gas constant, $T$ is the current temperature, and $Q$ is the reactant quotient:
+$$\Delta G = \Delta G^\circ + RT\ln Q$$
+
+Therefore, at equilibrium:
+$\Delta G = -RT\ln K_c$
+
+## Acids and bases
+
+!!! definition
+    - An **amphoteric** chemical may act as an acid or base depending on the situation.
+    - An **amphiprotic** chemical can **either accept or donate** $\ce{H+}$ depending on the situation.
+    - A **monoprotic** acid/base is one that can only accept/ionise one $\ce{H+}$ ion.
+    - An **alkali/alkaline** solution is an aqueous solution of a base, which may **not** necessarily be a **basic solution**.
+
+An **acid** and **base** are any two corrosive chemicals that react to form water and a salt. They also dissociate/ionise (depending on theory) in water to form electrolytes that conduct electricity.
+
+Acids:
+
+ - taste sour
+ - have a pH less than 7 in aqueous solutions at 25°C
+ - stain litmus paper **red**
+ - react with active metals to produce $\ce{H2_{(g)}}$ based on the activity series
+ - react with carbonates to form $\ce{CO2 + H2O}$
+
+Bases:
+
+ - taste bitter
+ - have a pH greater than 7 in aqueous solutions at 25°C
+ - feel slippery as they react with fats/oils to form soap
+ - stain litmus paper **blue**
+ - react with ammonium salts to product $\ce{NH3 + H2O}$
+
+### Arrhenius theory
+
+An acid **dissociates** in water to produce $\ce{H+}$ ions (protons).
+
+A base **dissociates** in water to produce $\ce{OH-}$ ions.
+
+### Bronsted-Lowry theory
+
+The Bronsted-Lowry theory focuses on reactions with water and less the acid and base ions themselves, so they **ionise** instead of **dissociate**.
+
+An acid is any compound that can **donate/ionise a proton ($\ce{H+}$) to water** to form a hydronium ion.
+$$\ce{acid + H2O -> acid- + H3O+}$$
+
+!!! info
+    In practice, the acid must contain a hydrogen atom attached by an easy-to-break bond (usually $\ce{H-O}$), but any high electronegativity difference polar bond would work as well.
+
+A base is any compound capable of **accepting/removing a proton ($\ce{H+}$) from an acid**.
+$$\ce{acid + base -> acid- + base+}$$
+
+!!! info
+    The proton usually comes from water. The base must be able to accept an $\ce{H+}$ ion to form a **dative covalent bond**, so they must contain **lone pairs**.
+
+Polyprotic acids ionise their $\ce{H+}$s one by one **sequentially**.
+
+!!! example
+    $\ce{
+    H3PO4 + H2O <=> H2PO4- + H3O+ \\
+    H2PO4- + H2O <=> HPO4^2- + H3O+ \\
+    HPO4^2- + H2O <=> PO4^3- + H3O+
+    }$
+    
+#### Conjugate acids/bases
+
+The result of a base obtaining a proton is a **conjugate acid**.
+
+The result of an acid losing a proton is a **conjugate base**.
+
+!!! example
+    In the reaction
+    $$\ce{NH3 + H2O -> NH4+ + OH-}$$
+    
+    $\ce{NH3}$ is a base that becomes a conjugate acid while $\ce{H2O}$ is an acid that becomes a conjugate base.
+
+### Louis theory
+
+A Lewis **acid** is any species that **accepts** an electron pair to form a dative covalent bond.
+
+A Lewis **base** is any species that **donates** an electron pair to form a dative covalent bond.
+
+### Strong/weak acids/bases
+
+**Strong** acids/bases will **completely** dissociate/ionise in an aqueous solution. This means that the initial concentration of acid will be equal to the end concentration of $\ce{H+ or H3O+}$.
+
+All strong polyprotic acids initially have a one-way reaction then follow with equilibrium reactions.
+
+!!! warning
+    Strength is a property of an acid and has nothing to do with its concentration.
+
+**Weak** acids/bases will only **partially** dissociate/ionise in an aqueous solution, leaving behind most of the initial acid ($\ce{[acid] > [H+]}$ at equilibrium).
+
+!!! warning
+    Measuring pH only returns $\ce{[H+] or [H3O+]}$, so it cannot be used to determine the concentration, identity, or strength of an acid.
+
+All weak polyprotic acid reactions are equilibrium reactions.
+
+!!! example
+    The following is a list of strong and weak acids:
+    
+    | Strong acid | Weak acid | Strong base | Weak base |
+    | --- | --- | --- | --- |
+    | $\ce{HClO4}$ | any $\ce{COOH}$ | $\ce{LiOH}$ | $\ce{NH3}$ |
+    | $\ce{HCl}$ | $\ce{CO2}$ | any $\ce{group\ 1 + OH}$ | $\ce{Al(OH)3}$ |
+    | $\ce{HBr}$ | $\ce{SO2}$ | any $\ce{group\ 2 + (OH)2}$ | |
+    | $\ce{HI}$ | $\ce{HF}$ | | |
+    | $\ce{H2SO4}$ | $\ce{HCN}$ | | |
+    | $\ce{HNO3}$ | $\ce{H3PO4}$ | | |
+
+To experimentally distinguish between a strong or weak acid/base, if their concentrations are equal, total **ion** concentration or $\ce{H3O+}$ concentration can be compared since the stronger acid ionises more.
+
+Practically, this means comparing the rate of reaction with a metal or water or measuring conductivity as they reflect total ion count.
+
+### pH and pOH
+
+This section will assume Bronsted-Lowry theory.
+
+pH represents $\ce{[H3O+]}$ logarithmically on a scale from 0 to 14.
+
+$$
+\ce{pH = -\log\big[H3O+_{(aq)}\big]} \\
+\ce{pOH = -\log\big[OH-_{(aq)}\big]}
+$$
+
+!!! warning
+    The number of sigfigs in pH is equal to the number of digits **after the decimal place**.
+
+A solution is **neutral** (neither acidic nor basic) when $\ce{[H3O+] = [OH-]}$. This happens to be $\ce{pH = 7}$ at SATP. In pure water, this is true as a small number of water molecules react with each other.
+
+In an equilibrium reaction between an acid and a base, $\ce{K_c = \frac{[H3O+][OH-]}{[H2O]}}$, but water has a constant concentration, so the equilibrium of the two ions is represented with the **water ionisation constant** $K_w$ is used.
+$$K_w = \ce{[H3O+][OH-] = 1.00\times10^{-14} @ SATP}$$
+
+As temperature **increases**, $K_w$ increases, therefore changing the pH of neutrality, but this may not necessarily change the acidity of the solution as the ion concentration is still the same.
+
+As pH increases, $\ce{[H3O+]}$ decreases, so $\ce{[OH-]}$ must increase to keep $K_w$ constant and maintain equilibrium.
+$$\ce{pK_w = pH + pOH}$$
+
+At 25°C, $\ce{pK_w = 14.0000}$, so:
+$$\ce{14 = pH + pOH}$$
+
+### Acid/base dissociation
+
+An equilibrium will be reached when a weak acid or base dissociates/ionises in water. The extent that the acid or base has dissociated/ionised can be quantified with **percent dissociation/ionisation**.
+$$\text{% ionisation} = \frac{\text{[acid ionised]}}{\text{[original acid]}}\times 100\%$$
+
+!!! note
+    When performing an approximation assumption in an ICE table, the assumption is also valid if the % ionisation is less than 5%.
+
+The $K_c$ of acid ionisation/dissociation is known as $K_a$, the **acid dissociation constant**.
+$$
+\ce{H2O_{(l)} + HX_{(aq)} <=> H3O+_{(aq)} + X-_{(aq)}}$$
+$$K_a = \ce{\frac{[X-][H3O+]}{[HX]}}$$
+
+The $K_c$ of base ionisation/dissociation is known as $K_b$, the **base dissociation constant**.
+$$\ce{H2O_{(l)} + X_{(aq)} <=> OH-_{(aq)} + HX+_{(aq)}}$$
+$$K_b = \ce{\frac{[HX+][OH-]}{[X]}}$$
+
+!!! example
+    $$\ce{NH3_{(aq)} + H2O_{(l)} <=> NH4+_{(aq)} + OH-_{(aq)}}$$
+    $$K_b = \ce{\frac{[NH4+] [OH-]}{[NH3]}}$$
+
+!!! warning
+    $K_a$ and $K_b$ only apply to acids and bases, respectively. Morphine, a base, does not have a $K_a$, but its conjugate acid does.
+
+At all temperatures:
+$$K_a \times K_b = K_w$$
+$$pK_a + pK_b = pK_w$$
+
+### Acid strength
+
+A **higher** $K_a$ or $K_b$ indicates that the acid or base is **stronger**, increasing percent ionisation. 
+
+**Strong acids/bases** have an effectively infinite $K_a$/$K_b$ in water, so they are all practically equally strong in water (this may not be true in other solvents).
+
+As $K_a$ and $K_b$ are inversely correlated, an **increase** in $K_a$ leads to a **decrease** in $K_b$.
+
+The conjugate acid/base of a **strong** acid/base is effectively infinitely weak such that it does not affect pH at all.
+
+Contrarily, the conjugate of a **weak** acid/base is measurably weak, strong enough to have $K_a$/$K_b$ that affect the pH and act as an acid or base.
+
+As $p$ indicates negative log, $\ce{pK_{\{a, b\}}}$ is **inversely** correlated with $\ce{K_{\{a, b\}}}$ so that none of the variables can be directly compared without conversion.
+
+### Acidity of salt solutions
+
+!!! definition
+    - A **salt** is an ionic compound that dissociates in water.
+
+The pH of a salt solution depends on the combination of the acidity of each of its dissociated ions. Whichever is **stronger** pushes the acidity of a solution in its direction.
+
+An ion originating from a **strong** acid/base is immeasurably weak and **has no effect**.
+
+An ion originating from a **weak** acid/base is measurably weak and **has an effect**.
+
+!!! example
+    For the salt NaCl: $\ce{HCl_{(aq)} + NaOH_{(aq)} -> H2O_{(l)} + NaCl_{(aq)}}$  
+    As both $\ce{HCl}$ and $\ce{NaOH}$ are strong, their conjugate acids/bases are both immeasurably weak, having no effect on the pH of the solution. Therefore, NaCl is a **neutral** salt.
+
+If both dissociated ions have a measurable effect, the acidity depends on whichever is stronger via $K_a$/$K_b$.
+
+### Titration curves
+
+!!! definition
+    - A **titrant** or **standard** is a solution with known properties that goes in the burette.
+    - A **sample** or **analyte** is a solution with potentially unknown properties that goes in the sample flask.
+    - The **equivalence point** of a titration is the point at which the solution is neutral $\ce{[H+] = [OH-]}$.
+
+A **titration curve** is generated in a titration where the pH of the solution is graphed against the volume of titrant added. Depending on the type or strength of the sample and titrant, different graphs can be generated.
+
+Unlike the diagrams below, in a sketch, the following information is needed:
+
+ - the initial pH of the solution (the pH of the sample)
+ - the pH after a lot of titrant is added (assumed to be the pH of the titrant)
+ - the volume of titrant required to neutralise the sample ($c_\ce{H}v_\ce{H}=c_\ce{OH}v_\ce{OH}$)
+ - the relative pH at the equivalence point (the relative pH of the salt solution)
+
+The graph can be split into two halves: the acid half and the base half. In the following diagram, both the acid and base are **strong**, so their lines are identically shaped:
+
+ - the line starts at the initial pH, hugging the line, until,
+ - it sharply curves to the vertical, crossing the equivalence point, and
+ - continues vertically.
+
+The same applies to the base but it instead ends at the final pH.
+
+<img src="/resources/images/s-s-titration.png" width=700>(Source: Kognity)</img>
+
+In scenarios where the **sample** is a **weak** acid/base, instead:
+
+ - the line immediately briefly rapidly rises from the initial pH, then
+ - gradually increases, until
+ - a sudden curve to the vertical (but less sudden than a strong acid/base), and
+ - continues vertically.
+
+<img src="/resources/images/w-s-titration.png" width=700>(Source: Kognity)</img>
+
+In scenarios where the **titrant** is a **weak** acid/base, it will take much more titrant to bring the pH of the sample to the level of the titrant. As such, the "brief rapid rise" is ignored and the line only gradually approaches but **clearly does not reach** the final pH.
+
+<img src="/resources/images/s-w-titration.png" width=700>(Source: Kognity)</img>
+
+In scenarios where the sample is a **polyprotic** acid/base, as its ions dissociate sequentially, it can be treated as multiple consecutive titrations where the first sample is **strong** but any subsequent titrations are **weak**.
+
+Each equivalent point volume after the first is a direct multiple of that first equivalent point volume.
+
+<img src="/resources/images/polyprotic-titration.png" width=700>(Source: Kognity)</img>
+
+### Titration curve analysis
+
+The **aha!** equation, also known as the Henderson-Hasselbach equation, is derived from the equilibrium equation.
+
+$$\ce{pH = pK_a + \log\frac{[A-]}{[HA]}}$$
+
+To graphically determine the $pK_a$ of a sample given a titration curve, the pH at the volume at **half** of the equivalence point can be identified. At that point, $pH = pK_a$.
+
+!!! warning
+    - If a weak **base** is the sample, this will return the $pK_a$ of the conjugate acid. The pH of the base can be determined by $14-pK_a$.
+    - If numbers are not given, drawing a line through the straight bits can give pH and equivalence volume values. However, **none** of these lines should be **parallel** to the axes.
+    - In titrations involving **polyprotic** compounds, as they are effectively multiple titrations, half of the equivalence point is actually half the distance between two equivalence points.
+    - This equation can **also be used to determine the pH of a buffer between a salt/acid and acid/base** as it can be assumed that there is no change to concentrations, but **cannot be used to determine the initial pH between an acid/base and water**.
+
+### pH indicators
+
+A pH indicator is a **weak acid/base** that is at one colour in a certain pH range and another in another pH range. Where $X$ is the indicator, it will form an equilibrium with the hydrogen ions in the solution:
+$$\ce{HX <=> H+ + X-}$$
+
+The indicator is **protonated** on the left and **deprotonated** on the right. The titrant can be viewed as an external stress on this equilibrium: if a base is added to an acid, the equilibrium will shift to the right to free up hydrogen ions, and vice versa.
+
+If the difference in concentration of $\ce{X-}$ and $\ce{HX}$ is **greater than** approximately 10:1, the solution will appear to be the colour of the higher concentration, meaning that pH indicators will change colour at a pH in the range of their $\ce{pK_a}\pm 1$.
+
+In choosing a good pH indicator, it must change colour in the **vertical** section of the titration curve to see the greatest effect, and it must be easily observable.
+
+As the weak curve has less of a vertical section than a strong curve, it is best to pick an indicator that changes **after** the equivalence point, which will require the **relative pH** at the equivalence point.
+
+The observability of an indicator depends on the colour it is changing to (or the **direction** the pH is changing). In general, humans are much better at noticing the **appearance** of **red** and **blue**.
+
+A **universal indicator** is a mixture of different pH indicators to change colours multiple times over the pH range. In this case, the colour wheel can used to determine the colour that will be formed (e.g., blue + blue + yellow = green). The shade of the colour does not matter.
+
+### Buffers
+
+!!! definition
+    - A **buffer solution** is one that can resist pH change when small quantities of a strong acid or base are added.
+    - An **acidic buffer** is one where an acid and extra of its conjugate base as a salt are present in the solution.
+    - A **basic buffer** is one where a base and extra of its conjugate acid as a salt are present in the solution.
+    - A **protonated** compound contains its proton.
+    - A **deprotonated** compound has lost its proton.
+    - The **buffering capacity** of a buffer is the quantity of strong titrant that can be added to the buffer without a significant change in pH.
+
+The **buffer region** is the pH range of a **weak** acid/base before the equivalence point that requires a large volume of titrant for a gradual pH change. In this region, there is sufficient undissociated acid/base to replenish those neutralised via Le Chatelier's principle.
+
+In the equilibrium between a weak acid and its component ions:
+$$\ce{HA <=> H+ + A-}$$
+
+A buffer solution is created when **excess** $\ce{A-}$ is added (the salt of the conjugate base) such that the position of equilibrium is shifted to the left to the point that **none of the original acid has dissociated** such that $\ce{[equilibrium HA] = [added HA]}$ and $\ce{[equilibrium A-] = [added A-]}$. It is used to **maintain** a certain pH in a solution.
+
+When the titrant is added to an **acidic buffer**:
+
+ - if an acid is added, $\ce{[H+]}$ increases, shifting the position to the left. This can be done repeatedly because of the excess $\ce{A-}$ present to react with the protons.
+ - if a base is added, $\ce{[H+]}$ decreases as they react, shifting the position to the right. This can be done repeatedly because of the excess $\ce{HA}$ from the original shift to the left from the salt addition.
+
+!!! example
+    To form the **acetic acid/acetate buffer** $\ce{CH3COOH <=> H+ + CH3COO-}$, if 1 mol/L $\ce{CH3COONa}$ is added to 0.1 mol/L $\ce{CH3COOH}$:
+    
+    The addition of $\ce{CH3COO-}$ will shift the position to the left, protonating it such that there will be 0.1 mol/L $\ce{CH3COOH}$ and 1 mol/L $\ce{CH3COO-}$.
+    
+    - If an acid is added, it will **shift left** and further react to form more $\ce{CH3COOH}$, reducing the change in pH.
+    - If a base is added, it will **shift right** by reacting with hydrogen ions to reduce their concentration, releasing more $\ce{H+ + CH3COO-}$ to replenish the lost hydrogen ions, reducing the change in pH.
+    
+This naturally occurs without a buffer, but a buffer significantly increases the quantity of titrant that can be added before the pH changes rapidly.
+
+The same applies to a **basic buffer** but in opposite directions. The salt of the conjugate acid is used instead.
+$$\ce{B + H2O <=> HB + OH-}$$
+
+!!! example
+    To form the **ammonia/ammonium buffer** $\ce{NH3 + H2O <=> NH4+ + OH-}$, if 1 mol/L $\ce{NH4+}$ is added to 0.1 mol/L $\ce{NH3}$:
+    
+    The addition of $\ce{NH4+}$ will shift the position to the left, deprotonating it such that there will be 0.1 mol/L $\ce{NH3}$ and 1 mol/L $\ce{NH4+}$.
+    
+    - If an acid is added, it will **shift right** by reacting with hydroxide ions to reduce their concentration, releasing more $\ce{NH4+ + OH-}$ to replenish the lost hydroxide ions, reducing the change in pH.
+    - If a base is added, it will **shift left** and further react to form more $\ce{NH3 + H2O}$, reducing the change in pH.
+
+To make an effective buffer, salt of the conjugate base/the conjugate acid is required to initially shift the position left. Adding more salt/acid increases the titrant that can be buffered.
+
+A buffer only acts over a certain pH. In order for it to be effective, the ratio of $\ce{[A-]}$ to $\ce{[HA]}$ must be within 10x or 0.1x, although usually buffers are made with 90% excess salt/acid + 10% acid/base or vice versa. Using the **aha!** equation, this means that the **range of a buffer** is equal to $pK_a\pm 1$, where $pK_a$ is that of the **acid/conjugate acid of the base**.
+

