# 17: Boltzmann Factor and Partition Functions

- Page ID
- 11813

Statistical Mechanics provides the connection between microscopic motion of individual atoms of matter and macroscopically observable properties such as temperature, pressure, entropy, free energy, heat capacity, chemical potential, viscosity, spectra, reaction rates, etc. Statistical Mechanics provides the microscopic basis for thermodynamics, which, otherwise, is just a phenomenological theory. Microscopic basis allows calculation of a wide variety of properties not dealt with in thermodynamics, such as structural properties, using distribution functions, and dynamical properties – spectra, rate constants, etc., using time correlation functions. Because a statistical mechanical formulation of a problem begins with a detailed microscopic description, microscopic trajectories can, in principle and in practice, be generated providing a window into the microscopic world. This window often provides a means of connecting certain macroscopic properties with particular modes of motion in the complex dance of the individual atoms that compose a system, and this, in turn, allows for interpretation of experimental data and an elucidation of the mechanisms of energy and mass transfer in a system.

- 17.1: The Boltzmann Factor
- The proportionality constant k (or kB) is named after him: the Boltzmann constant. It plays a central role in all statistical thermodynamics. The Boltzmann factor is used to approximate the fraction of particles in a large system. The Boltzmann factor is given by: \(\exp(-\beta E_i)\).

- 17.2: The Thermal Boltzman Distribution
- The Boltzmann distribution represents a thermally equilibrated most probable distribution over all energy levels. There is always a higher population in a state of lower energy than in one of higher energy.

- 17.3: The Average Ensemble Energy
- The probability of finding a molecule with energy \(E_i\) is equal to the fraction of the molecules with energy \(E_i\). The average energy is obtaining by multiplying \(E_i\) with its probability and summing over all \(i\): \[ \langle E \rangle = \sum_i E_i P_i \]

- 17.4: Heat Capacity at Constant Volume
- The heat capacity at constant volume, denoted \(C_V\), is defined to be the change in thermodynamic energy with respect to temperature.

- 17.5: Pressure in Terms of Partition Functions
- Pressure can also be derived from the canonical partition function.

- 17.6: Partition Functions of Distinguishable Molecules
- A system such as a gas can consist of a large number of subsystem. How is the partition function of the system built up from those of the subsystems? This depends on whether the subsystems are distinguisable or indistinguishable.