Effusion

The image on the left shows effusion, where the image on the right shows diffusion. Effusion occurs through an orifice smaller than the mean path of the particles in motion where diffusion occurs through an opening in which multiple particles can flow through simultaneously.

Effusion is the process in which a gas escapes through a small hole. This occurs if the diameter of the hole is considerably smaller than the mean free path of the molecules.^[1] According to Graham's law, the rate at which gases effuse (i.e., how many molecules pass through the hole per second) is dependent on their molecular weight. Gases with a higher molecular weight effuse more slowly than gases with a lower molecular weight; the number of smaller molecules escaping will be greater but the mass of the larger molecules will be greater. By the kinetic theory of gases, the kinetic energy for a gas at a temperature $T$,

$\frac{3}{2}k_{\rm B} T = \frac{1}{2}m v_{\rm rms}^2$

where $k_{\rm B}$ is the Boltzmann constant. Their mass $m$ and the square of the root mean square molecular speed, $v_{\rm rms}$, are thus inversely proportional. At any given temperature, lighter molecules have a higher speed, which results in more molecules passing through the hole per unit time. This is why a balloon filled with low molecular weight helium deflates faster than an equivalent balloon full of higher molecular weight oxygen.

Graham's law

Thomas Graham (1805–1869), a Scottish chemist, found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles.^[2] In other words, the relative rates of effusion of two gases at the same temperature and pressure are given by the inverse ratio of the square roots of the masses of the gas particles. The equation is given by

${\mbox{Rate of effusion of gas}_1 \over \mbox{Rate of effusion of gas}_2}=\sqrt{M_2 \over M_1}$

where $M_1$ and $M_2$ represent the molar masses of the gases. This equation is known as Graham's law of effusion.

The effusion rate for a gas depends directly on the average velocity of its particles. Thus, the faster the gas particles are moving, the more likely they are to pass through the effusion orifice. A figure of Graham's law of effusion linked below shows the rate of effusion (the rate at which the gas is transferred across the barrier through the pin hole) is inversely proportional to the square root of the mass of the gas molecules.^[3]

References

1. K.J. Laidler and J.H. Meiser, Physical Chemistry, Benjamin/Cummings 1982, p.18
2. Zumdahl, Steven S. (2008). Chemical Principles. Boston: Houghton Mifflin Harcourt Publishing Company. p. 164. ISBN 978-0-547-19626-8.
3. Effusion of a gas into an evacuated chamber