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
We can simplify this by dividing both sides by 1/2, yielding
dividing both sides by mB gives:
Now dividing both sides by vA2 gives:
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
Lastly, we can take the square root of both sides, giving us:
This is Graham's Law. It is important to note that the gas on the top of the left fraction (in this case A) is on the bottom of the right fraction, and vice-verse.
The question that may be running through your mind now is "what good is this?"