Why helium balloons go flat first
Fill one balloon with helium and another with air, and the helium balloon shrinks first. Small helium atoms move faster and slip through tiny gaps more often than the heavier nitrogen and oxygen molecules in air. Thomas Graham measured this effect in the 1840s and found a simple rule.
The law
rate₁ ÷ rate₂ = √(M₂ ÷ M₁)
where M₁ and M₂ are the molar masses of the two gases. Because time is the inverse of rate, the time for the same amount of gas to escape follows the opposite ratio:
time₁ ÷ time₂ = √(M₁ ÷ M₂)
Where it comes from
At a given temperature, every gas has the same average kinetic energy per molecule, ½mv². If m is smaller, v must be larger. The root-mean-square speed is:
v_rms = √(3RT ÷ M)
with R = 8.314 J/(mol·K), T in kelvins and M in kg/mol. Hydrogen at 68 °F moves at about 1,900 m/s; oxygen at about 480 m/s.
Example
Compare hydrogen (2.016 g/mol) with oxygen (32.00 g/mol):
- Rate ratio: √(32.00 ÷ 2.016) = 3.98, so hydrogen escapes about four times faster.
- If a given amount of oxygen takes 60 seconds to escape through a pinhole, the same amount of hydrogen takes 60 ÷ 3.98 ≈ 15 seconds.
Using it to find an unknown molar mass
Measure the effusion time of an unknown gas and of a known gas through the same opening under the same conditions. Then M_unknown = M_known × (t_unknown ÷ t_known)². This was one of the early methods to estimate molar masses.
Limits
Graham’s law is exact only for effusion through holes much smaller than the distance molecules travel between collisions, and only when both gases are at the same temperature and pressure. Leaks through rubber or plastic also depend on how the gas dissolves in the material, so balloons are a good illustration but not a precise test.