docs/sph3u7.md

@@ -1,4 +1,4 @@
-# SL Physics - A
+# SL Physics - 1
 
 The course code for this page is **SPH3U7**.
 
@@ -72,7 +72,7 @@ Every SI unit can be expanded with metric prefixes. Note that the difference bet
 Scientific notation is written in the form of $m×10^{n}$, where $1 \leq m < 10, n \in \mathbb{Z}$. All digits before the multiplication sign in scientific notation are significant.<br>
 
 !!! example
-    The speed of light is 300 000 000 ms<sup>-1</sup>, or $3×10^8$ ms<sup>-1</sup>.
+    The speed of light in a vacuum ($c$) is effectively 300 000 000 ms<sup>-1</sup>, or $3×10^8$ ms<sup>-1</sup>.
 
 ### Orders of magnitude
 
@@ -316,18 +316,789 @@ By the formula of the gradient and the formula for the area underneath an accele
 
 **Projectile motion** is uniformly accelerated motion that does not leave the vertical plane (is two-dimensional). Note that the two directions (horizontal and vertical) that the projectile moves in are independent of one another. This means that variables such as average velocity can be calculated by breaking up the motion into vector **components**, then finding the resultant vector.
 
-Projectiles move at a constant horizontal velocity and move at a uniformly accelerated velocity (usually $9.81 \text{ ms}^2 \text{ [down]}$).
+Projectiles move at a **constant** horizontal velocity and move at a uniformly accelerated velocity (usually $9.81 \text{ ms}^2 \text{ [down]}$).
+
+Therefore the horizontal velocity of a projectile ($v_x$) is constant and is equal to $v \cos \theta$ and the vertical velocity ($v_y$) is equal to $v \sin \theta$ where $\theta$ is the direction of $\vec{v}$ above the horizontal axis.
 
 ## 2.2 - Forces
 
+A force is a "push" or "pull" exerted by one object acting on another object and is always represented in the form of a vector. The SI unit for force is the newton ($\text{N}$), equivalent to $\text{kg} \cdot \text{m} \cdot \text{s}^{-2}$.
+
+Forces do not cause motion but instead cause *changes* in motion; therefore a constant force on an object results in **uniformly accelerated motion**.
+
+A force must have:
+
+ - A magnitude and direction
+ - An object on which it acts and a location on that object where it acts
+ - An object providing the force
+ - An object acting and an object acted upon that are *not* the same
+
+### Types of forces
+
+A force must be classified as one of the following types of forces:
+
+Contact (the two objects are touching) forces:
+
+ - **Tension** ($\vec{F_T}$) is the pulling force from one object on another. Note that tension goes both ways: whichever object is pulling is also being pulled in the opposite direction.
+ - **Normal force** ($\vec{F_N}$) is the "resistance" or "pushback" force from one object on another pushing on it and occurs because of Newton's third law of motion. It is always perpendicular to the surfaces of the objects.
+ - **Kinetic and static friction** ($\vec{F_{fK}}$ and $\vec{F_{fS}}$, respectively) is the force resisting the motion of two surfaces sliding against each other. Kinetic friction opposes an object in motion while static friction opposes an object from moving in the first place and is equal to the force applied that would cause the object to move. The vector of friction is parallel to the surface and opposite the direction of motion.
+ - **Drag** ($\vec{F_D}$) is the resistive force caused by fluids in the path of an object. Its vector is always pointing in the opposite direction of the object's *instantaneous* velocity.
+ - **Thrust** ($\vec{F_T}$) is the propulsive force caused by mass (usually fluids) being ejected from an object. The force vector always points opposite the direction of the ejected mass.
+
+### Friction
+
+Static friction is always equal to the force applied by another object until an object begins to slip, at which point kinetic/dynamic friction applies. Kinetic friction, also known as **dynamic friction**, is a constant force that is lower than the peak force of static friction. 
+
+The **coefficient of friction** ($\mu$) represents the ratio between friction force and normal force, and has a different value for kinetic and static friction. The **maximum** force of static friction is equal to its coefficient multiplied by the normal force ($|F_{fS}|_{\max} = \mu_s F_N$), while the force of kinetic friction is equal to its coefficient multiplied by the normal force ($F_{fK} = \mu_k F_N$).
+
+<img src="/resources/images/static-kinetic-friction.png" width=500>(Source: Kognity)</img>
+
+Non-contact forces:
+
+ - **Weight** ($\vec{W}$) is the attraction between a mass and a planet. Its vector always points toward the planet. The magnitude of weight is equal to the mass of the object times the acceleration due to gravity ($mg$).
+ - **Electromagnetic forces** ($\vec{F_M}$ and $\vec{F_E}$) are not covered in this course.
+
+The origin and target of a force are placed in the subscript of that force.
+
+!!! example
+    If a rock is pushed across the ground by a hand at a constant velocity, the forces acting on the rock would include:
+
+    - the normal force from the hand, or $F_{T \text{ hR}}$
+    - the kinetic friction from the ground, or $F_{fK \text{ gR}}$
+    - the weight of the rock from the planet, or $W_\text{ ER}$
+    - the normal force from the ground pushing against the rock's weight, or $F_{N \text{ gR}}$
+
+### Force diagrams
+
+Also known as **free body diagrams**, force diagrams are used to list the forces acting on a single object represented as a point.
+
+<img src="/resources/images/free-body-diagram.png" width=700>(Source: Kognity)</img>
+
+!!! warning
+    The labels of the vectors in the graphs should use symbols and subscripts as per the example in [#Friction](./#friction).
+
+Force diagrams must include:
+
+ - The object as a point
+ - Either $\vec{a}=0$ or a squiggly arrow labeled $\vec{a}$ in the direction of acceleration
+ - All contact forces with their **subscript-labeled** vectors pointing away from the object and their lengths representative of their magnitude relative to the other vectors
+	- If there are multiple forces acting on an object in the same direction, those vectors should be stacked
+ - Weight as a vector following the rules above
+
+When dealing with forces on an **inclined plane**, it can be easier to work with if the axes are rotated such that the horizontal axis is parallel to the surface between the objects. The force of weight always points downward even in this scenario, and can be broken up into its vector **components**.
+
+<img src="/resources/images/rotated-fbd.png" width=700>(Source: Kognity)</img>
+
+<img src="/resources/images/rotated-fbd-explanation.png" width=700>(Source: Kognity)</img>
+
+### Balanced and unbalanced forces
+
+If an object as a **zero net force** in all dimensions, the forces acting on it are **balanced** and motion does not change. Otherwise, there is a change in motion and the forces acting on the object are **unbalanced**.
+
+### Newton's laws of motion
+
+Newton's first law, also known as the **law of inertia**) states that every object will remain at rest or at constant velocity unless acted upon by an unbalanced force.
+
+Newton's second law states that the **net** force on an object is equal to its mass multiplied by its acceleration, such that $\Sigma \vec{F} = m\vec{a}$.
+
+Newton's third law states that every force applied on an object as an equal and opposite reaction force on another object. These pairs of forces are known as **action/reaction pairs**. Forces are *only* part of the same pair when they:
+
+ - have the same magnitude but opposite directions
+ - are of the same type (e.g., both are weight or normal force)
+ - start and stop at the same time
+ - act on **two different objects**, thus requiring **two** force diagrams to show both of them
+
+!!! example
+    If a block is lying on the ground, the force from the weight of the Earth on the block is the same as the force of the block on the Earth.
+
+!!! warning
+    The weight of an object and the normal force resisting that weight is *not* an action/reaction pair as they appear on the *same* force diagram and act on the *same* object.
+
+### Pulleys
+
+An ideal pulley changes the **direction** of force without affecting its magnitude. The tension on and acceleration of both objects on either end of pulley are always **equal**.
+
+If both objects on both sides of a pulley have the same mass, Newton's third law applies and tension is equal to one of the objects' weight. Otherwise, two free body diagrams can be drawn to represent the forces on both objects. As pulleys only change the direction of force, one pulley system can be straightened out and axes set relative to the direction of acceleration.
+
+!!! example
+    The direction $\text{[clockwise]}$ is often used for this purpose.
+
 ## 2.3 - Work, energy, and power
 
