Why Do Helium Balloons Deflate Faster?

Written by paul dohrman
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Why Do Helium Balloons Deflate Faster?
Graham's Law of Effusion

Helium gas effuses, or escapes from, balloons faster than air. The reason for this is because it travels much faster than heavier gas molecules at the same temperature. A common misconception is that a difference in atomic size is what explains the difference in effusion rates, but this relation is not borne out by experiments.

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A Misconception

A common misconception is that helium escapes from balloons faster than air because helium atoms are smaller, and can therefore fit through holes in the elastic better.

This might even seem to be backed up experimentally. For example, hydrogen escapes more slowly than helium, and hydrogen molecules are larger than helium molecules because they're diatomic.

However, when the sizes of gas atoms are measured, the size differences do not coherently fit the observed effusion rate differences.

Experimental Results

Given what we know about atomic sizes, there is not much variation in size between air molecules. For example, nitrogen, the leading constituent of air, differs in radius from helium by a factor of only about 50%, but differs from helium in effusion rate by significantly more than 50%. Neon is practically the same size as helium but effuses much more slowly than helium.

Graham's Law of Effusion

Before atomic radii were known, Scottish chemist Thomas Graham discovered an empirical relation between rate of effusion and molecular weight. The relation can be stated as seen in the diagram, where M1 and M2 stand for the molecular masses of gas 1 and gas 2.

Why Do Helium Balloons Deflate Faster?
Graham's Law of Effusion

Kinetic Theory of Gases

The explanation of Graham's theory had to wait until the kinetic theory of gases was developed in the late 19th century. One of the greatest contributions of the kinetic theory of gases was the revelation that the temperature of a gas, as measured on the Kelvin scale, is proportional to the kinetic energy of the gas, and therefore to its mass and the square of its average velocity. This means that, if two gases are at the same temperature, their kinetic energies are the same. If one is twice as massive, the other will average a speed four times as fast. Since effusion is a measure of particles traversing an area in a period of time, velocity differences would drive the difference in effusion rates between different molecules.

Derivation

Graham's law can therefore be easily derived from the kinetic theory. Start off with the observation above that effusion rates are proportional to average velocity because effusion is the rate a gas molecule passes through an area. v(1) ∝ effusion rate 1 and v(2) ∝ effusion rate 2. As stated above, two gases of equal temperature have equal kinetic energy. So KE1 = KE2. So m1 --- v1^2 = m2 --- v2^2. We can avoid having to worry about proportionality constants by merely dealing in ratios, so that the proportionality constants cancel out. Therefore, √[m1/m2] = v2 / v1 = effusion rate 2 ÷ effusion rate 1, thus giving us Graham's Law of Effusion.

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