Verifying the speed distribution: Stern’s molecular-beam experiment

Otto Stern’s 1920 molecular-beam apparatus gave the first direct experimental measurement of a gas’s actual molecular speeds, rather than only their statistical consequences (pressure, diffusion, viscosity) – a rotating-drum time-of-flight technique later refined by Zartman and Ko (1930-1934) into a quantitative confirmation of the full Maxwell-Boltzmann functional form, not just its mean. sample() draws exactly such a synthetic “beam” of molecular speeds; binning it into a histogram and comparing it to the exact analytic pdf reproduces the kind of check Stern, and later Zartman and Ko, made directly against a real gas.

import matplotlib.pyplot as plt
import numpy as np

from chemistrykit.statmech import MaxwellBoltzmannSpeedDistribution

mass = 2.18e-25  # potassium-like, kg -- close to the alkali-metal beams Stern's and Zartman/Ko's apparatus actually used
temperature = 470.0  # K, a plausible oven temperature for an alkali-metal beam source

dist = MaxwellBoltzmannSpeedDistribution(mass=mass, temperature=temperature)
sampled_speeds = dist.sample(200_000, rng=0)

fig, ax = plt.subplots(figsize=(7, 5))
ax.hist(sampled_speeds, bins=80, density=True, alpha=0.5, color="steelblue", label="synthetic 'beam' (200,000 draws)")
v = np.linspace(0.0, float(sampled_speeds.max()), 400)
ax.plot(v, dist.pdf(v), color="crimson", linewidth=2, label="exact Maxwell-Boltzmann pdf")
ax.set_xlabel("speed (m/s)")
ax.set_ylabel("probability density")
ax.set_title("Sampling the Maxwell-Boltzmann distribution (a Stern-type beam measurement)")
ax.legend()
fig.tight_layout()

The sampled mean speed converges to the exact analytic mean speed as the number of “molecules” in the synthetic beam grows – exactly the statistical convergence a real, finite molecular-beam measurement is limited by:

for n in (100, 10_000, 1_000_000):
    speeds = dist.sample(n, rng=0)
    print(f"n = {n:>9,}: sampled mean speed = {speeds.mean():.2f} m/s  (exact: {dist.mean_speed():.2f} m/s)")

plt.show()

n =       100: sampled mean speed = 284.19 m/s  (exact: 275.32 m/s)
n =    10,000: sampled mean speed = 274.88 m/s  (exact: 275.32 m/s)
n = 1,000,000: sampled mean speed = 275.24 m/s  (exact: 275.32 m/s)