+### Work
+
+Work is a scalar value measured in **joules** ($\text{J}$) and represents the energy transferred to an object or system. It is equal to the dot product (a special type of multiplication between vectors) of force and displacement, where $\theta$ is the **angle between the two vectors**, such that:
+$$W=\vec{F}\cdot\vec{d}\cdot\cos\theta$$
+
+<img src="/resources/images/work-equation.png" width=700>(Source: Kognity)</img>
+
+When work is done **by** exactly one force, the above is formula is used. However, when calculating for work **on** an object, or **total work**, the sum of all forces ($\Sigma F$) should be used instead of $F$.
+
+The **work-energy theorem** states that the **net work on** an object is also equal to the difference in kinetic energy of an object.
+$$W=\Delta E_k = \frac{1}{2}mv^2$$
+
+### Kinetic energy
+
+Kinetic energy is the energy of an object from its motion, or when its force is in the same direction as its displacement.
+$$\Delta E_k = \frac{1}{2}mv^2$$
+
+!!! warning
+    The above formula assumes that no work is transformed into potential energy (e.g., a spring).
+
+### Gravitational potential energy
+
+Gravitational potential energy is the **stored energy** from the movement of an object away from a body with a large gravitational attraction. It is equal to, where $g$ is the positive acceleration due to gravity, and $h$ is the vertical displacement from a **reference point**:
+$$E_g = mg\Delta h$$
+
+!!! warning
+    Whether an object has gravitational potential energy largely depends on the reference height. For example, by setting the reference height to be the same as the object's, its potential energy is $0$. If the reference height is above the object's, its potential energy is negative.
+
+### Elastic potential energy
+
+Springs always want to return to equilibrium. In an ideal spring, Hooke's law states that the force needed to extend or compress a spring by distance $x$ from equilibrium scales linearly, such that:
+$$\vec{F}_s = k \Delta x$$
+
+The **spring constant** $k$ represents the "stiffness" of the spring. When returning to equilibrium, the **spring restoring force** is equal to the opposite of Hooke's law, or $\vec{F}_s = -k \Delta x$.
+
+The difference in elastic **potential** energy is the **negative** of the work done by elastic force, and is equal to:
+$$\Delta E_e = \frac{1}{2}k \Delta x^2$$
+
+### Conservation of energy
+
+The law of conservation of energy states that mechanical energy (the total kinetic and potential energies) in a **closed system** remains constant. Therefore, the net energy initially must be the same in the end.
+$$\Sigma E_i = \Sigma E_f$$
+
+The types of energy involved may have varying values as long as their sum is fixed. **Work** determines how energy is distributed among the types of energy.
+
+### Power
+
+Power is a a scalar unit represented in **watts** ($\text{1 W}=\frac{\text{1 J}}{\text{1 s}}$), and it is **not energy**, but instead the work done over a time interval, or the rate of doing work.
+$$P=\frac{\Delta E}{\Delta t}$$
+
+By rearranging units, power can also be represented in terms of force and velocity::
+$$P=Fv$$
+
+### Efficiency
+
+Efficiency measures how much work, energy, or power is retained in a process, and is represented as a percentage ($1$ being 100% efficient, $0$ being 0% efficient).
+$$\text{Efficiency = }\frac{\text{useful work/energy/power output}}{\text{total work/energy/power input}}$$
+
+The second law of thermodynamics states that some energy is always lost to a non-useful form when converted, so the efficiency of a process is **always less than** $100%$.
+
 ## 2.4 - Momentum and impulse
 
+### Momentum
+
+The momentum of an object describes the difficulty of changing its motion. Its unit is **Newton seconds** ($\text{Ns}$) and it is expressed mathematically as $\vec{P}$ such that $\vec{P}$ is equal to mass times velocity, or:
+$$\vec{P} = m\vec{v}$$
+
+Newton's second law can be manipulated so that momentum of an object — if its mass is constant — is equal to the net average force on that object multiplied by time.
+$$
+\Sigma \vec{F}_\text{avg} = m\vec{a} \\
+ = m(\frac{\vec{v_2}-\vec{v_1}}{\Delta t}) \\
+ = \frac{m \vec{v_2} - m \vec{v_1}}{\Delta t}) \\
+ = \frac{\vec{P_2} - \vec{P_1}}{\Delta t} \\
+\Sigma \vec{F}_\text{avg} = \frac{\Delta \vec{P}}{\Delta t} \\
+$$
+
+This means that to achieve the *same momentum*, a smaller force can be applied for a longer period of time or a larger force can be applied in a smaller period of time.
+
+As momentum is a **vector**, it must have a direction, and these differences in direction and therefore sign must be taken under consideration.
+
+!!! example
+    Bouncing balls change direction, so one of the velocities is in a direction opposite of the other.
+
+### Conservation of momentum
+
+Assuming there are no external forces, or if they are negligible, momentum between two objects must be conserved:
+$$\Sigma\vec{P}_i=\Sigma\vec{P}_f$$
+
+In all collisions, momentum is conserved **only** if external forces are zero or are negligible.
+
+**Perfectly elastic** collisions conserve all kinetic energy and transform it into potential energy. These do not exist in reality because of the second law of thermodynamics.
+
+!!! example
+    A ball bouncing infinitely.
+
+**Inelastic** collisions lose kinetic energy ($E_f < E_i$).
+
+!!! example
+    Most collisions, e.g., a car pushing another car.
+
+**Completely inelastic collisions** lose the maximum amount of kinetic energy while still conserving momentum.
+
+!!! example
+    Two cars hitting each other and stopping.
+
+Conservation of momentum applies even across dimensions. As it is a vector, it can be split up into components, which are independent of each other and can be acted on separately.
+
+### Impulse
+
+Impulse is the change in momentum of an object because of a force. It is expressed mathematically as $\vec{J}$, and is also represented in Newton seconds.
+$$\vec{J} = \Sigma \vec{F}_\text{avg} · \Delta t$$
+
+Note that that is also equal to $\Delta \vec{P}$ above.
+
 ## 3.1 - Thermal concepts
 
+**Heat** is the transfer of thermal energy due to a difference in temperature between two objects.
+
+The main postulates of **kinetic molecular theory** state that,
+
+ - matter is made of constantly moving particles,
+ - particles exert repulsive and attractive forces when too close or too far to other particles, respectively, and,
+ - the interparticular distances and forces account for the three states of matter
+
+<img src="/resources/images/states-of-matter.png" width=700>(Source: Kognity)</img>
+
+The **temperature** of an object is the **average** kinetic energy of particles in that object, and is linearly related to the **pressure** of its container $p$. Kelvin ($K$) is used to represent temperature, which starts at **absolute zero** — the point at which the pressure, and thus the vibration and movement of particles, is zero — and increases at equal intervals as the Celsius scale ($-273°\text{C}=0\text{ K}$).
+$$T(K)=T(°C)+273$$
+
+The average kinetic energy in joules of a gas can be found by relating the temperature of the object $T$ and the Boltzmann constant $k_B$:
+$$\frac{1}{2}m\overline{v^2}=\frac{3}{2}k_BT$$
+
+The **internal energy** of a substance is the sum of the kinetic energy of the particles (temperature) and the potential energy of the particles due to their changing attractive/repulsive forces. The potential energy of an object is highly dependent on its state. A more solid state results in lesser potential energy as the particle and closer and have weaker intermolecular forces.
+
+!!! note
+    When changing state, the temperature of a substance does not change as the energy is used to break the attractive intermolecular forces instead of increasing kinetic energy. In otder to change state, the particles must have enough energy to overcome the intermolecular forces holding them together.
+
+### Specific heat capacity
+
+The amount of energy transferred to change the temperature of an object is **independent** on whether it increases or decreases. The **specific heat capacity** $c$ of a substance is the energy required to change 1 kg of that substance by 1 K and such has the units $\text{J}\cdot\text{kg}^{-1}\cdot\text{K}^{-1}$. Where $Q$ is the quantity of thermal energy needed (no idea why it's not $\Delta Q$) , $m$ is the mass of a substance, $c$ is the specific heat capacity of the substance, and $\Delta T$ is the difference in temperature:
+$$Q=mc\Delta T$$
+
+When two objects reach thermal equilibrium together, the end temperature they reach is the same and the thermal energy one loses is equal to the one the other gains in a closed system.
+$$Q_\text{lost}+Q_\text{gained}=0$$
+
+A **temperature-energy** graph shows the energy required to heat a mass. If its mass is 1 kg, the slope of the graph is equal to $\frac{1}{c}$.
+
+### Latent heat
+
+!!! definition
+    - **Boiling** occurs when liquid becomes a gas in the whole liquid at a temperature known as the boiling point.
+    - **Evaporation** occurs only at the surface of a liquid at any temperature and can be enhanced by increasing surface area, heat, or by increasing air movement at the surface.
+    
+    Both evaporation and boiling remove the same amount of energy for a given mass.
+
+When changing phase, a substance's temperature does not increase, but instead the energy goes toward expanding intermolecular spacing. As such, it can be identified on a temperature-energy graph by areas of no change in temperature as more energy is added.
+
+The **specific latent heat** $L$ of a substance's fusion/vapourisation the amount of energy required for it to change phase, and is different between all substances and each difference in phase between substances. As such, it can only be used when $\Delta T=0$.
+$$Q=mL$$
+
 ## 3.2 - Modelling a gas
 
