17: Boltzmann Factor and Partition Functions
- Page ID
- 11813
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
- This page discusses the connection between microscopic atomic behavior and macroscopic properties in statistical mechanics, highlighting the role of ensembles and averages. It explains how temperature relates to average kinetic energy, using the Boltzmann factor to illustrate the impact of temperature on the likelihood of particle states and the rates of physical processes.
- 17.2: The Boltzmann Distribution represents a Thermally Equilibrated Distribution
- This page explores the relationship between energy states and the total number of systems in an ensemble, highlighting the Boltzmann distribution's role in predicting system occupancy based on energy levels. It introduces the molecular partition function, \(q\), showing its dependence on temperature for determining thermally accessible states.
- 17.3: The Average Ensemble Energy is Equal to the Observed Energy of a System
- This page explains the canonical ensemble in statistical mechanics, highlighting the probability of molecules at specific energy levels based on the Boltzmann distribution. It details how to compute average energy using the partition function and introduces the variable \(\beta\) for simplification. The notation may vary, for instance, using \(Z\) instead of \(Q\). The focus is on foundational concepts relating to energy distributions.
- 17.4: Heat Capacity at Constant Volume is the Change in Internal Energy with Temperature
- This page discusses heat capacity at constant volume (\(C_V\)), emphasizing Dulong and Petit's findings on solid heat capacity and its limitations at low temperatures. Einstein's theory addresses these variations with atomic vibrations but also approaches zero near absolute zero. While effective for lead's heat capacity below 15 K, it diverges from experimental values at lower temperatures, suggesting the need for additional corrections to improve accuracy.
- 17.5: Pressure can be Expressed in Terms of the Canonical Partition Function
- This page explains the derivation of pressure from the canonical partition function in statistical mechanics, linking it to thermodynamic principles and the ideal gas law. It includes equations that relate average pressure to energy and the partition function, along with a thought experiment illustrating gas compression with a piston.
- 17.6: The Partition Function of Distinguishable, Independent Molecules is the Product of the Molecular Partition Functions
- This page discusses the derivation of the partition function for a system of distinguishable subsystems, such as gas molecules. It highlights that energy is additive, allowing the total energy to be expressed as the sum of individual molecule energies. For distinguishable, independent molecules, the overall partition function is the product of individual partition functions.
- 17.7: Partition Functions of Indistinguishable Molecules Must Avoid Over Counting States
- This page discusses the partition function in statistical mechanics, comparing calculations for distinguishable and indistinguishable particles. Using a two-particle model, it shows that distinguishable particles have four states, leading to a squared partition function. In contrast, indistinguishable particles have three states, requiring a modification to the partition function through a factor of \(N!\) to prevent overcounting.
- 17.8: Partition Functions can be Decomposed into Partition Functions of Each Degree of Freedom
- This page explains the partition function for indistinguishable, independent molecules, linking total average energy to the average energy of a single particle. It details how a particle's energy comprises contributions from different degrees of freedom—translational, rotational, vibrational, and electronic.
- 17.E: Boltzmann Factor and Partition Functions (Exercises)
- This page presents a series of homework problems from Chapter 17 of "Physical Chemistry: A Molecular Approach," focusing on energy, heat capacity, and partition functions for various gas models, including ideal and van der Waals gases. It details derivations for pressure and molar heat capacity through partition function evaluations.