17.S: Boltzmann Factor and Partition Functions (Summary)
- Page ID
- 550571
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This chapter covers statistical thermodynamics: the link between the microscopic properties and bulk properties of matter. At its heart, the foundational principle of statistical thermodynamics is that any observable thermodynamic or mechanical property of a system is the time average of its instantaneous value at equilibrium. By considering all the possible configurations of a system that are consistent with the overall state of the system, we can find that the most likely configuration is consistent with a Boltzmann distribution, and that the quantity that serves as a “thermodynamic wavefunction” for that distribution is the partition function. If the partition function for a system is known, then all mechanical and thermodynamic properties can be calculated, and it is possible to determine how the particles that make up the system are statistically distributed among the system’s quantum states.
We will begin by deriving the Boltzmann distribution from considering quantum states with equal a priori probabilities. After deriving the Boltzmann distribution, we will derive partition functions for general systems, and for systems consisting of independent particles. We will see that the overall partition function of a system can be expressed in terms of a product of molecular partition functions, and that the molecular partition function can in turn be expressed as a product of partition functions for each degree of freedom of the molecule. As we go along, we will show how to derive several mechanical and thermodynamic quantities from the partition functions of these systems. We will make connections between statistical thermodynamics and the relative intensities of spectroscopic transitions to show how spectroscopy can be used to measure temperatures and abundances of molecules in a thermal environment.
By the end of this chapter, you should know the following:
- The principle of equal a priori probabilities
- The configuration of a system of \(N\) particles in \(k\) states
- How the Boltzmann distribution relates to the weights of configurations
- The physical interpretation of the molecular partition function
- The difference between the molecular partition function \(q\) and the system partition function \(Q\)
- When it is possible to calculate \(Q\) from \(q\)
- How Bose-Einstein and Fermi-Dirac statistics impact the relationship between \(Q\) and \(q\)
- How the molecular partition function can be used to describe the various degrees of freedom of a molecule
Skill goals (what you should be able to do/calculate):
- Calculate the weight of a configuration.
- Given a set of states and their energies (or levels, degeneracies, and energies), calculate the molecular partition function
- Use the Boltzmann distribution to calculate the probability that a system or molecule will be found with a given energy at a given temperature
- Use simple models for the energies of an ideal diatomic gas to calculate state populations
- Use partition functions to calculate total energy, heat capacity, pressure, and entropy
- Use the relative intensities of a molecular spectrum to calculate temperature and number density