Physical Chemistry

1. Statistical Mechanics
   1. Boltzmann Distribution
      1. Derivation and Formulation
         1. Boltzmann's entropy formula
         2. Statistical interpretation of entropy
      2. Maxwell-Boltzmann Statistics
         1. Applicability to classical systems
         2. Relations to kinetic theory of gases
      3. Deviations and Corrections
         1. Quantum corrections
         2. Extensions to include quantum statistics
   2. Partition Functions
      1. Definition and Mathematical Representation
      2. Canonical Ensemble: NVT Ensemble
         1. Role in determining thermodynamic properties
         2. Connection with Helmholtz free energy
      3. Microcanonical Ensemble: NVE Ensemble
         1. Energy distribution
         2. Derivation of thermodynamics in isolated systems
      4. Grand Canonical Ensemble: μVT Ensemble
         1. Exchange of particles and its significance
         2. Connection with chemical potential
      5. Applications in Calculating Thermodynamic Properties
         1. Derivation of internal energy, entropy, and heat capacity
         2. Evaluating reaction equilibria
      6. Quantum Partition Functions
         1. Quantum harmonic oscillator
         2. Quantum rigid rotor and statistical sums
   3. Ensemble Theory
      1. Fundamentals of Ensemble Theory
         1. Concept of an ensemble and probabilities
         2. Differences between various ensembles
      2. Comparison of Ensembles
         1. Equivalence in the thermodynamic limit
         2. Usage in different physical situations
      3. Time Averages vs Ensemble Averages
         1. Ergodic hypothesis
         2. Implications for statistical mechanics
   4. Fluctuations and Correlation
      1. Fluctuation-Dissipation Theorem
         1. Relationship between response functions and correlation functions
      2. Thermodynamic Fluctuations
         1. Analysis of large deviation theory
         2. Impacts on heat capacity and compressibility
      3. Correlation Functions
         1. Pair correlation function and its significance
         2. Structure and dynamic correlations in systems
   5. Applications to Thermodynamic Quantities
      1. Calculation of Macroscopic Properties
         1. Pressure, volume, and temperature (PVT) relations
         2. Exergonic and endergonic reactions
      2. Phase Transitions and Critical Phenomena
         1. Order parameters and spontaneous symmetry breaking
         2. Role in delineating critical exponents and universality classes
      3. Non-ideal Systems
         1. Corrections for interacting particles
         2. The virial expansion
   6. Quantum Statistical Mechanics
      1. Fermionic and Bosonic Systems
         1. Fermi-Dirac statistics
         2. Bose-Einstein statistics
      2. Quantum Gases
         1. Ideal gas behavior and degeneracies
         2. Bose-Einstein condensation
      3. Applications in Quantum Systems
         1. Quantum phase transitions
         2. Superfluidity and superconductivity phenomena
   7. Irreversible Processes
      1. Non-equilibrium Statistical Mechanics
         1. Introduction to Boltzmann's H-theorem
         2. Understanding of relaxation times and transport properties
      2. Approach to Equilibrium
         1. Entropy production
         2. Role of the Liouville theorem in time evolution