Physical Chemistry

Physical chemistry is the fundamental branch of chemistry that applies the principles, theories, and techniques of physics to the study of chemical systems. It seeks to explain the macroscopic and microscopic phenomena of chemistry by exploring the underlying physical laws governing matter and energy, utilizing core concepts such as thermodynamics, quantum mechanics, and kinetics. By doing so, physical chemistry provides the theoretical foundation for understanding why chemical reactions occur, the rates at which they proceed, and the structural properties of molecules, thereby bridging the gap between the abstract world of physics and the practical applications of chemical science.

1. Foundational Concepts for Physical Chemistry
   1. Essential Mathematical Tools
      1. Differential Calculus
         1. Functions and Limits
            1. Continuity
            2. Limit Theorems
            3. L'Hôpital's Rule
         2. Derivatives and Differentiation Rules
            1. Power Rule
            2. Product Rule
            3. Quotient Rule
            4. Chain Rule for Single Variables
         3. Partial Derivatives
            1. Notation and Interpretation
            2. Mixed Partial Derivatives
            3. Applications in Thermodynamics
            4. Gradient and Directional Derivatives
         4. Exact and Inexact Differentials
            1. Criteria for Exactness
            2. Integrating Factors
            3. Physical Significance in Thermodynamics
         5. Multivariable Chain Rule
            1. Applications to Composite Functions
         6. Taylor Series Expansions
            1. First-Order Expansions
            2. Second-Order Expansions
            3. Applications in Approximations
            4. Error Analysis
      2. Integral Calculus
         1. Indefinite Integrals
            1. Basic Integration Rules
            2. Integration by Parts
         2. Definite Integrals
            1. Fundamental Theorem of Calculus
            2. Properties of Definite Integrals
         3. Techniques of Integration
            1. Substitution Method
            2. Trigonometric Substitution
            3. Partial Fractions
         4. Line Integrals
            1. Path Dependence and Independence
            2. Conservative Vector Fields
         5. Multiple Integrals
            1. Double Integrals
            2. Triple Integrals
            3. Change of Variables
            4. Jacobian Determinant
      3. Differential Equations
         1. Classification of Differential Equations
            1. Order and Degree
            2. Linear vs Nonlinear
            3. Ordinary vs Partial
         2. First-Order Ordinary Differential Equations
            1. Separable Equations
            2. Linear First-Order Equations
            3. Integrating Factor Method
            4. Exact Equations
         3. Second-Order Ordinary Differential Equations
            1. Homogeneous Linear Equations
            2. Nonhomogeneous Linear Equations
            3. Characteristic Equation Method
            4. Method of Undetermined Coefficients
         4. Boundary and Initial Conditions
            1. Initial Value Problems
            2. Boundary Value Problems
         5. Separation of Variables for Partial Differential Equations
            1. Wave Equation
            2. Heat Equation
            3. Schrödinger Equation
      4. Linear Algebra
         1. Vectors and Vector Spaces
            1. Vector Addition and Scalar Multiplication
            2. Dot Product and Cross Product
            3. Linear Independence
            4. Basis and Dimension
            5. Orthogonal and Orthonormal Bases
         2. Matrices and Determinants
            1. Matrix Addition and Multiplication
            2. Transpose and Symmetric Matrices
            3. Determinant Properties
            4. Matrix Inverse
            5. Rank of a Matrix
         3. Eigenvalue Problems
            1. Eigenvalues and Eigenvectors
            2. Characteristic Polynomial
            3. Diagonalization
            4. Applications in Quantum Mechanics
            5. Hermitian Matrices
      5. Probability and Statistics
         1. Basic Probability Concepts
            1. Sample Spaces and Events
            2. Probability Rules
            3. Conditional Probability
            4. Bayes' Theorem
         2. Probability Distributions
            1. Discrete Distributions
               1. Binomial Distribution
               2. Poisson Distribution
            2. Continuous Distributions
               1. Normal Distribution
               2. Exponential Distribution
         3. Statistical Measures
            1. Mean
            2. Median
            3. Mode
            4. Variance
            5. Standard Deviation
         4. Stirling's Approximation
            1. Derivation
            2. Applications in Statistical Mechanics
   2. Review of Classical Physics
      1. Classical Mechanics
         1. Newton's Laws of Motion
            1. First Law (Inertia)
            2. Second Law (F = ma)
            3. Third Law (Action-Reaction)
            4. Applications to Chemical Systems
         2. Energy Concepts
            1. Kinetic Energy
            2. Potential Energy
            3. Conservation of Energy
            4. Work-Energy Theorem
         3. Simple Harmonic Motion
            1. The Harmonic Oscillator
            2. Classical Solution
            3. Energy in Harmonic Motion
         4. Rotational Motion
            1. The Rigid Rotor
            2. Moment of Inertia
            3. Angular Momentum
         5. Electrostatic Interactions
            1. Coulomb's Law
            2. Electric Potential Energy
      2. Electromagnetism
         1. Electric and Magnetic Fields
            1. Electric Field Definition
            2. Magnetic Field Definition
            3. Lorentz Force Law
         2. Maxwell's Equations
            1. Gauss's Law for Electricity
            2. Gauss's Law for Magnetism
            3. Faraday's Law of Induction
            4. Ampère-Maxwell Law
         3. Electromagnetic Waves
            1. Wave Equation Derivation
            2. Properties of Electromagnetic Waves
            3. Light as an Electromagnetic Wave
            4. Energy and Momentum of Light