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Molecular Speeds

Level 1 - Physics topic page in Matter at Thermal Scale.

Principle

Gas molecules do not all move at the same speed.

At a fixed temperature, molecular speeds form a distribution, and useful averages describe different features of that distribution.

Notation

\(v\)
molecular speed
\(\mathrm{m\,s^{-1}}\)
\(v_{\mathrm{mp}}\)
most probable speed
\(\mathrm{m\,s^{-1}}\)
\(\bar v\)
mean speed
\(\mathrm{m\,s^{-1}}\)
\(v_{\mathrm{rms}}\)
root-mean-square speed
\(\mathrm{m\,s^{-1}}\)
\(M\)
molar mass
\(\mathrm{kg\,mol^{-1}}\)
\(T\)
absolute temperature
\(\mathrm{K}\)
\(\lambda\)
mean free path
\(\mathrm{m}\)
\(r\)
molecular radius in a collision model
\(\mathrm{m}\)

Method

Derivation 1: Compare common speed averages

The Maxwell-Boltzmann distribution gives a curve of how many molecules lie in each speed interval. Three speed measures are commonly used.

Most probable speed
\[v_{\mathrm{mp}}=\sqrt{\frac{2RT}{M}}\]
Mean speed
\[\bar v=\sqrt{\frac{8RT}{\pi M}}\]
RMS speed
\[v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}\]
Ordering
\[v_{\mathrm{mp}}<\bar v<v_{\mathrm{rms}}\]

Increasing temperature shifts the distribution toward larger speeds and broadens it. Increasing molar mass shifts the speeds lower.

Derivation 2: Estimate collision spacing

The mean free path is the average distance a molecule travels between collisions. A simple hard-sphere model says collisions become more likely when molecules are larger or number density is higher.

Number density
\[\frac{N}{V}\]
Collision cross-section
\[\pi(2r)^2=4\pi r^2\]
Mean free path
\[\lambda=\frac{1}{4\sqrt{2}\pi r^2(N/V)}\]
Mean free time
\[t_{\mathrm{mean}}=\frac{\lambda}{v}\]

Rules

These are the compact speed-distribution and collision estimates.

Most probable
\[v_{\mathrm{mp}}=\sqrt{\frac{2RT}{M}}\]
Mean speed
\[\bar v=\sqrt{\frac{8RT}{\pi M}}\]
RMS speed
\[v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}\]
Mean free path
\[\lambda=\frac{1}{4\sqrt{2}\pi r^2(N/V)}\]
Mean free time
\[t_{\mathrm{mean}}=\frac{\lambda}{v}\]

Examples

Question
Estimate
\[v_{\mathrm{rms}}\]
for nitrogen molecules at
\[300\,\mathrm{K}\]
using
\[M=2.8\times10^{-2}\,\mathrm{kg\,mol^{-1}}\]
Answer
\[v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3(8.31)(300)}{2.8\times10^{-2}}}=5.2\times10^2\,\mathrm{m\,s^{-1}}\]

Checks

  • Use kelvins in speed formulas.
  • The RMS speed is larger than the mean speed for the Maxwell-Boltzmann distribution.
  • At the same temperature, lighter molecules move faster on average.
  • Mean free path depends on number density and molecular size, not directly on molecular speed.

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Phases of Matter