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 is used to Approximate the Fraction of Particles in a Large System
- The proportionality constant \(k\) (or \(k_B\)) is named after Ludwig Boltzmann. 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: \(e^{-\beta E_i}\).
- 17.2: The Boltzmann Distribution represents a Thermally Equilibrated 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 is Equal to the Observed Energy of a System
- 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 \(i\): \[\langle E \rangle=\sum_i{E_iP_i} \nonumber\]. Using the Boltzmann distribution for \(P_i\) allows us to show that the average ensemble energy is equal to the observed energy of the system.
- 17.4: Heat Capacity at Constant Volume is the Change in Internal Energy with Temperature
- 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 can be Expressed in Terms of the Canonical Partition Function
- The canonical partition function can be used to derive an equation of state for pressure, \(P\).
- 17.6: The Partition Function of Distinguishable, Independent Molecules is the Product of the Molecular Partition Functions
- 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 depends on whether the subsystems are distinguishable or indistinguishable. For distinguishable systems, the partition function is the product of the molecular partition functions.
- 17.7: Partition Functions of Indistinguishable Molecules Must Avoid Over Counting States
- For indistinguishable particles, the number of possible states are reduced compared to distinguishable particles. The partition function must be modified to avoid over counting identical states.
- 17.8: Partition Functions can be Decomposed into Partition Functions of Each Degree of Freedom
- Just as the partition function of a system of \(N\) particles can be decomposed into the product of partition functions for each molecule, a molecular partition function can be decomposed into the product of the partition functions for each degree of freedom.