+### Ideal gases
+
+An ideal gas is an imaginary gas used to model real gases. The following assumptions are made:
+
+ - Gas particles are all identical and perfectly spherical
+ - Gas particles do not exert any intermolecular forces — as such, their potential energy does not change
+ - Gas particles collide perfectly elastically
+ - Gas particles are so small that their volume is significantly smaller than that of the container
+
+These assumptions do not hold for real gases as gases have real volume and there are intermolecular forces between them. As such, the ideal gas model is best used if the gas is at a **low** pressure, **higher** temperature, and **low** density.
+
+The average kinetic energy of the gas particles is proportional to the temperature of a gas. A greater speed of gas particles also increases the **collisions** between gas particles and the walls of the box, therefore increasing **pressure**, therefore increasing **temperature**. The pressure of a gas on a surface can be expressed by the force $F$ over its area $A$.
+$$p=\frac{F}{A}$$
+
+The state of an ideal gas in a container can be modelled by the following equation, where $p$ is the pressure of the pressure in pascals or newtons per square metre ($\text{Pa}$ or $\text{Nm}^{-2}$), $V$ is the volume in cubic metres, $n$ is the number of moles of gas particles, $R$ is the universal gas constant ($R=8.31\text{J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}$), and $T$ is the temperature of the gas in Kelvin:
+$$pV=nRT$$
+
+### Gas laws
+
+!!! definition
+    - An **isobaric**, **isothermal**, or **isochloric**/**isovolumetric** process have a constant pressure, temperature, or volume, respectively.
+    - A **thermodynamic process** induces a change in a gas's temperature, pressure, or volume.
+
+A **thermodynamic cycle** consists of thermodynamic processes that return a gas to its initial state, and appear in pressure-volume graphs as closed shapes. In such a cycle, the work done **on the gas** is equal to the pressure and change in volume:
+$$W=p\Delta V$$
+
+If the pressure is not constant, the work done is instead the area under the graph in that process.
+
+Please see [HL Chemistry#Gas laws](/sch3uz/#gas-laws) for more information.
+
+In a pressure-volume graph, the curves of constant temperature are known as **isotherms**.
+
+## 4.1 - Oscillations
+
+!!! definition
+    - Oscillation is defined as a cyclical motion of an object around a central point.
+    - The **amplitude** ($A$) of an oscillation is the greatest displacement from its equilibrium ($\Delta x$).
+    - An object at an **equilibrium position** is at rest.
+
+### Simple harmonic motion
+
+In **simple harmonic motion** (SHM), an object oscillates in a fixed time interval around a central **equilibrium point** with a **linear restoring force** directed toward that equilibrium point.
+$$\vec{F} \propto -\Delta \vec{x}$$
+
+Alternatively, simple harmonic motion can be motion from a linear restoring force always directed to a fixed equilibrium point.
+
+!!! example
+    A spring has a restoring force equal to $\vec{F}=-k\vec{x}$.
+
+The position/velocity/acceleration-time graphs of an object in this type of motion are all **sinusoidal**.
+
+<img src="/resources/images/shm-graphs.png" width=700>(Source: Kognity)</img>
+
+According to the law of conservation of energy, the total amount of energy in the system during simple harmonic motion remains constant, so graphing kinetic and potential energy against displacement from equilibrium returns two opposite parabolas centred on the equilibrium position.
+$$\Sigma E \propto x^2$$
+
+The only factor that affects the amplitude of SHM is the energy provided. The only factor that affects the period/frequency of SHM is the mass of the object and the restoring force.
+
+## 4.2 - Travelling waves
+
+!!! definition
+    - A **wave** is a disturbance that propagates energy but does not transfer matter.
+    - **Vibration** is the oscillation around a central equilibrium point.
+
+A travelling wave is a wave transmits energy without transmitting physical material. Waves are generally periodic in both time and space. A wave is **in phase** if the displacement from equilibrium at any point is the same after each period. A (discrete) pulse is energy sent not as part of simple harmonic motion while a continuous wave causes particles to move in simple harmonic motion.
+
+ - **Transverse** waves have their particles move **perpendicular** to the propagation of energy.
+ - **Longitudinal** waves have their particles move **parallel** to the propagation of energy.
+
+!!! example
+    - A slinky stretched horizontally and being shaken up and down sends transverse waves down its length.
+    - A slinky stretched horizontally and being pushed back and forth sends longitudinal waves down its length.
+
+!!! example
+    A longitudinal wave travelling through gas:
+    
+    <img src="/resources/images/longitudinal-wave.gif">(Source: Kognity)</img>
+
+### Graphing travelling waves
+
+A **displacement-time** graph shows the displacement *from equilibrium*  of **one** particle in simple harmonic motion. It can be used to identify the frequency of a wave as all particles oscillate at the same frequency in a wave.
+
+<img src="/resources/images/dt-wave-graph.png" width=700>(Source: Kognity)</img>
+
+A **displacement-distance** graph shows the displacement *from equilibrium* of **each** particle at a given time and can be used to find the wavelength ($\lambda$) of a wave.
+
+<img src="/resources/images/dd-wave-graph.png" width=700>(Source: Kognity)</img>
+
+The speed of a wave, or the speed at which energy propagates, in a medium is related to its frequency ($f$) and wavelength ($\lambda$) by the universal wave equation.
+$$v=f\lambda$$
+
+As all electromagnetic waves travel at the same speed — the speed of light — only either frequency or wavelength are needed to find the other property.
+$$c=f\lambda$$
+
+The only factor that affects the frequency of a wave is the source of energy, so changes in media do not change the frequency of a wave.
+
+The only factor that affects the speed of the wave is the media it is travelling in.
+
+## 4.3 - Wave characteristics
+
+A **wavefront** is a line that connects different waves at points with the same phase. The distance between wavefronts is the wavelength. Straight wavefronts indicate waves that travel in the same direction at every point on the wavefront. **Rays** are arrows that indicate the direction of motion of the **wavefront**.
+
+<img src="/resources/images/wavefront.gif">(Source: Kognity)</img>
+
+!!! example
+    Wavefronts of waves emanating from a point in the centre:
+    
+    <img src="/resources/images/radial-wave.png">(Source: Kognity)</img>
+
+The **energy** of a wave is proportional to the square of its amplitude.
+$$E \propto A^2$$
+
+If a wave reflects off a **fixed** boundary it changes phase by inverting due to Newton's third law.
+
+### Superposition and interference
+
+The **superposition principle** states that the displacement of a particle of more than one wave at any time is the **sum** of the displacements caused by each individual wave. After superposition, waves continue on as if the other was not present. The result of combining waves via superposition is known as **interference**.
+
+ - If the waves have displacements in the same direction, the interference is **constructive**.
+ - If the waves have displacements in opposite directions, the interference is **destructive**.
+
+<img src="/resources/images/superposition.png" width=700>(Source: Kognity)</img>
+
+When waves of the same frequency meet, if their crests or troughs overlap, a **supercrest** or **supertrough** is formed, respectively. Waves that result in a net displacement of particles of zero form a **node**.
+
+### Polarisation
+
+**Unpolarised** waves are those that oscillate in every direction **perpendicular** to the direction of energy propagation, while those that are polarised only do so in one plane. Only **transverse** waves can be polarised.
+
+<img src="/resources/images/polarise-head-on.png" width=700>(Source: Kognity)</img>
+
+Waves are polarised by a **polariser**: a material that only allows charged particles to oscillate in one plane. When unpolarised light passes through a polariser, only one plane can progagate fully. The other directions of oscillation have their amplitudes reduced. **Linearly polarised** electromagnetic waves have a single plane of polarisation.
+
+A **polarising filter** can be used to polarise light, and an **analyser** is a second polariser used to determine if light is polarised.
+
+<img src="/resources/images/polariser.png" width=700>(Source: Kognity)</img>
+
+The energy of a wave is proportional to its intensity which is proportional to the square of the amplitude.
+$$E\propto I\propto A^2$$
+
+**Malus's law** states that for a polarised wave of energy $E_0$, the amplitude from the second filter, where $\theta$ is the angle between the polariser and the analyser, such that:
+$$E=E_0\cos\theta$$
+
+And so, where $I$ is the intensity of the polarised/analysed light, $I_0$ is the intensity of the plane-polarised light, and $\theta$ is the angle between the polariser and the analyser:
+$$I=I_0\cos^2\theta$$
+
+When **unpolarised light** passes through a polariser, the average result of $I\cos^2\theta$ is $\frac{1}{2}$, so the intensity of polarised light is **half** of the intensity of unpolarised light.
+
+When unpolarised light reflects off of a **smooth non-metallic** surface it will be at least partially polarised.
+
+<img src="/resources/images/reflect-polarise.png" width=700>(Source: Kognity)</img>
+
+## 4.4 - Wave behaviour
+
+### Reflection
+
+When waves meet a flat surface and are not diffused or refracted, they are reflected at the same angle they entered relative to the **normal**.
+
+!!! definition
+    - The **incident** wave/angle refers to the wave hitting the surface.
+    - The **reflected** wave/angle refers to the wave moving away from the surface.
+    - The **normal** is an imaginary line perpendicular to the surface at the arrival point of the incident ray.
+
+<img src="/resources/images/reflection.png" width=700>(Source: Kognity)</img>
+
+### Refraction
+
+When light travels between two media that have different **optical densities**, it refracts and changes its direction as well as its speed. The **frequency** of the wave does not change but its speed does, therefore the **wavelength** changes as well.
+
+!!! definition
+    **Refraction** is the change in the direction and speed of light from a change in the optical density of the medium it travels through.
+
+When a ray travels from a **less** optically dense to **more** optically dense medium (fast to slow), the ray bends **toward** the normal and its wavelength decreases.
+
+<img src="/resources/images/refraction.gif" width=700>(Source: Kognity)</img>
+
+### Snell's law and total internal reflection
+
+The **refractive index** of a medium represents the ratio of wave speeds in the media, where $v$ is the speed of light in the medium:
+$$n=\frac{c}{v}$$
+
+Air has an index of refraction of $1$.
+
+Snell's law states that the incident and refracted angles are related by the index of refraction of the media.
+$$\frac{n_1}{n_2}=\frac{\sin\theta_2}{\sin\theta_1}=\frac{v_2}{v_1}$$
+
+or
+$$n_1\sin\theta_1=n_2\sin\theta_2$$
+
+If a ray travels from a **slower** to a **faster** medium, the ray bends away from the normal. If the ray would refract so much that it does not leave the incident medium at all, **total internal reflection** (TIR) occurs and the ray follows the rules of **reflection**. The critical angle denotes the incident angle required for the refracted angle to be exactly $90°$, and any angle greater than it results in TIR.
+$$\sin\theta_c = \frac{n_2}{n_1}$$
+
+<img src="/resources/images/total-internal-reflection.png" width=700>(Source: Kognity)</img>
+
+### Diffraction
+
+!!! definition
+    An **aperture** is a small opening in a barrier.
+
+Diffraction is the bending of waves around an opening or obstacle.
+
+<img src="/resources/images/diffraction.png" width=700>(Source: Kognity)</img>
+
+Huygen's wave theory says that every point on a wavefront is a source of circular waves/wavelets that move outward from that point.
+
+<img src="/resources/images/huygen-wavelets.png" width=700>(Source: Kognity)</img>
+
+As a wavefront passes an aperture, there are fewer points beside it and the wavefront becomes rounder.
+
+For observable diffraction, the wavelength must be greater than or equal to the width of the aperture.
+
+### Two-point interference
+
+When two points create waves, their wavefronts spread out in all directions. If waves are emanated from two points at the same speed and frequency, they form **nodal** and **anti-nodal** lines.
+
+!!! definition
+    - A **nodal** line is a set of points where waves interfere **destructively** and remain at equilibrium while other parts of the medium vibrate.
+    - An **anti-nodal** line is a set of points where waves interfere **constructively** and have the maximum vibration in the medium.
+
+<img src="/resources/images/point-interference.png" width=700>(Source: Kognity)</img>
+
+For all points on **anti-nodal** lines, the path difference between the point sources is $n\lambda$, where $n$ is an integer indicating the number of anti-nodal lines away from the central anti-nodal line ($n=0$).
+
+For all points on **nodal** lines, the path difference between the point sources is $(n+\frac{1}{2})\lambda$, where $n$ is an integer indicating the number of nodal lines away from the nodal lines ($n=0$) adjacent the central anti-nodal line (the diagram below mistakenly has one- instead of zero-indexed nodal line numbers).
+
+<img src="/resources/images/point-interference-busy.png" width=700>(Source: Kognity)</img>
+
+### Double slit interference of electromagnetic waves
+
+A light wave that is **coherent** is **monochromatic** (of the same wavelength) with a constant phase difference. Because light from different sources is incredibly difficult to ensure constant phase differences, a double-slit test can be performed by using a point source at $S_0$ to ensure that coherent waves pass at $S_1$ and $S_2$. The darker and brighter **fringes** are due to anti-nodal and nodal lines from wave interference.
+
+<img src="/resources/images/double-slit.png" width=700>(Source: Kognity)</img>
+
+By using mafs involving rays along a nodal line, the separation, $s$ between **nodal/anti-nodal** lines on the distant surface can be related to the distance from the slits to the screen, $D$, the separation between the slits, $d$, and the wavelength $\lambda$.
+$$s=\frac{\lambda D}{d}$$
+
+The number of nodal lines depends on the ratio $\frac{d}{\lambda}$.
+
+**Dark fringe** positions on the screen are equally spaced at the same separation $s$ between the bright fringes. The perpendicular bisector with a length of $D$ is a **bright fringe**.
+
+<img src="/resources/images/fringe-slit-derive.png" width=700>(Source: Kognity)</img>
+
+## 4.5 - Standing waves
+
+A standing wave does not travel, so no energy is transferred. They are caused by the interference of symmetrical travelling waves moving in opposite directions.
+
+!!! definition
+    **Resonance** is when an external driver oscillating at a system's natural frequency transfers energy into the system, in this case a standing wave.
+
+<img src="/resources/images/standing-wave.gif" width=700></img>
+
+The locations of zero amplitude are called **nodes** and the locations of maximum amplitude are **anti-nodes**. Adjacent nodes are separated by $\frac{\lambda}{2}$. The wave speed is determined by the medium and the wavelengths of standing waves that fit on the medium are determined by the length of the medium.
