States of Matter and Phase Changes

States of Matter

• Solid matter has definite shape and volume
• Liquid matter has definite volume but takes the shape of its container
• Gaseous matter lacks definite shape or volume
• Plasma is ionized gas at a high temperature

Phase Changes

• Melting is the transition from solid to liquid
• Freezing is the transition from liquid to solid
• Vaporization is the transition from liquid to gas
• Condensation is the transition from gas to liquid
• Sublimation is the transition from solid to gas
• Deposition is the transition from gas to solid

Chemical vs. Physical Properties

• Chemical properties describe a substance's ability to change its composition, such as flammability and reactivity
• Physical properties describe the substance itself, such as color, density, and melting point

Intensive vs. Extensive Properties

• Intensive properties are independent of the amount of substance, like temperature, pressure, and density
• Extensive properties depend on the amount of substance, like mass and volume

Atomic Structure

• Protons have a positive charge and are located in the nucleus
• Neutrons have no charge and are located in the nucleus
• Electrons have a negative charge and orbit the nucleus

Atomic Number and Mass Number

• The atomic number (Z) is the number of protons in an atom
• The mass number (A) is the sum of protons and neutrons in an atom

Isotopes

• Isotopes are atoms with the same number of protons but different numbers of neutrons

Ions

• A cation is a positive ion that has lost electrons
• An anion is a negative ion that has gained electrons

Molecules and Compounds

• A molecule consists of two or more atoms held together by chemical bonds
• A compound is a molecule containing two or more different elements

Balancing Equations

• Balancing chemical equations involves ensuring that the number of atoms of each element is the same on both sides of the equation

Types of Reactions

• Combination (Synthesis): A + B → AB
• Decomposition: AB → A + B
• Single Replacement: A + BC → AC + B
• Double Replacement: AB + CD → AD + CB
• Combustion: A reaction with oxygen that produces heat and light

Stoichiometry

• Mole Concept: 1 mole contains $6.022 \times 10^{23}$ entities (Avogadro's number)
• Molar Mass: Mass of one mole of a substance
• Mass Percent Composition: (Mass of element / Mass of compound) x 100%
• Empirical Formula: Simplest whole-number ratio of atoms in a compound
• Limiting Reactant: Reactant that is completely consumed first
• Percent Yield: (Actual Yield / Theoretical Yield) x 100%

Gas Laws

• Boyle's Law: $P_1V_1 = P_2V_2$ (constant T, n)
• Charles's Law: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$ (constant P, n)
• Avogadro's Law: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$ (constant P, T)
• Ideal Gas Law: PV = nRT

Dalton's Law of Partial Pressures

• $P_{total} = P_1 + P_2 + P_3 +...$

Graham's Law of Effusion

• $\frac{Rate_1}{Rate_2} = \sqrt{\frac{M_2}{M_1}}$

Solutions

• Molarity (M): Moles of solute per liter of solution
• Molality (m): Moles of solute per kilogram of solvent
• Solubility: Maximum amount of solute that can dissolve in a solvent at a given temperature

Colligative Properties

• Colligative properties depend on the concentration of solute particles
• Freezing Point Depression: $\Delta T_f = K_f \cdot m$
• Boiling Point Elevation: $\Delta T_b = K_b \cdot m$

Acid-Base Definitions

• Arrhenius: Acid produces $H^+$ in water; base produces $OH^-$ in water
• Bronsted-Lowry: Acid is a proton donor; base is a proton acceptor
• Lewis: Acid is an electron pair acceptor; base is an electron pair donor

pH Scale

• $pH = -log[H^+]$
• $pOH = -log[OH^-]$
• $pH + pOH = 14$

Strong vs. Weak Acids and Bases

• Strong acids and bases completely dissociate in water
• Weak acids and bases partially dissociate in water

Titration

• Titration determines the concentration of a solution by reacting it with a solution of known concentration

Equilibrium Constant (K)

• Equilibrium constant (K) is the ratio of products to reactants at equilibrium

Le Chatelier's Principle

• If conditions change for a system in equilibrium, the system shifts to relieve the stress

Laws of Thermodynamics

• 1st Law: Energy is conserved
• 2nd Law: Entropy of an isolated system increases

Enthalpy (H)

• Enthalpy is the heat content of a system

Entropy (S)

• Entropy measures disorder

Gibbs Free Energy (G)

• $G = H - TS$
• $\Delta G < 0$: Spontaneous
• $\Delta G > 0$: Non-spontaneous
• $\Delta G = 0$: Equilibrium

Oxidation and Reduction

• Oxidation involves the loss of electrons
• Reduction involves the gain of electrons

Electrochemical Cells

• Electrochemical cells convert chemical energy into electrical energy (galvanic cells) or vice versa (electrolytic cells)

Radioactive Decay

• Alpha Decay: Emission of an alpha particle ($^4_2He$)
• Beta Decay: Emission of a beta particle ($^0_{-1}e$)
• Gamma Decay: Emission of a gamma ray ($\gamma$)

Half-Life

• Half-life is the time taken for half of the radioactive nuclei to decay

Algorithmic Decoding

• Algorithm 1 illustrates autoregressive decoding for generative models

PageRank Algorithm

• PageRank assigns a numerical weight to linked documents to measure relative importance
• A link from one page to another counts as a vote of importance

PageRank Formula

• $PR(A) = (1-d) + d \left( \frac{PR(T_1)}{C(T_1)} + \dots + \frac{PR(T_n)}{C(T_n)} \right)$
• $PR(A)$ is the PageRank of page A
• $d$ is the damping factor, typically 0.85
• $T_1, \dots, T_n$ are pages linking to page A
• $PR(T_i)$ is the PageRank of page $T_i$
• $C(T_i)$ is the number of outgoing links from page $T_i$

PageRank Implementation

• Initialize PageRank values (e.g., 1/N where N is the number of pages)
• Iterate and update each PageRank based on the formula
• Repeat until the difference in PageRank values between iterations is below a threshold