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 \(U_{\text{classical}}(T) = \frac{3}{2}Nk_BT\).
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.”
Potential energyHeat capacityThermal 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.
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.
Show that for a harmonic oscillator the free energy is
\begin{equation}
F = k_BT\log\left(1 - e^{-\frac{\hbar\omega}{k_BT}}\right)
\end{equation} Note that at high temperatures such that
\(k_BT\gg\hbar\omega\) we may expand the argument of the logarithm
to obtain \(F\approx k_BT\log\left(\frac{\hbar\omega}{kT}\right)\).
From the free energy above, show that the entropy is
\begin{equation}
\frac{S}{k_B} =
\frac{\frac{\hbar\omega}{kT}}{e^{\frac{\hbar\omega}{kT}}-1}
- \log\left(1-e^{-\frac{\hbar\omega}{kT}}\right)
\end{equation}
Entropy of a simple harmonic oscillatorHeat capacity of a simple harmonic oscillator
This entropy is shown in the nearby figure, as well
as the heat capacity.
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.
How much heat (in joules) is added to the gas in each of these two
processes?
What is the temperature at the end of the second process?
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?
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.
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.