+
+<img src="/resources/images/harmonic-waves.png" width=700>(Source: Kognity)</img>
+
+The frequencies that would produce standing waves are known as **natural frequencies**, and are individually known as **harmonics** The image above shows the first four harmonics. The first harmonic is also known as the **fundamental frequency**.
+
+Inside an **air column**, the difference in pressure allows for the formation of standing waves as waves reflect off of the interface where air changes pressure. A closed end must have nodes while open ends must have antinodes.
+
+Therefore:
+
+ - In a stretched string or a **closed-ended** air column, both ends must be **nodes**.
+ - In an open-ended air column, both ends must be **anti-nodes**.
+ - In a half-open air column, the closed end must be a node while the open end must be an anti-node.
+
+!!! example
+    The wavelength of any half-open air column is $\lambda=\frac{L}{4}$, and $\lambda=\frac{L}{2}$ in a stretched string.
+
+<img src="/resources/images/standing-tube.png" width=700>(Source: Kognity)</img>
+
+Resonance occurs in an air column when the length of the open tube is a multiple of half the wavelength of the sound. **Resonant lengths** occur at each antinode.
+
+In a **open-ended** or **closed-ended**  air column:
+$$\lambda_n=\frac{n}{4}L$$
+
+## 5.1 - Electric fields
+
+### Charge
+
+A **charge** is a scalar that allows a mass to experience force from an electromagnetic field, and it may only be positive or negative. It is both **quantised** — the smallest charge possible is the one on a single proton/electron and all charges are multiples of that value — and **conserved** in a closed system.
+
+Charges are expressed in **Coulombs** ($\text{C}$) with the symbol $q$. One electron has a charge of $-1.60×10^{-19}\text{ C}$.
+
+Similar to gravitation, the forces between charges is proportional to their charges and inversely proportional to the square of the distance between them.
+$$F_E \propto q_1q_2 \propto \frac{1}{r^2}$$
+
+The equation that describes the force between charges is as follows, where $k$ is a constant with the value $8.99×10^9\text{ N m}^{2}\text{C}^{-2}$:
+$$F_E = \frac{kq_1q_2}{r^2}$$
+
+The **electrical permittivity**, $\epsilon$, of a material is its resistance in creating an electric field in it, and has the units $\text{farad}\cdot\text{m}^{-1}$. Free space/a vaccum has a permittivity of $1$ and its value is known as $\epsilon_0$.
+$$k=\frac{1}{4\pi\epsilon}$$
+
+As $\epsilon$ changes depending on the medium, the constant $k$ changes as well.
+
+### Actual electric fields
+
+Similar to a gravitational field, the force an object experiences is dependent on both charges but the **field** around an object is only reliant on one. The **electric field strength** $\vec{E}$ is the force per unit of charge (with units $N\cdot C^{-1}$) on a tiny positive test charge and is equal to:
+$$\vec{E}=\frac{\vec{F_e}}{q}$$
+
+The direction of the field depends on its charge — by convention, all test charges are assumed to be **positive**, so vector arrows will point **away** from a positive source and **toward** a negative source.
+
+### Electric potential
+
+**Electric potential energy** ($E_E$) is the energy a charge in an electric field has in joules. It depends on the polarity of the charge, the magnitude of the charge, and the strength of the electric field.
+$$\Delta E_E = W = F_E\Delta x$$
+
+**Electric potential** ($V$), also known as voltage, is the potential energy a charge in an electric field has **per unit charge** in volts ($\text{V, 1 V = 1}\frac{\text{J}}{\text{C}}$). It depends only on the electric field strength.
+$$
+V=\frac{E_E}{q} \\
+\Delta V = \frac{W}{q} = \frac{\Delta E_E}{q}
+$$
+
+The **electron-volt** ($\text{eV}$) is energy required to move an electron through a potential difference of one volt ($1\text{ eV}=1.60×10^{-19}\text{ J}$). One electron has a mass of $9.110×10^{-31}\text{ kg}$.
+
+When $\Delta V=0$ the points are equipotential to each other, and no work is done moving between them. The images below outline equipotential lines — note that they are always **perpendicular** to electric field lines.
+
+<img src="/resources/images/equipotential-charge.png" width=700>(Source: Kognity)</img>
+
+<img src="/resources/images/equipotential-two-charges.png" width=700>(Source: Kognity)</img>
+
+### Uniform electric fields
+
+Electric field lines like the ones shown below show the direction that an electric field acts in.
+
+<img src="/resources/images/electric-field.png" width=500>(Source: Kognity)</img>
+
+If two charges establishing a field are parallel "plates", they form a **uniform electric field** between them, where the magnitude and direction of the electric field strength is identical at all points in the field, similar to a gravitational field.
+
+<img src="/resources/images/uniform-electric-field.png" width=700>(Source: Kognity)</img>
+
+Electric field lines never intersect and their density represents the strength of the electric field.
+
+In a uniform electric field, equations similar to the ones used in gravitational fields can be applied.
+$$
+\Delta E_E = q\vec{E}\Delta x \\
+\Delta E_E = q\Delta V \\
+\vec{E} = \frac{\Delta V}{\Delta x} \\
+$$
+
+where any deltas refer to differences between the two plates.
+
+As the electric field strength is constant, the equipotential lines in the field are also evenly distributed.
+
+## 5.2 - Heating effect of electric currents
+
+!!! definition
+    A **charge carrier** is any mass that enables charges to be moved over a potential difference.
+
+Electric currents ($I$) are the flow of electrons due to a potential difference between both ends of a conductor. They are measured in **amperes** ($\text{A}$) which represent coulombs per second ($1\text{ A}=\frac{1\text{ C}}{1\text{ s}}$).
+$$I=\frac{q}{t}$$
+
+The current of each element in series is equal.
+$$I_1=I_2=I_3$$
+
+The current of each element in parallel adds up to the total current.
+$$I_1+I_2+I_3$$
+
+The **drift velocity** is the average velocity of electrons in a given volume.
+
+<img src="/resources/images/drift-velocity.png" width=700>(Source: Kognity)</img>
+
+As such, where $I$ is the current, $n$ is the number of electrons, $A$ is the cross-sectional area of the conductor, $q$ is the charge of one electron, and $v$ is the electron drift velocity:
+$$I=nAvq$$
+$$\Delta Q = nAv\Delta tq$$
+
+In an electric circuit, electrons flow from the negative terminal to the positive terminal — however historically current was described as from the positive to the negative terminal and so that is known as **conventional current**, opposite the direction of electron flow. In a cell, the **longer side** is the positive terminal while the shorter is the negative terminal, unlike a typical AA battery.
+
+The **electric power** transformed across a circuit is eqial to the potential difference across it and the current through it.
+$$P=IV$$
+
+### Resistance
+
+Resistance ($R$) is inversely proportional to current with the unit of ohms ($\Omega$).
+$$R=\frac{V}{I}$$
+
+This can be rearranged to form Ohm's law:
+$$V=IR$$
+
+The **effective** resistance of a series of resistors in series is the sum of their resistances.
+$$R_\text{eff}=\Sigma R$$
+
+The **effective** resistance of a series of resistors in parallel is the inverse of the sum of the inverse of their resistances.
+$$\frac{1}{R_\text{eff}}=\Sigma\frac{1}{R}$$
+
+A **resistor** is a conductor that limits the current that can flow through it.
+
+The **resistivity** ($\rho$) of a substance is its resistance per unit length with a unit cross-section with units $\pu{\Omega\cdot m}$. At a onstant temperature:
+$$\rho=\frac{RA}{L}$$
+
+The conductivity ($\sigma$) of a substance is the inverse of its resistivity.
+$$\sigma=\frac{1}{\rho}$$
+
+### Circuits
+
+**Electromotive force** (emf, $\epsilon$) is not a force but is the voltage gained when charges pass through a cell, representing the total electric potential energy provided to the whole circuit.
+
+All cells have **internal resistance**, which is the resistance of th cell itself. It is depicted as the resistance being in series with the electromotive force. Where $P$ is the power dissipated in the cell, $I$ is the current the cell produces, and the internal resistance of the cell $r$:
+$$P=I^2r$$
+
+The voltage across the terminals of a battery is equal to, where $I_r$ is the current across the internal resistance:
+$$V_{AB}=\epsilon-I_r$$
+
+The total voltage (emf) of a circuit can be found with Ohm's law:
+$$\epsilon=I(R+r)$$
+
+An **ammeter** measures the current of a load and must be in series with the load. Ideally it should have the lowest resistance possible.
+
+A **voltmeter** measures the voltage of a load and must be in parallel with the load. Ideally it should have the greatest resistance possible. If the voltmeter and load have similar resistances to an order of magnitude or two, its resistance should be considered by treating it as a load in parallel to the load.
+
+Kirchoff's **junction rule** states that the current entering a junction must be equal to the current exiting a junction.
+$$I_\text{in}=I_\text{out}$$
+
+Kirchoff's **loop rule** states that sum of the changes in potential around a closed circuit must equal zero.
+
+A **potential divider** splits the voltage across a resistor anywhere from 0 V to emf by changing its resistance by either using 2+ resistors or by being a variable resistor.
+
+**Light-dependent resistors** (LDRs) decrease their resistance as they are exposed to greater light intensities.
+
+**Thermistors** decrease their resistance as they are exposed to greater temperatures.
+
+### Ohmic resistors
+
+An ohmic resistor (also known as an **ideal resistor** is one that follows Ohm's law at all voltages, and so results in a straight line on a V-I graph.
+
+Non-ohmic resistors exist due to the heating effect from the electric current passing through the resistor changing its resistance.
+
+<img src="/resources/images/filament-lamp-vi.png" width=700>(Source: Kognity)</img>
+
+Non-ohmic resistors may be ohmic at a certain range of voltages.
+
+## 6.1 - Circular motion
+
+Circular motion is any motion along a circular path. The instantaneous velocity of an object in circular motion at any point is always tangent to its point on the circle, so the difference between any two velocities ($\Delta \vec{v}$) is radial (lies along the radius).
+
+!!! definition
+    - An object is in **uniform circular motion** when all instantaneous velocities are equal in magnitude or when all accelerations are equal in magnitude.
+    - **Period** ($T$) represents the time required to complete one revolution ($360°$ or $2\pi$ radians).
+    - **Frequency** ($f$) represents the revolutions per second represented in hertz ($\text{1 Hz=1s^{-1}}$), and is the inverse of period.
+
+!!! warning
+    As velocity and acceleration are vectors, direction must be considered when manipulating them. This means that the **kinematic equations** cannot be used.
+
+### Angular thingies
+
+**Angular displacement** ($\Delta \theta$) is the change in angle an object has rotated relative to a reference position. By convention, angular displacement anti-clockwise is positive.
+
+**Angular speed/velocity** ($\omega$) is the rate of angular displacement, and is expressed in radians per second. It is also known as angular frequency.
+$$\omega=\frac{\Delta \theta}{\Delta t}=\frac{2\pi}{T}$$
+
+An object in uniform circular motion has an angular speed of $2\pi f$. Since distance travelled in one period is the circumference of a circle, the **speed** of an object is:
+$$v=\omega r$$
+$$v=\frac{2\pi r}{T}$$
+
+### Centripetal force/acceleration
+
+!!! warning
+    **Centrifugal** force is a **fictitious** (psuedo) force used to explain movement opposite the predicted acceleration in a circular frame of reference. The phenomenon is clearly explained from an external field of reference by Newton's first law of motion.
+
+Centripetal acceleration ($\vec{a}_c$) is the acceleration causing an object to move in a circle. An object with uniform circular motion has a centripetal acceleration with constant **magnitude**.
+$$|\vec{a}_c|=\frac{v^2}{r}$$
+
+Therefore, the acceleration of an object in uniform circular motion is:
+$$|\vec{a}_c|=\frac{4\pi^2r}{T^2}$$
+
+The centripetal **force** on an object is the net force on that object causing it to move in a circle. In some cases, only some **components** of forces are included as part of centripetal force.
+$$\Sigma \vec{F}_c=m\vec{a}_c$$
+
+!!! example
+   When a car turns in a circle, static friction is one factor in centripetal force. When riding a roller coaster, weight and normal force are included as centripetal force. When spinning around on the *WindSeeker* at an amusement park, only the horizontal component of tension ($\vec{F}_{Tx}$) is responsible for centripetal force.
+
+## 6.2 - Newton's law of gravitation
+
+All masses exert gravitational force on each other. The universal law of gravitation states that the gravitational force, $F_G$ between any two objects is attractive along the line joining them and equal to:
+$$\vec{F}_G=G\frac{mM}{r^2}$$
+
+where $G=6.67×10^{-11} \text{ N}\cdot\text{m}^2\cdot\text{kg}^{-2}$ is Newton's constant of universal gravitation, the **orbital radius** $r$ is the distance between the objects' **centre of mass**, and $m$ and $M$ are the masses of the objects acted on and acting, respectively. This indicates that the force on a given mass is proportional to its distance: $F_G \propto \frac{1}{r^2}$.
+
+The net gravitational force acting on an object is equal to the resultant vector formed by components of each force acting on it.
+
+### Gravitational fields
+
+A **force field** is a model representing a region of space where a mass or charge experiences a **non-contact** force. Its units are newtons per kilogram (for mass) or newtons per coulomb (for charge). A gravitational field surrounds every mass and exerts a force equal to that given by Newton's law of gravitation, extending infinitely into space.
+
+The **gravitational field strength** at any point in the field represents the force per unit of mass experienced by any mass—it is constant for any mass at the same distance $r$.
+$$|\vec{g}|=G\frac{M}{r^2}$$
+
+Its units also make it equivalent to the acceleration experienced by that mass. On Earth, $\vec{g}=9.81 \text{ N/kg [down]}$.
+$$\vec{g}=\frac{\vec{F}_G}{m}$$
+
+!!! note
+    Only the distance between the centre of mass of the objects and the mass of the **body acting** on another affect gravitational field strength of the acting body.
+
+<img src="/resources/images/grav-radial-lines.png" width=700>(Source: Kognity)</img>
+
+Gravitational field lines equidistantly point radially to the centre of a mass to indicate strength—a greater density of field lines in a given area indicates greater strength.
+
+### Orbital motion
+
+In space, only gravity acts on **satellites**—objects that orbit around another object, effectively as if in constant free fall. As gravity is the only force, it is also the only force contributing to centripetal force.
+$$\vec{F}_G=\Sigma\vec{F}_c$$
+
+So the orbital speed of a satellite must be independent of its own mass, such that:
+$$v=\sqrt{\frac{GM}{r}}$$
+
+Additiionally, Kepler's third law, derived from centripetal force and the universal law of gravitation,  states that the square of the period of an orbit is proportioanl to the cube of the radius.
+$$T^2 \propto r^3$$
+
+A **geosynchronous** orbit is one that returns to its original position relative to the orbitee after one rotation of the orbitee.
+
+A **geostationary** orbit is a geosynchronous orbit that always remains in the same position relative to the orbitee..
+
+
 ## Resources
 
