## The Kinetic-Molecular Theory Explains the Behavior of Gases, Part II

According to Graham’s law, the molecules of a gas are in rapid motion and the molecules themselves are small. The average distance between the molecules of a gas is large compared to the size of the molecules. As a consequence, gas molecules can move past each other easily and diffuse at relatively fast rates.

The rate of effusion of a gas depends directly on the (average) speed of its molecules:

\(\text{effusion rate}\phantom{\rule{0.2em}{0ex}}\propto \phantom{\rule{0.2em}{0ex}}{u}_{\text{rms}}\)

Using this relation, and the equation relating molecular speed to mass, Graham’s law may be easily derived as shown here:

\({u}_{\text{rms}}=\sqrt{\cfrac{3RT}{M}}\)

\(M=\phantom{\rule{0.2em}{0ex}}\cfrac{3RT}{{u}_{\text{r}\text{m}\text{s}}^{2}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\cfrac{3RT}{{\overline{u}}^{2}}\)

\(\cfrac{\text{effusion rate A}}{\text{effusion rate B}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\cfrac{{u}_{\text{r}\text{m}\text{s}\phantom{\rule{0.2em}{0ex}}\text{A}}}{{u}_{\text{r}\text{m}\text{s}\phantom{\rule{0.2em}{0ex}}\text{B}}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\cfrac{\sqrt{\frac{3RT}{{M}_{\text{A}}}}}{\sqrt{\frac{3RT}{{M}_{\text{B}}}}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\sqrt{\cfrac{{M}_{\text{B}}}{{M}_{\text{A}}}}\)

The ratio of the rates of effusion is thus derived to be inversely proportional to the ratio of the square roots of their masses. This is the same relation observed experimentally and expressed as Graham’s law.