Rate of Effusion Calculator

Two gases, one small hole, one question: given how heavy each gas is, how much faster does the lighter one escape — or spread — than the heavier one? This page answers that from the molar masses alone, and it answers the reverse question too: hand it a measured pair of rates and it returns the molar mass of the gas you have not identified yet.

Graham's law is a single statement containing four quantities — the two rates and the two molar masses. Any three of them fix the fourth, so the same page predicts a rate from the masses, checks a ratio you measured in the lab, or turns a rate pair into an identification. You choose the process first: effusion through a small opening, or diffusion of one gas into another. The two molar masses describe the gas pair and stay put when you switch; each process keeps its own pair of rate boxes.

That makes it useful in three fairly different situations: coursework and lab work where an unknown gas has to be narrowed down; reasoning about leaks and gas handling, where the practical question is whether one gas escapes appreciably faster than another; and the industrial case below, where two molecules differ by three mass units and a single pass through a barrier separates them by less than half a percent.

Graham's law says that the rate at which a gas effuses is inversely proportional to the square root of its molar mass. Put two gases through the same opening under the same conditions and the constants drop out, leaving a ratio that depends on nothing but the two masses.

$r_1$ and $r_2$ are the two rates in moles per second, and $M_1$ and $M_2$ the two molar masses in grams per mole. Squaring both sides gives $r_1^{2} M_1 = r_2^{2} M_2$, which is the form that shows why the mass direction is the sensitive one: the ratio of the rates enters twice.

Why the ratio has no units of its own

Both rates are read on the same scale and both molar masses on the same scale, so the conversion factors sit above and below the line and cancel. That is why the page asks only that the two rates match each other and the two molar masses match each other — as long as you are consistent, the ratios behave.

Both processes, one ratio — but not one rate

Effusion and diffusion are described by the same square-root law, and for a given pair of gases they give the same ratio. The absolute rates are not interchangeable: an effusion rate counts molecules crossing a hole per second, while a diffusion measurement follows one gas spreading into another over a distance, and the two numbers for the same gas are different. That is why the process choice in this calculator swaps in its own pair of rate boxes rather than reusing one, and why an effusion rate should never be typed next to a diffusion rate.

The three other directions

Rearranging the same relation gives every box in terms of the other three. The two rate directions are gentle; the two molar-mass directions square the rate ratio:

Because that last pair squares the ratio, a small error in a measured rate roughly doubles in the molar mass it produces. A 1% slip shows the effect: with rates of 0.99 and 1.00 mol/s and a reference gas of 28.014 g/mol, the page returns 27.45652 g/mol instead of 28.014 — a 1% rate error turned into a 2% mass error. When you are identifying an unknown gas from two measured rates, the rates are the part worth repeating.