  - [IB Physics Data Booklet](/resources/g11/ib-physics-data-booklet.pdf)

docs/sph4u7.md

@@ -0,0 +1,574 @@
+# SL Physics - 2
+
+The course code for this page is **SPH4U7**.
+
+## Magnetism
+
+Magnets are objects with north/south dipoles that create a field around them. Although ferromagnetic substances can repel each other, paramagnetic substances are always attracted to a magnetic field. See [HL Chemistry#Physics properties of transition elements](/sch3uz/#physical-properties-of-transition-elements) for more details.
+
+### Magnetic fields
+
+<img src="/resources/images/magnetic-field.png" width=700>(Source: Kognity)</img>
+
+Similar to electric and gravitational fields, magnetic fields (also known as **B-fields**) are drawn by their effect on a north pole. Since magnetic poles always appear in equal magnitude pairs, all magnetic field lines for a magnet must form closed loops from north to south **outside** and south to north **inside** the magnet. Much like electric field lines, magnetic field lines never touch
+
+<img src="/resources/images/more-magnetic-fields.png" width=700>(Source: Kognity)</img>
+
+Atoms in ferromagnetic materials are tiny magnets with **dipoles**. These dipoles act on neighbouring dipoles and can cause the whole object to align — this is known as an **electric domain**.
+
+!!! note
+    Nickel, cobalt, or any alloy with nickel, cobalt, or iron can become magnetised this way.
+
+**Unmagnetised** domains have dipoles pointing in random directions that are aligned when exposed to a magnetic field where they become **magnetised** domains. As such, bar magnets are always broken into smaller magnets, each with two poles — a monopole is impossible to create.
+
+### Straight-line electromagnets
+
+Moving electric charges produce magnetic fields. A circle filled with an "x" indicates that the current is moving away from the viewer in the third dimension while a dotted circle indicates it is moving toward the viewer.
+
+These magnetic fields are centred on the conductor, are in a plane perpendicular to the conductor, and have decreasing magnetic field strength over distance.
+
+<img src="/resources/images/straight-line-electromagnet.png" width=700>(Source: Kognity)</img>
+
+The **right-hand rule** for straight-line conductors indicates that when the conductor would be grasped by the right hand, the thumb would point in the direction of current and the fingers pointing in the direction of the magnetic field.
+
+<img src="/resources/images/right-hand-rule.png" width=700>(Source: Kognity)</img>
+
+### Solenoid electromagnets
+
+A **solenoid** is a conductor coil in a tight helix. Current passed through a solenoid will generate a **uniform magnetic field** inside the coil with a pattern identical to that of a bar magnet outside it.
+
+<img src="/resources/images/selenoid-electromagnet.png" width=700>(Source: Kognity)</img>
+
+The right-hand rule can be applied again to a solenoid to identify the direction of the north pole or direction of magnetic field in the coil:
+
+<img src="/resources/images/selenoid-right-hand-rule.png" width=700>(Source: Kognity)</img>
+
+### Properties of moving charges
+
+As only moving electric charges generate magnetic fields, stationary electric charges are **unaffected** by external magnetic fields. Moving charges are affected by Newton's third law of motion — equal and opposite forces are exerted on the charge and the magnet. As such, where $q$ is the charge of the particle and $\vec{v}\times \vec{B}$ is the **cross product** (vector multiplication) of the velocity of the particle and the magnetic field strength in Teslas:
+$$\vec{F_m}=q\vec{v}\times \vec{B}$$
+
+**Magnetic field strength** ($B$) represents the force acting per unit current in a conductor of unit length perpendicular to the field with the unit Tesla ($\pu{T}$)
+
+The **magnetic force** is always plane **perpendicular** to both $\vec{v}$ and $\vec{B}$. Just the magnitude of the force can be found by using the angle between the two vectors ($\theta$):
+$$|F_m|=qvB\sin\theta$$
+
+Regardless of $\theta$, the force vector is always perpendicular to both $B$ and $v$,
+
+The above equation can be rearranged to find **electromagnetic** force in terms of current and wire length in a **uniform magnetic field**:
+$$|F_{em}|=BIL\sin\theta$$
+
+<img src="/resources/images/magnet-on-wire.png" width=700>(Source: Kognity</img>
+
+The **right-hand-rule** can be used to determine the direction of force — the thumb points in the direction of current/velocity, the fingers point in the direction of the magnetic field, and the palm points in the direction of force. Alternatively, just three fingers can be used.
+
+<img src="/resources/images/right-hand-rule-forces.png" width=700>(Source: Kognity)</img>
+
+When two straight-line conductors with current flowing through them are brought together, they either mutually attract or repel. The ampere is defined based on the current required to flow through a scenario involving two parallel straight-line conductors.
+
+<img src="/resources/images/parallel-current-conductors.png" width=700>(Source: Kognity)</img>
+
+Inside a **uniform magnetic field**, charges move in **uniform circular motion** at a constant velocity. If the particle did not enter the field at a perfect right angle, some of the velocity is used to change the path of the particle to be in a spiral — still perfectly circular, but additionally moving in the third dimension perpendicular to the circle.
+$$\Sigma F_c = F_m$$
+
+## Nukes
+
+### Atomic structure
+
+!!! definition
+    - A **photon** is a particle that represents light.
+    - A **nucleon** is a subatomic particle in an atomic nucleus — that is, a proton or neutron.
+    - A **nuclide** is a nucleus with a specific number of protons and neutrons.
+
+Please see [HL Chemistry#2.1 - Atoms](/sch3uz/#21-atoms) and [HL Chemistry 1#2.2 - Electrons in atoms](/sch3uz/#22-electrons-in-atoms) for more information.
+
+An electron in an atom will only become excited if a photon of exactly the right amount of energy strikes it. That energy can be found using the formula:
+$$\Delta E=hf$$
+
+where $E$ is the energy of the photon at frequency $f$, and $h$ is Planck's constant ($\pu{6.63\times 10^{-34} Js}$ or $\pu{4.14\times10^{-15} eVs}$).
+
+An electron that de-excites will emit a photon of that exact energy and thus frequency to return to its previous state.
+
+### Binding energy
+
+According to Einstein's theory of special relativity:
+$$\Delta E=\Delta mc^2$$
+
+**Neutrons** in the nucleus hold the protons together via **strong nuclear forces** that somewhat act like glue. An increase in neutrons increases the strong nuclear force. In smaller nuclei, $N=Z$, but in larger nuclei, $N>Z$ as more neutrons are required to keep the nucleus stable as the number of protons increases.
+
+The mass of a stable nucleus is always less than the sum of the masses of the individual nucleons (the **mass defect**) as some of the mass is converted to energy during the formation of a nucleus. The energy used is known as the **binding energy** of a nucleus.
+$$E_\pu{binding} = \pu{mass defect}\times c^2$$
+
+As such, the binding energy of a nucleus is also the energy required to separate it completely into individual nucleons.
+
+Atomic mass is measured relative to the mass of a carbon-12 atom, which is exactly 12 u (unified atomic mass units).
+$$\pu{1 u}=\pu{1.661\times10^{-27} kg}=\pu{931.5 MeVc^-2}$$
+
+A higher **binding energy per nucleon** results in more energy required to break it apart and thus it being more stable.
+
+<img src="/resources/images/binding-energy-curve.png" width=700>(Source: Kognity)</img>
+
+!!! note
+    It is required to know the general shape of the curve, that $~8.8 \pu{ MeV}$ is the maximum, and that the end boundaries are $0$ and $~7.5 \pu{ MeV}$. It is also required to know the elements at each of those points (hydrogen-1, iron-56/nickel-62, and uranium-238).
+
+Since a greater binding energy per nucleon is more energetically favourable, nuclei to the right of iron-56 fission (split) while those to the left fuse (combine) to release energy — changes that would increase binding energy per nucleon are likely to occur because of this.
+
+### Radioactive decay
+
+!!! definition
+    - An **alpha particle** is a helium-4 nucleus (2 protons, 2 neutrons).
+    - A **beta particle** is an electron.
+    - A **gamma ray** is a photon.
+
+Radioactivity is the emission of **ionising** (will make ions) radiation due to changes of a nucleus. It is **random** and spontaneous — it is unaffected by external factors such as other nuclei decaying.
+
+**Nuclear equations** are similar to chemical equations but show how nuclei change in a nuclear process by tracking the atomic and mass numbers. A nuclear equation is balanced so that the sum of the atomic and the sum of the mass numbers on both sides are equal.
+
+$$A\to B+C$$
+
+**Alpha decay** occurs when the strong nuclear force is unable to hold a large nucleus together and emits an alpha particle. The alpha particle is positive and can barely penetrate paper. The two particles move in opposite directions with momentums equal in magnitude.
+
+<img src="/resources/images/alpha-decay.png" width=700>(Source: Kognity)</img>
+
+$$\ce{^A_Z N → ^{A-4}_{Z-2} N' + ^4_2 He}$$
+
+$\ce{^4_2 He}$ may be replaced by $\ce{^4_2\alpha}$.
+
+!!! example
+    Radium-226 alpha decays to radon-222.
+
+**Beta-minus decay** ($\beta^-$) occurs when a neutron decays into a proton and releases a beta-minus particle (an electron and an electron antineutrino). It can penetrate up to 3 mm of aluminum. Where $\overline{v}_e$ is the antineutrino:
+
+<img src="/resources/images/beta-minus-decay.png" width=700>(Source: Kognity)</img>
+
+$$\ce{^1_0n → ^1_1p + ^0_{-1}e + ^0_0\overline{v}_e}$$
+
+!!! note
+    The bar over the electron antineutrino identifies it as an **antiparticle**.
+
+The beta-minus particle can be written explicitly over the electron as $^0_{-1}\beta$.
+
+In terms of the mother nucleus, the reaction results in the mass number staying the same while the atomic number increases by one.
+
+**Beta-plus decay** ($\beta^+$) occurs when a proton decays into a neutron and releases a **positron** (an antielectron with a positive charge) and an electron neutrino ($v_e$)
+$$\ce{^1_1p → ^1_0n + ^0_1e + ^0_0v_e}$$
+
+The positron can be written as a beta-plus particle as $^0_1\beta$.
+
+In terms of the mother nucleus, the reaction results in the mass number staying the same while the atomic number decreases by one. 
+
+**Gamma decay** occurs when an excited nucleus transfers its energy to a high-energy photon with frequencies in the gamma region of the electromagnetic spectrum. There is no change in mass nor atomic number. A nuclide with an asterisk $*$ indicates it as excited. This emits *ionising radiation* which is not good for living beings.
+
+<img src="/resources/images/gamma-decay.png" width=700>(Source: Kognity)</img>
+
+$$\ce{^A_ZX^* → ^A_ZX + ^0_0\gamma}$$
+
+### Detecting radiation
+
+As radiation cannot be seen, it must be detected experimentally.
+
+A **Geiger counter** utilises a gas-filled tube with a wire in the centre at high voltage. When a charged particle passes through, gas is ionised which cascade onto the wire to produce a pulse.
+
+A **cloud chamber** contains vapour that turns into liquid droplets when ionising particles pass though, resulting in visible lines showing the path of the particles. A magnetic field can spiral the particle in a certain direction which allows for its charge to be identified.
+
+### Half-life
+
+The **half-life** of an element is the time required for half of the nuclides in a sample to decay — it is always the same no matter the number of initial nuclides.
+
+As such, this means that the number of parent nuclei decreases by 50% of its current value each half-life.
+
+<img src="/resources/images/half-life.png" width=700>(Source: Kognity)</img>
+
+!!! example
+    **Radioactive dating** analyses the ratio between carbon-14 and carbon-12 to determine the age of plant nmatter. As the ratio of C-14 and C-12 in the atmosphere has been relatively comstant, living plants maintain that balance by constantly exchanging carbon. Dead plants' carbon-14 slowly decay at a known rate, so the ratio can be used to determine the time since the plant died.
+
+### Nuclear reactions
+
+A nuclear reaction occurs when a nucleus is hit by another nucleus or subatomic particle and a different nuclide is formed (nuclear **transmutation**). In such a collision, energy and momentum must be conserved. Generally, neutrons are most commonly used in these reactions as they are not affected by Coulomb force exerted by the protons or electrons.
+
+The **reaction energy** $Q$ is the difference in mass between the initial and final states multiplied by $c$ squared. In the sample reaction $a+X → Y+b$:
+$$Q=[(M_a + M_X) - (M_Y + M_b)]c^2$$
+
+ - If $Q$ is positive, the reaction is **exothermic** and will occur at any amount of initial kinetic energy.
+ - If $Q$ is negative, an initial kinetic energy equal to $Q$ is required (the activation energy).
+
+Nuclear **fusion** occurs when two lighter nuclei combine into a heavier one, releasing energy in the process.
+
+Nuclear **fission** occurs when a heavy nucleus splits into two lighter nuclei. Along with some excess neutrons, energy is released. The two split pieces are usually somewhat unequal.
+
+### Fission in reactors
+
+The energy release of nuclei is very large — the energy density per unit mass is much higher than any other conventional source.
+
+As nuclei get smaller, their stablility increases as the number of neutrons also decreases, so excess neutrons can set off a chain reaction by reacting with more nuclei.
+
+<img src="/resources/images/fission-chain.gif" width=700>(Source: Kognity)</img>
+
+However, neutrons that are ejected often have too much energy and must be **moderated** to slow down to prevent a critical mass where the number of reactions is self-sustaining, leading to overheating and reactor meltdown. A **moderator** is a material surrounding fuel rods to slow down incoming neutrons — usually heavy water.
+
+**Control rods** are also inserted into the reaction core to control the rate of reaction. These absorb the neutrons from the moderator and the amount absorbed can be adjusted by raising the rods partially up to all the way from the reactor.
+
+Nuclear power is superior to other types of energy generation in that:
+
+ - it has a high power output due to high energy density
+ - there are large reserves of nuclear fuel on Earth
+ - there are no greenhouse gases emitted to generate power
+
+Nuclear power has the following issues in that:
+
+ - waste is highly radioactive with long half-lives, rendering removal and storage of nuclear waste a major issue
+ - initial startup costs are expensive
+ - strict maintenance is required due to the risk of nuclear meltdown
+ - fissionable fuel can be recovered and used for destructive weapons
+ - mining uranium is unhealthy — miners are exposed to harmful radiation and waste material from mines deemed not pure enough is not easy to dispose
+
+### Nuclear fusion
+
+Nuclear fusion generates energy per unit mass an order of magnitude greater than can be achieved with fission. The sun takes hydrogen and fuses it into helium. Heavier stars can fuse elements up to iron-56.
+$$\ce{4 ^1_1 H → ^4_2 He + 2 e^+ + 2 v_e + 2\gamma}$$
+
+Nuclear fusion power has the following issues in that it is currently unsustainable for more than a few seconds because:
+
+ - the temperature required for the reaction is greater than 100 million degrees Celsius
+ - it requires more energy input to heat the sample than is obtained from the fusion reaction
+ - materials currently known cannot withstand the temperature making containment difficult — currently magnetic fields are used to hold the particles in place
+
+## The Standard Model
+
+<img src="/resources/images/standard-model.png" width=700>(Source: Kognity)</img>
+
+### Elementary particles
+
+An elementary particle is a subatomic particle that is not composed of other particles.
+
+Particles currently thought to be elementary as of January 2021 include bosons, quarks, and leptons:
+
+### Bosons/Force exchange particles
+
+!!! definition
+    - **Virtual particles** are bosons that do not have infinite range.
+
+Bosons are particles that carry/exchange forces between particles. The theory of exchange forces posits that all forces are due to particles exchanged between elementary particles. There are four types of bosons that can be roughly categorised by their effect.
+
+**Gluons** ($\pu{g}$) are exchanged for matter to feel **strong nuclear** force: the strongest interaction between particles. These particles are heavy (120 MeV/c<sup>2</sup>) and short-lived, thus giving the force a very short range.
+
+**Photons** ($\pu{\gamma}$) are exchanged for matter to feel **electromagnetic** force: the second strongest force responsible for magnetism and electric force that only act on charged particles. These particles have a rest mass of zero and travel for an infinite distance until they are absorbed.
+
+The **W<sup>+</sup>, W<sup>-</sup>, and Z<sup>0</sup>** bosons are together responsible for the weak nuclear force and are the third strongest force. These particles have a heavy rest mass (80 GeV/c<sup>2</sup> for Ws, 91 GeV/c<sup>2</sup> for Z) and so are even more limited in range than gluons.
+
+**Gravitons** are responsible for gravitational force: the weakest force. These particles are massless and so they have infinite range.
+
+The Higgs field and Higgs boson are responsible for elementary particles obtaining their mass because of magical fields and rainbows.
+
+### Quarks
+
+!!! definition
+    - A **hadron** is any particle made of quarks.
+    - A **baryon** is any hadron made of three quarks. An **antibaryon** is any particle made of three antiquarks.
+    - A **meson** is a hadron made of exactly one quark and one antiquark involved in the **strong** interaction.
+    - A **fermion** is any particle with mass (hadrons or leptons)
+
+Gluons (strong force) only interact with quarks, which are heavier, more tightly bound elementary particles. There are six quarks with different properties:
+
+| Charge | | | |
+| --- | --- | --- | --- |
+| $\frac{2}{3}$e | up (u) | charm (c) | top (t) |
+| $-\frac{1}{3}$e | down (d) | strange (s) | bottom (b) |
+
+!!! reminder
+    e is the elementary charge ($\pu{1.6\times10^{-19} C}$).
+
+ - All quarks have a **baryon number** of $\frac{1}{3}$.
+ - All quarks have a **strangeness number** of 0 except for the strange quark, whose number is equal to -1.
+ - All quarks have their own respective **antiquark**: an antiparticle with opposite charge and baryon number but otherwise identical mass.
+ - The **quark confinement theory** states that a singular quark cannot be isolated from its hadron.
+
+Every particle has its own **antiparticle** with the same properties but with opposite quantum numbers. In practice, this indicates that mass stays the same while baryon number, lepton number, and charge are opposite. When a corresponding quark and antiquark meet, annihilate each other to become energy. They are denoted by a bar over their letter.
+
+!!! example
+    An up antiquark (also known as "u-bar") is written as ū.
+
+!!! note
+    - Protons are composed of two up quarks and one down quark (uud).
