## Activity: Equipartition theorem

Contemporary Challenges 2022 (4 years)
This lecture introduces the equipartition theorem.
Recall that $U=\frac32 Nk_BT$ for a monatomic gas. \begin{align} \Omega &= CV^NU^{\frac32N} \\ & \vdots \\ U &= \frac32 Nk_BT \end{align} The equipartition theorem is an elegant shortcut, but takes some steps to practice.
Step 1
Choose a set of parameters that can describe any arbitrary state of the material (gas or otherwise). This will include positions and velocities, and could also include angular momenta. If the molecule contains “springs” you might want to include the lengths of those springs.
Step 2

Write an explicit expression for the total classical Newtonian energy of the material in terms of the independent free variables.

For the ideal monatomic gas, we have $E = \frac12mv_1^2 + \frac12mv_2^2 + \frac12mv_3^2 + \cdots \text{(total of N atoms)}$ where if we write things in terms of Cartesian coordinates gives us $\frac12mv_1^2 = \frac12mv_{1x}^2 + \frac12mv_{1y}^2 + \frac12mv_{1z}^2$ so you end up with $3N$ terms that depend on independent free variables.

Step 3

Count the number of indepdendent free variables that are squared in the expression for the energy (“quadratic terms”). We call this the number of degrees of freedom $f$.

For the ideal monatomic gas $f=3N$.

Step 4

The equipartition theorem predicts that \begin{align} U_{\text{classical}}(T) &= \frac{f}{2}k_BT \end{align}

For the ideal monatomic gas $U_{\text{classical}}(T) = \frac{3}{2}Nk_BT$.

For the curious, here is a 3.5 minute video explaining ineptly why only the quadratic terms in the energy get equipartition, recorded in Spring 2021.

After this we do something else.

• group Heat capacity of N$_2$

group Small Group Activity

30 min.

##### Heat capacity of N2
Contemporary Challenges 2022 (4 years)

Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
• assignment Potential energy of gas in gravitational field

assignment Homework

##### Potential energy of gas in gravitational field
Potential energy Heat capacity Thermal and Statistical Physics 2020 Consider a column of atoms each of mass $M$ at temperature $T$ in a uniform gravitational field $g$. Find the thermal average potential energy per atom. The thermal average kinetic energy is independent of height. Find the total heat capacity per atom. The total heat capacity is the sum of contributions from the kinetic energy and from the potential energy. Take the zero of the gravitational energy at the bottom $h=0$ of the column. Integrate from $h=0$ to $h=\infty$. You may assume the gas is ideal.
• group Applying the equipartition theorem

group Small Group Activity

30 min.

##### Applying the equipartition theorem
Contemporary Challenges 2022 (4 years)

Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature $T$.
• assignment Ideal gas in two dimensions

assignment Homework

##### Ideal gas in two dimensions
Ideal gas Entropy Chemical potential Thermal and Statistical Physics 2020
1. Find the chemical potential of an ideal monatomic gas in two dimensions, with $N$ atoms confined to a square of area $A=L^2$. The spin is zero.

2. Find an expression for the energy $U$ of the gas.

3. Find an expression for the entropy $\sigma$. The temperature is $kT$.

• assignment Free energy of a harmonic oscillator

assignment Homework

##### Free energy of a harmonic oscillator
Helmholtz free energy harmonic oscillator Thermal and Statistical Physics 2020

A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with $\varepsilon_n = n\hbar\omega$, where $n$ is an integer $\ge 0$, and $\omega$ is the classical frequency of the oscillator. We have chosen the zero of energy at the state $n=0$ which we can get away with here, but is not actually the zero of energy! To find the true energy we would have to add a $\frac12\hbar\omega$ for each oscillator.

1. Show that for a harmonic oscillator the free energy is $$F = k_BT\log\left(1 - e^{-\frac{\hbar\omega}{k_BT}}\right)$$ Note that at high temperatures such that $k_BT\gg\hbar\omega$ we may expand the argument of the logarithm to obtain $F\approx k_BT\log\left(\frac{\hbar\omega}{kT}\right)$.

2. From the free energy above, show that the entropy is $$\frac{S}{k_B} = \frac{\frac{\hbar\omega}{kT}}{e^{\frac{\hbar\omega}{kT}}-1} - \log\left(1-e^{-\frac{\hbar\omega}{kT}}\right)$$

This entropy is shown in the nearby figure, as well as the heat capacity.

• assignment Ideal gas calculations

assignment Homework

##### Ideal gas calculations
Ideal gas Entropy Sackur-Tetrode Thermal and Statistical Physics 2020

Consider one mole of an ideal monatomic gas at 300K and 1 atm. First, let the gas expand isothermally and reversibly to twice the initial volume; second, let this be followed by an isentropic expansion from twice to four times the original volume.

1. How much heat (in joules) is added to the gas in each of these two processes?

2. What is the temperature at the end of the second process?

3. Suppose the first process is replaced by an irreversible expansion into a vacuum, to a total volume twice the initial volume. What is the increase of entropy in the irreversible expansion, in J/K?

• assignment Entropy and Temperature

assignment Homework

##### Entropy and Temperature
Energy Temperature Ideal gas Entropy Thermal and Statistical Physics 2020

Suppose $g(U) = CU^{3N/2}$, where $C$ is a constant and $N$ is the number of particles.

1. Show that $U=\frac32 N k_BT$.

2. Show that $\left(\frac{\partial^2S}{\partial U^2}\right)_N$ is negative. This form of $g(U)$ actually applies to a monatomic ideal gas.

• group Heat and Temperature of Water Vapor

group Small Group Activity

30 min.

##### Heat and Temperature of Water Vapor

In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.
• face Thermal radiation and Planck distribution

face Lecture

120 min.

##### Thermal radiation and Planck distribution
Thermal and Statistical Physics 2020

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.
• assignment Entropy, energy, and enthalpy of van der Waals gas

assignment Homework

##### Entropy, energy, and enthalpy of van der Waals gas
Van der Waals gas Enthalpy Entropy Thermal and Statistical Physics 2020

In this entire problem, keep results to first order in the van der Waals correction terms $a$ and \$b.

1. Show that the entropy of the van der Waals gas is \begin{align} S &= Nk\left\{\ln\left(\frac{n_Q(V-Nb)}{N}\right)+\frac52\right\} \end{align}

2. Show that the energy is \begin{align} U &= \frac32 NkT - \frac{N^2a}{V} \end{align}

3. Show that the enthalpy $H\equiv U+pV$ is \begin{align} H(T,V) &= \frac52NkT + \frac{N^2bkT}{V} - 2\frac{N^2a}{V} \\ H(T,p) &= \frac52NkT + Nbp - \frac{2Nap}{kT} \end{align}

Learning Outcomes