Compare molecular speed distributions and threshold fractions

Predict a fraction, not just an average

Two hypothetical dilute gases are at 300 K. A has molar mass 16 g/mol; B has 64 g/mol. Predict which has more of its molecules above 600 m/s. Then compare the shaded areas. These are theoretical equilibrium distributions for stipulated masses, not measured histograms of particular chemical species.

The molecular-motion lesson already establishes RMS speed and mean translational energy. This investigation adds the whole speed distribution: most molecules do not move at exactly the RMS speed, and knowing one average does not mean every individual is identical.

Read area and height differently

The vertical axis is probability density per m/s. A narrow interval of width Δv near speed v contains approximately fraction f(v)Δv; a wider interval needs the area under the curve. The height at one speed is not a fraction, and the continuous model assigns probability zero to one exact speed. The total area over all nonnegative speeds is one.

At threshold zero, the fraction above is one. Increasing the threshold cannot increase that fraction. Compare 600 and 1200 m/s for both gases. The taller, narrower heavy-gas peak does not imply a larger high-speed population. The chart uses one fixed horizontal and vertical scale across every comparison.

The curve continues beyond 3000 m/s. The interface reports that off-chart tail separately and includes it in the numerical threshold fraction. Shading stops at the chart edge only because the drawing has finite width, not because the physical model has a maximum molecular speed.

Three summaries of the same distribution

Let s = √(2RT/M), with temperature T in kelvin, molar mass M in kg/mol and R = 8.314462618 J/(mol K). Convert 16 g/mol to 0.016 kg/mol. With dimensionless x = v/s, the Maxwell speed density is

f(v) = 4x² exp(−x²)/(√π s).

The most probable speed is s, the mean speed is 2s/√π and RMS speed is √(3/2)s. The marker rows and table show their ordering. RMS is larger than the arithmetic mean because it squares speeds before averaging and taking a square root.

At zero bulk flow the mean velocity vector is zero because directions cancel. Mean speed is positive. Random microscopic motion can therefore coexist with a stationary macroscopic gas. This activity displays distributions, not simulated trajectories or a transient collision process.

Change one cause, then challenge an inference

The first pair holds temperature fixed and changes mass. Increasing mass fourfold halves the speed scale. The entire distribution shifts to smaller speeds; its area remains one. Its fraction above a threshold v equals the lighter distribution’s fraction above 2v. Test this by saving the heavy fraction at 600 m/s and comparing the light fraction at 1200 m/s.

The second pair holds mass fixed and increases temperature fourfold. The speed scale doubles, while mean translational energy increases fourfold. A wider curve has a lower peak because its total area stays one.

The third pair holds T/M fixed while changing both. The speed curves and every threshold fraction coincide, but mean translational energy per mole, 3RT/2, differs fourfold. Thus an observed speed distribution alone would not identify temperature without information about molecular mass. Save this as a counterexample to “same speeds means same energy.”

Translational energy is not automatically the total internal energy of a polyatomic gas, which can also have rotational and vibrational contributions. Equal gas speeds also do not imply equal diffusion times, effusion rates or reaction yields without further assumptions and models.

Source and numerical boundaries

The OpenStax Maxwell-distribution reference supplies the density and speed summaries. The AP Chemistry CED, 3.5.A.1–4, connects kinetic motion, temperature and graphical representations. The diagrams, comparison cases and transfer task here are original.

Tail fractions use a bounded numerical integration in dimensionless speed, with 512 Simpson panels and upper cutoff x=12; the omitted tail beyond that cutoff is less than 4×10⁻⁶². This cutoff bound is distinct from quadrature error and from any experimental uncertainty. No measurements or uncertainty intervals are inferred from the displayed precision. The model assumes classical equilibrium ideal gases and makes no claim that an arbitrary real gas remains ideal at every stipulated condition.