Overview#

• Molecules in a gas have a range of speeds—they are not all moving at \(v_{\text{rms}}\).

• The Maxwell speed distribution \(P(v)\) gives the probability density: \(P(v)\,dv\) = fraction of molecules with speeds in \(dv\) at speed \(v\).

• Three characteristic speeds: most probable \(v_P\), average \(v_{\text{avg}}\), rms \(v_{\text{rms}}\), with \(v_P < v_{\text{avg}} < v_{\text{rms}}\).

Maxwell speed distribution#

\(P(v)\) is a probability distribution; the fraction of molecules with speeds between \(v_1\) and \(v_2\) is

• The total area under \(P(v)\) is 1.

• \(P(v)\) depends on temperature \(T\) and molar mass \(M\): at lower \(T\), the distribution narrows and shifts to lower speeds.

Three characteristic speeds#

Order on the \(P(v)\) curve: \(v_P\) is at the peak; \(v_{\text{avg}}\) is to the right (average pulled by high-speed tail); \(v_{\text{rms}}\) is farthest right (squaring emphasizes high speeds).

The high-speed tail#

A small fraction of molecules have speeds much larger than \(v_P\). This tail is important for:

• Evaporation: Molecules in the high-speed tail have enough energy to escape the liquid surface.

• Nuclear fusion: In the Sun, protons in the high-speed tail can overcome Coulomb repulsion and fuse; typical speeds are too low.

Summary#

• \(P(v)\,dv\) = fraction of molecules with speeds in \(dv\) at \(v\).

• \(v_P < v_{\text{avg}} < v_{\text{rms}}\); all \(\propto \sqrt{T/M}\).

• \(v_P = \sqrt{2RT/M}\), \(v_{\text{avg}} = \sqrt{8RT/(\pi M)}\), \(v_{\text{rms}} = \sqrt{3RT/M}\).

• High-speed tail: evaporation, fusion.