+    - Neutrons are composed of one up quark and two down quarks (udd).
+
+### Leptons
+
+Leptons are lighter and more loosely bound elementary particles compared to quarks. They do not participate in the strong interaction. All leptons have a **lepton generation/family** which is based on their relative mass. A higher mass indicates a higher generation.
+
+| Charge | Generation 1 (L<sub>I</sub>) | Generation 2 (L<sub>II</sub>) | Generation 3 (L<sub>III</sub>) |
+| --- | --- | --- | --- |
+| -1e | electron (e) | muon (µ) | tau ($\tau$) |
+| 0 | electron neutrino ($\pu{v_e}$) | muon neutrino ($\pu{v_\mu}$) | tau neutrino ($\pu{v_\tau}$) |
+
+ - All leptons have a **lepton number** of 1.
+
+### Elementary particle interactions
+
+In any interaction, the following are true:
+
+ - **charge** is conserved
+ - the **baryon number** is always conserved
+ - the **lepton number** of each family is always conserved
+ - the **strangeness number** is always conserved in *strong and electromagnetic interactions*
+
+!!! example
+    A lepton number of $\pu{L_{III} = 1}$ on one side becoming $\pu{L_{II} = 1}$ on the other is impossible as lepton family must be kept consistent during interactions.
+
+## Feynman diagrams
+
+A Feynman diagram provides a graphic representation of particle interactions to predict the outcome of a particle collision.
+
+Generally, the time axis is left-to-right but can be specified to be otherwise. The following assumes time moves from left to right.
+
+Fermions are represented by **straight lines with arrows**. Particles have their arrows pointing *forward* in time while antiparticles point backward (even though they still move in the direction of time).
+
+<img src="/resources/images/fermion-feynman.png" width=700>(Source: Kognity)</img>
+
+Bosons/force exchange particles are represented by wiggly lines with no arrow.
+
+<img src="/resources/images/boson-feynman.png" width=700>(Source: Kognity)</img>
+
+Particles only interact at a **vertex** where left refers to the state before the interaction while the right refers to the state afterward. A vertex must have one arrow pointing **toward** and one **away** from the vertex. Conservation laws apply at each vertex.
+
+<img src="/resources/images/vertex-feynman.png" width=700>(Source: Kognity)</img>
+
+Contents of hadrons must be shown. (See the last example for an example.)
+
+### Feynman diagram examples
+
+!!! example
+    An electron being repelled by another electron due to Coulomb repulsion:
+    <img src="/resources/images/electron-repulsion.png" width=700>(Source: Kognity)</img>
+
+!!! example
+    Beta decay:
+    <img src="/resources/images/beta-decay-feynman.png" width=700>(Source: Kognity)</img>
+
+!!! example
+    Some weak interaction that violates strangeness:
+    <img src="/resources/images/weird-feynman.png" width=700>(Source: Kognity)</img>
+
+## Energy sources
+
+!!! definition
+    - A **primary** energy source is one that is not transformed and used directly by the consumer, such as burning wood for heat.
+    - A **secondary** energy source is one that is converted from a primary source, such as electricity.
+    - **Proved reserves** are the resources that are known to be obtainable.
+    - **Production** are the actual reserves and placed in the market in a certain time interval.
+    - The **expectancy** of a product is the estimated time its proved reserves will last given its production (proved reserves ÷ production).
+    - The **specific energy** ($E_{SP}$) of a source is the energy obtained per unit mass of fuel (J/kg).
+    - The **energy density** ($E_D$) of a source  is the energy obtained per unit volume (J/m<sup>3</sup>).
+
+If a fuel source can be replenished with a "reasonable" amount of time — one human generation, or 50–100 years — it is considered to be **renewable**. The world still largely uses non-renewable energy sources.
+
+<img src="/resources/images/renewable-energy-pie.png" width=700>(Source: Kognity)</img>
+
+Electricity is the most common secondary energy source due to its convenience and ease of transport.
+
+### Sankey diagrams
+
+Sankey diagrams show the transfer of energy in a system via directed lines proportional to quantity of energy. Arrows pointing away indicate energy **degradation** — losses in energy transformation.
+
+<img src="/resources/images/sankey-diagram.png" width=700>(Source: Kognity)</img>
+
+To minimise electrical losses during transportation, high voltage, low resistance wires with high cross-sectional areas are used to reduce resistance to reduce power loss.
+
+### Power generation
+
+A moving magnetic field produces an electromotive force as alternating current via **induction** and is how the large majority of power generation is handled.
+
+<img src="/resources/images/power-plant-generation.png" width=700>(Source: Kognity)</img>
+
+In a nutshell, a source of thermal energy such as burning fossil fuels is used to boil water whose steam is then used to turn a turbine to generate power before condensing and repeating the cycle.
+
+Coal and oil-powered power plants have efficiencies of around 40% while natural gas is slightly higher at 50% as the gas itself can be used to turn a turbine.
+
+In **nuclear** power plants, the coolant fluid is instead used to turn a turbine.
+
+<img src="/resources/images/nuclear-power-generation.png" width=700>(Source: Kognity)</img>
+
+As only uranium-235 is fissile, but uranium-238 is significantly more common (99.3% U-238 to 0.7% U-235), uranium must first be enriched until the concentration of U-235 is ~3%. **Gaseous diffusion** is a form of enrichment by forming uranium hexafluoride gas and then spinning it in a centrifuge to force separation of U-238 and U-235 based on mass. The energy used in enriching uranium is substantial and should be included in Sankey diagrams.
+
+Moderators are used to encourage fission as they slow down neutrons that are going too fast to fission (most) to a speed more suitable for fission.
+
+### Wind
+
+The kinetic energy of wind can be harnessed to generate power. As convection currents provide the greatest airflow near large bodies of water, wind farms are often constructed there. The wind turns rotor blades which turn a turbine to generate power. It is a source of clean and renewable energy.
+
+Assuming **all wind kinetic energy** is converted to mechanical energy, where $P$ is the power generated, $A$ is the area of the circle that the blades spin around, $\rho$ is the density of the air, and $v$ is the speed of the wind in the direction of the blades:
+$$P=\frac{1}{2}A\rho v^3$$
+
+<img src="/resources/images/wind-generator.png" width=700>(Source: Kognity)</img>
+
+| Advantages | Disadvantages |
+| --- | --- |
+| Renewable | Wind strength is inconsistent |
+| Wind is widely available | Turbine blades may kill birds |
+| Does not emit greenhouse gases | Many of them are needed to replace one fossil fuel plant, requiring lots of space so they don't interfere with each other |
+
+### Hydro
+
+In hydroelectric plants, a dam is often used to increase the height of a reservoir so that it falls and spins a turbine to generate power. As such, the energy generated is roughly equal to the gravitational potential energy of the water. Where $\Delta h$ is the **average height** of the water from the turbine:
+$$E=mg\Delta h$$
+
+<img src="/resources/images/hydro-generator.png" width=700>(Source: Kognity)</img>
+
+During times of lower demand, dams often have a **pumped storage** system that pumps water back into the reservoir for use during higher demand.
+
+### Tidal
+
+A **tidal barrage** generates energy via the kinetic energy of water moving during changes in tide using a multi-directional turbine.
+
+### Photovoltaic
+
+!!! definition
+    **Intensity** is the power delivered per unit area (watts per square metre).
+
+Photovoltaic (PV) cells are made of silicon doped with phosphorus and boron impurities to convert sunlight directly into electricity. Light from the sun frees electrons in the silicon to produce a current.
+
+<img src="/resources/images/photovoltaic-generation.png" width=700>(Source: Kognity)</img>
+
+The **solar constant** $S$ of $\pu{1.36\times10^3 W/m^2}$ determines the intensity of the sun's light that reaches the Earth. At different latitudes and between seasons, the intensity changes because the Earth is round and is tilted, respectively.
+
+### Solar heating
+
+Instead of converting between multiple forms of energy, solar heating directly converts the sun's energy to heat, increasing efficiency drastically. By using insulation, a black substance, and a glass top, the heat from the sun is trapped and absorbed into water where it is used to heat things.
+
+<img src="/resources/images/solar-heating.png" width=700>(Source: Kognity)</img>
+
+## Thermal energy transfer
+
+!!! definition
+    - **Radiation** is the transfer of energy via electromagnetic waves emitted away from an object. No medium is needed.
+    - **Convection** is the transfer of thermal energy via another medium away from an object.
+    - **Conduction** is the transfer of thermal energy via physical contact.
+
+### Black bodies
+
+!!! definition
+    **Emissivity** ($e$) is a dimensionless value from 0 to 1 indicating the ability of an object to emit radiation relative to a black body (which has an emissivity of 1). Darker and duller surfaces have an emissivity closer to 1 while shinier and whiter ones are closer to 0.
+
+All bodies with an absolute temperature will emit radiant energy in the form of electromagnetic waves. The temperature of the body determines the wavelengths and power of the radiation emitted. A **perfect emitter** has an emissivity of 1 and is known as a **black body**, absorbing all electromagnetic radiation.
+
+Generally, as the temperature of a body goes down, its peak power density is reduced and its peak wavelength increases.
+
+**Wien's displacement law** relates the temperature of a black body to the waves it emits. Where $\lambda_\text{max}$ is the peak wavelength in metres and $T$ is the temperature of the body in kelvin:
+$$\lambda_\text{max}\times T=\pu{2.9\times10^{-3} m\cdot K}$$
+
+The **Stefan-Boltzmann** law relates the specifications of a body to the power it emits. Where $P$ is the power emitted by the body, $A$ is its surface area, $T$ is its temperature, $e$ is its emissivity, and $\sigma$ is the Stefan-Boltzmann constant (equal to $\pu{5.67\times10^{-8} Wm^{-2}K^{-4}}$):
+$$P=e\sigma AT^4$$
+
+In problems where the environment temperature is **different** from the temperature of an object, there will be power loss. The net power emitted by a body will be:
+$$P_\text{net}=e\sigma A(T_1^4-T_2^4)$$
+
+The solar radiation reaching earth is equal to $\pu{S= 3.9\times10^{26} W}$ with the assumption that it is a black body.
+
+### Albedo
+
+Derived from $I=\frac{P}{A}$, the intensity at a point in space can be related to the power of the radiation emitted by the source ($P$) and the distance between the two ($d$):
+$$I=\frac{P}{4\pi d^2}$$
+
+!!! example
+    The solar constant is derived in this way by substituting $d$ as the distance from the Earth to the sun.
+
+As Earth and most other planetary bodies are not flat disks pointed at the sun, in reality the sun's intensity is reduced to a quarter due to the formula for a sphere. Therefore, the power absorbed/incident to the Earth is equal to, where $S$ is the solar constant:
+$$P_\text{in}=(1-\alpha)\frac{S}{4}A$$
+
+**Albedo** ($\alpha$) is the ratio of power from incident rays reflected or scattered to the power absorbed by a body, ranging from 0 to 1. A black body has albedo 0. On average, Earth's albedo is equal to $0.3$ due primarily to the atmosphere but also clouds and ice.
+$$\alpha=\frac{\text{energy scattered/reflected}}{\text{energy absorbed}}$$
+
+Greenhouse gases are responsible for remaining increases in temperature. By absorbing and then re-emittng their natural frequencies of electromagnetic radiation (infrared for greenhouse gases), they delay the release of radiation back into space and heat up the atmosphere.
+
+### Radiation absorption by greenhouse gases
+
+All molecules have a natural frequency at which they absorb radiation from the electromagnetic spectrum and **resonate** at. The natural frequency of greenhouse gases is in the infrared region, which is what the Earth re-emits solar radiation as. Therefore, greenhouse gases absorb this radiation and them re-emit it in all directions, "trapping" some of the radiation. Resonance is also the phenomenon responsible for the protection from ultraviolet radiation by the ozone layer.
+
+## Photoelectric effect
+
+**Wave-particle duality** posits that everything can be described as either a particle or a wave, and that all particles will show some wave characteristics and all waves will show some particle characteristics, so really what the hell is happening?
+
+The photoelectric effect is the phenomenon in which electrons are emitted when electromagnetic radiation hits a material. It was theorised that EM radiation travelled in **discrete** energy packets known as **quanta**, which held a defined amount of energy that could not be divided smaller. Where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength of the light:
+$$E=\frac{hc}{\lambda}$$
+
+!!! example
+    The application of the photoelectric effect in the real world has led to large advances in technology, such as in **photodiodes** in light sensors — semiconductors where electrons are freed by indicent light to raise the conductivity of the material, and **charged coupled devices** in cameras — the "film" of the modern digital camera.
+
+It was later observed that for a given material, electrons were ejected when light shone on a surface only if the light's energy/frequency was greater than a certain threshold. Below that threshold, electrons were not ejected **regardless** of the intensity of the light.
+
+Electrons have a "binding energy" that hold them to the nucleus of an atom. To release an electron from the nucleus, energy greater than that binding energy must be provided. This binding energy, known as the **work function** $W_0$, is therefore the minimum energy required for a surface to eject electrons.
+
+As such, where $E_k$ is the kinetic energy of the ejected electron, $E_{ph}$ is the energy of the incident photon/radiation, and $W_0$ is the work function of the surface:
+$$E_k=E_{ph}-W_0$$
+
+Electrons are ejected and thus the photoelectric effect observed **only** if $E_k>0$. The equation above shows that the kinetic energy of an ejected electron is determined **only** by the wavelength/frequency of the incident radiation and *not* by the intensity of the light.
+
+The intensity of incoming radiation effectively represents the number of photons striking per unit area of a surface, so while it does not affect whether electrons are ejected, it affects the **number** of electrons that are ejected *only if they are determined to be ejected*.
+
+In a light frequency-kinetic energy graph,
+
+ - the x-intercept represents the **threshold frequency** $f_0$: the minimum frequency required to liberate electrons at all.
+ - the y-intercept represents the **work function** $-W_0$: the "binding energy" of the electrons.
+
+### Momentum of photons
+
+It was also observed that during photon-electron collisions that momentum and kinetic energy were conserved, further reinforcing the idea of wave-particle duality of light. Where $h$ is Planck's constant, $f_i$ and $f_f$ are the initial and final frequencies of the light, $m$ is the mass of the electron, and $v_{ei}$ and $v_{ef} are the initial and final velocities of the electron:
+$$hf_i+\frac{1}{2}mv^2_{ei}=hf_f + \frac{1}{2}mv^2_{ef}$$
+
+In general, the momentum of a photon is equal to, where $h$ is Planck's constant and $\lambda$ is the wavelength of the light:
+$$p=\frac{h}{\lambda}$$
+
+### Matter waves
+
+Particles/waves cannot act like particles and waves at the same time. For a given observation, it adopts the property of one or the other. In reality, all particles exhibit wave properties *sometimes* and all waves exhibit particle properties *sometimes*. Each particle has a wave function that determines how likely it is to be somewhere at any point in time.
+
+By equating the equations for momentum of photons and particles, the (de Broglie) wavelength of a particle can be determined. Where $\lambda$ is the wavelength of the particle, $m$ is its mass, $v$ is its velocity, and $h$ is Planck's constant:
+$$\lambda=\frac{h}{mv}$$
+
+Therefore, wavelengths of "particles" are only really significant for small masses at high speeds rather than large masses at lower speeds.
+
+!!! example
+    A 50 kg mass moving at 16 m/s is has a wavelength many orders of magnitude smaller than a quark and will not display any observable wave behaviours.
+
+The discovery of wave-particle duality has led to advancements in technology such as the scanning electron microscope.
+
+## Special relativity
+
+Einstein's theory of special relativity states that time and space are relative depending on the **frame of reference** of the observer, and light travels at the *same speed* of $\pu{3.0\times10^8 m/s}$ in a vacuum no matter how it is observed in all inertial frame of reference.
+
+ - An **inertial** reference frame is one in which the law of inertia in the frame holds true. Only frames of reference moving at a constant velocity or at rest are inertial, and the same laws of physics apply in all inertial frames of reference.
+
+### Time dilation and length contraction
+
+The faster an observer moves, to ensure that it appears to them that light travels at $c$, time slows down for the observer. Observers in inertial frames of reference will experience time at a slower rate — this phenomenon is known as time dilation.
+
+For two **inertial** reference frames, where $t_s$ is the time observed between two events (stationary/**proper** time) at the same location and at rest relative to a stationary observer, $t_m$ is the time observed between two events in a different frame of reference (moving time), $v$ is the speed difference between the frames of reference, and $\gamma$ is the **Lorentz factor**:
+$$
+\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \\
+t_m=\gamma t_s
+$$
+
+Space is also relative. An observer moving at a higher constant velocity will have space contract — this phenomenon is known as **length contraction**.
+
+Where $L_s$ is the length/distance measured of an object at the same location and at rest relative to a stationary observer, and $L_m$ is the length/distance from a different reference frame:
+$$L_m=\frac{L_s}{\gamma}$$
+
+From different inertial reference points, there can never be agreement on whether events are simultaneous. Because time slows down for an observer moving faster, one event will also appear to happen first.
+
+### Matter-energy equivalence
+
+The total energy of an object in an inertial frame of reference relative to another is related to its velocity and its mass.
+$$
+E_t=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} \\
+E_k=E_t-E_\text{rest}
+$$
+
+At rest, the energy of an object is related to its mass.
+$$E_\text{rest}=mc^2$$
+
+## Resources
+
+ - [IB Physics Data Booklet](/resources/g11/ib-physics-data-booklet.pdf)
+ - [IB SL Physics Syllabus](/resources/g11/ib-physics-syllabus.pdf)
+ - [Dealing with Uncertainties](/resources/g11/physics-uncertainties.pdf)
+ - [External: IB Physics Notes](https://ibphysics.org)

