Chapter 13: Statistical Thermodynamics

The Boltzmann distribution forms the cornerstone, describing how molecular populations distribute across available energy levels at thermal equilibrium based on the principle of equal a priori probabilities. From this distribution emerges the molecular partition function, a mathematical construct that encodes complete thermodynamic information for systems of independent molecules. The chapter systematically develops partition functions for distinct molecular motions including translational movement through space, rotational motion around molecular axes (employing high-temperature approximations for both linear and nonlinear geometries), and vibrational oscillations modeled as harmonic oscillators. These partition functions serve as powerful tools for calculating crucial thermodynamic quantities starting with mean molecular energy, which connects directly to measurable heat capacity through summing contributions from each independent mode of motion. The statistical interpretation of entropy through the Boltzmann formula reveals the molecular basis for macroscopic disorder, introducing the important concept of residual entropy that persists at absolute zero. The partition function further enables calculation of Helmholtz and Gibbs free energies, providing a quantitative pathway from molecular structure and spectroscopic data to determine equilibrium constants. This framework demonstrates how statistical mechanics transforms information about energy levels and molecular geometry into predictions of chemical equilibrium, reaction spontaneity, and other observable thermodynamic behaviors without requiring direct measurement of individual molecular properties.