Graham's Law

Graham's Law, named after Thomas Graham, is simply a derivation based on the statement in Kinetic Molecular Theory that temperature is a measure of average kinetic energy.

Stated differently, two gases at the same temperature have the same average kinetic energy.

That means that for two gases, a and b, if they are at the same temperature, then

where KE_a is the kinetic energy of gas a and KE_b is the kinetic energy of gas b.

From physics, we know that kinetic energy is

where m is the mass of a gas particle and v is the velocity (NOT the volume)

So, substituting in we get

We can simplify this equation by multiplying by 2, leaving

Graham decided to further rearrange the equation putting the masses on one side of the equation and the velocities on the other.

Lastly, he decided to take the square root of both sides, leaving him

Only 2 small adjustments remain. Since the masses of individual particles are too small to measure, we cannot find values for the fraction on the left. However, the masses of individual particles are proportional to their molar masses. So, we can replace the masses on the left with molar masses.

In addition, we cannot measure the velocity of individual particles, but we know that velocity is proportional to the rate at which gases diffuse or effuse. So, we can replace the velocities with rates, leading to this formula:

This is the final formula for Graham's Law, which raises the question: Why does anyone care about this math?