docs/tok.md

@@ -0,0 +1,43 @@
+# Theory of Knowledge
+
+## Knowledge questions and claims
+
+There are two types of knowledge **claims**:
+
+ - First-order claims: claims that are made in areas of knowledge or by individual knowers about the world
+ - Second-order claims: claims that are made about **knowledge** justified using TOK, usually involving an examination of the nature of knowledge
+
+!!! example
+    - The sky is blue. (first-order)
+    - Logic is innate. (second-order)
+
+Knowledge **questions** are questions that examine or engage with knowledge claims, such as by including any of the following phrases:
+
+ - How can we know that…
+ - …knowledge…
+ - How far is it justified…
+
+!!! example
+    Is our knowledge in mathematics more certain than that of science?
+
+## Knowledge themes and areas
+
+The main areas of knowledge are:
+
+ - History
+ - Human sciences
+ - Natural sciences
+ - Mathematics
+ - Arts
+
+The main themes of knowledge are:
+
+ - Knowledge and technology
+ - Knowledge and language
+ - Knowledge and politics
+ - Knowledge and religion
+ - Knowledge and indigenous studies
+
+## Resources
+
+ - [External: TOK 2022](https://tok2022.weebly.com)

mkdocs.yml

@@ -1,31 +0,0 @@
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-  - Home: index.md
-  - SL Physics A: sph3u7.md
-  - HL Chemistry A: sch3uz.md
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-  - HL English A: eng3uz.md
-  - IB Resources: resources.md
-
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package.json

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+{
+  "name": "eifueo",
+  "version": "0.1.0",
+  "description": "A VitePress version of the Eifueo project",
+  "main": "index.js",
+  "scripts": {
+    "docs:dev": "vitepress dev docs",
+    "docs:build": "vitepress build docs",
+    "docs:serve": "vitepress serve docs",
+    "test": "echo \"Error: no test specified\" && exit 1"
+  },
+  "repository": {
+    "type": "git",
+    "url": "git+https://git.eggworld.tk/eggy/eifueo.git"
+  },
+  "author": "eggy",
+  "license": "GPLv3",
+  "bugs": {
+    "url": "https://git.eggworld.tk/eggy/eifueo/issues"
+  },
+  "homepage": "https://git.eggworld.tk/eggy/eifueo",
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