# 21: The Boltzmann Distribution Function

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

• 21.1: Finding the Boltzmann Equation
We previously introduced the principle of equal a priori probabilities, which asserts that any two microstates of an isolated system have the same probability. From the central limit theorem, we infer that an isolated system is functionally equivalent to a constant-temperature system when the system contains a sufficiently large number of molecules. From these ideas, we can now find the relationship between the energy values and the corresponding probabilities of difference states.
• 21.2: Lagrange's Method of Undetermined Multipliers
Lagrange’s method of undetermined multipliers is a method for finding the minimum or maximum value of a function subject to one or more constraints.
• 21.3: Deriving the Boltzmann Equation I
We can use Lagrange’s method to find the dependence of the quantum-state probability on its energy. This is an alternative way to derive the Boltzmann distribution.
• 21.4: Deriving the Boltzmann Equation II
This derivation of Boltzmann’s equation from maximum entropy is the most common introductory treatment and relies on the assumption that all of the Ni are large enough to justify treating them as continuous variables This assumption proves to be invalid for many important systems; but the result obtained is clearly correct and leads to satisfactory agreement between microscopic models and the macroscopic properties of a wide variety of systems.
• 21.5: Partition Functions and Equilibrium - Isomeric Molecules
In Section 20.11, we discuss chemical equilibrium between isomers from the perspective afforded by Boltzmann’s definition of entropy. Now, let us consider equilibrium in this system from the perspective afforded by the energy-level probabilities.
• 21.6: Finding ß and the Thermodynamic Functions for Distinguishable Molecules
All of a substance’s thermodynamic functions can be derived from the molecular partition function.
• 21.7: The Microscopic Model for Reversible Change
Now let us return to the closed (constant- N ) system to develop another perspective on the dependence of its macroscopic thermodynamic properties on the molecular energy levels and their probabilities. We undertake to describe the system using volume and temperature as the independent variables. In thinking about the energy-level probabilities, we stipulate that any parameters that affect the state of the system remain constant.
• 21.8: The Third Law of Thermodynamics
In some cases, however, the assumption that the entropy is zero at absolute zero leads to absolute entropy values that are not consistent with other experiments. In these cases, the absolute entropies can be brought into agreement with other entropy measurements by taking into account degeneracies.
• 21.9: The Partition Function for a System of N Molecules
The molecular origins of the energies of the system enter the ensemble treatment only indirectly. The theory deals with the relationships between the possible values of the energy of the system and its thermodynamic state. How molecular energy levels and intermolecular interactions give rise to these values of the system energy becomes a separate issue.
• 21.10: Problems

This page titled 21: The Boltzmann Distribution Function is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.