The Maxwell-Boltzmann Distribution of Molecular Speeds

A rotating-drum beam experiment, a velocity selector, and Boltzmann's H-theorem.

Objectives

On completing this experiment you should be able to:

Measure the speed distribution of atoms in an effusive beam with a rotating-drum apparatus.
Show that the beam distribution carries an extra factor of v relative to the distribution inside the oven, and explain why.
Determine the oven temperature from the shape of the distribution alone.
Verify the predicted ratios of the most probable, mean and root-mean-square speeds.
Identify an unlabelled charge from the mass dependence of its speed distribution.
Follow the relaxation of a gas to equilibrium and show that the H-function decreases monotonically.

Theory

In equilibrium the number of molecules with speed between v and v + dv is

F(v)\,dv = 4 \pi \left(\frac{m}{2 \pi k T}\right)^{3/2} v^2 \exp(-m v^2 / 2 k T)\,dv .

The flux through a small hole is proportional to v F(v), because faster molecules reach the hole more often, so the beam that emerges is distributed as $v^3$ exp(-v^2/alpha^2) with alpha = $\sqrt{2kT/m}$ . Writing the beam distribution in that form and fitting it to the measured deposit is a determination of T that uses no thermometer at all.

Boltzmann's H-theorem states that for a dilute gas evolving by binary collisions, H = integral of f ln f $d^{3v}$ never increases, and is stationary only for the Maxwell-Boltzmann distribution. Since S = -k N H + constant, this is the microscopic counterpart of the principle of increasing entropy, and it links the Boltzmann relation $S = k$ ln W to the second law.

Key relations

$F(v)=4\pi\left(\frac{m}{2\pi kT}\right)^{3/2}v^{2}e^{-mv^{2}/2kT}$
Maxwell-Boltzmann speed distribution in the gas.
$f_{\mathrm{beam}}(v)=\frac{2}{\alpha^{4}}v^{3}e^{-v^{2}/\alpha^{2}}$
Flux-weighted distribution in an effusive beam.
$v_{mp}:\bar{v}:v_{rms}=1:\sqrt{4/\pi}:\sqrt{3/2}$
Ratios of the characteristic speeds.
$\frac{\mathrm{d}H}{\mathrm{d}t}\le 0,\qquad S=-kNH+\text{const}$
Boltzmann's H-theorem.

Procedure by part

Part A - Rotating-drum beam experiment

Expose the drum to the beam for a fixed time and read the number of atoms deposited on each detector strip.

Select the charge and the oven temperature, and set the drum speed so that the deposit falls well inside the strip array.
Expose the drum. The apparatus reports the counts on all sixty strips.
Convert each strip position s to a speed using $v = 2 R^2$ omega / s, and the counts to a probability density using the Jacobian.
Fit the beam distribution and obtain alpha, and hence the oven temperature.
Repeat at three oven temperatures and confirm that alpha scales as the square root of T.
Repeat with the unlabelled charge and identify it.

Part B - Velocity selector

Set the selector speed and measure the transmitted beam intensity.

Select the charge and oven temperature.
Step the selector speed over its whole range, recording the detector count rate at each setting.
Convert each speed setting to a transmitted molecular speed using $v = 2$ pi nu L / phi.
Plot the count rate against v, fit the beam distribution, and compare the temperature obtained with that from Part A.

Part C - Relaxation to equilibrium and the H-theorem

Start a gas of hard spheres from a chosen non-equilibrium state and follow the speed distribution and the H-function.

Choose the gas, the temperature and the initial condition.
Run the simulation and inspect the speed histogram at the end together with the Maxwell-Boltzmann curve.
Inspect the H-function trace and confirm that it decreases monotonically to a constant.
Repeat from a different initial condition at the same energy and confirm that the final state is the same.
Comment on what this implies about the approach to equilibrium and about the Boltzmann relation $S = k$ ln W.

Questions to answer in your report

Derive the beam distribution from the Maxwell-Boltzmann distribution and explain physically why the extra factor of v appears. Quote the value of the exponent you measured. [8]
Your H-function falls monotonically and then flattens. State the H-theorem, relate H to the entropy, and explain the apparent conflict with the time-reversibility of the underlying collisions. [10]
Two runs of Part C started from very different initial conditions at the same total energy and reached the same final distribution. What does this illustrate, and how does it connect to $S = k$ ln W? [6]