Diffusion occurs when gas molecules disperse throughout a container. Effusion occurs when a gas passes through an opening that is smaller than the mean free path of the particles, that is, the average distance traveled between collisions. Effectively, this means that only one particle passes through at a time.
If a mixture of gases is placed in a container with porous walls, the gases effuse through the small openings in the walls. The lighter gases pass through the small openings more rapidly (at a higher rate) than the heavier ones (
[link] ). In 1832, Thomas Graham studied the rates of effusion of different gases and formulated
Graham’s law of effusion :
The rate of effusion of a gas is inversely proportional to the square root of the mass of its particles :
This means that if two gases A and B are at the same temperature and pressure, the ratio of their effusion rates is inversely proportional to the ratio of the square roots of the masses of their particles:
A balloon filled with air (the blue one) remains full overnight. A balloon filled with helium (the green one) partially deflates because the smaller, light helium atoms effuse through small holes in the rubber much more readily than the heavier molecules of nitrogen and oxygen found in air. (credit: modification of work by Mark Ott)
Applying graham’s law to rates of effusion
Calculate the ratio of the rate of effusion of hydrogen to the rate of effusion of oxygen.
Solution
From Graham’s law, we have:
Using molar masses:
Hydrogen effuses four times as rapidly as oxygen.
Check your learning
At a particular pressure and temperature, nitrogen gas effuses at the rate of 79 mL/s. Using the same apparatus at the same temperature and pressure, at what rate will sulfur dioxide effuse?
Here’s another example, making the point about how determining times differs from determining rates.
Effusion time calculations
It takes 243 s for 4.46
10
−5 mol Xe to effuse through a tiny hole. Under the same conditions, how long will it take 4.46
10
−5 mol Ne to effuse?
Solution
It is important to resist the temptation to use the times directly, and to remember how rate relates to time as well as how it relates to mass. Recall the definition of rate of effusion:
and combine it with Graham’s law:
To get:
Noting that
amount of A =
amount of B , and solving for
time for Ne :
and substitute values:
Finally, solve for the desired quantity:
Note that this answer is reasonable: Since Ne is lighter than Xe, the effusion rate for Ne will be larger than that for Xe, which means the time of effusion for Ne will be smaller than that for Xe.
Check your learning
A party balloon filled with helium deflates to
of its original volume in 8.0 hours. How long will it take an identical balloon filled with the same number of moles of air (ℳ = 28.2 g/mol) to deflate to
of its original volume?
