Graham’s Law of Diffusion and Effusion
Graham’s law states that the rate of effusion or diffusion of a gas is inversely proportional to the square root of its molar mass: Rate1/Rate2 = √(M2/M1). In brief: hydrogen (2.016 g/mol) effuses about 4 times faster than oxygen (31.998 g/mol), since √(31.998 / 2.016) = 3.98.
Light gas molecules move faster than heavy ones at the same temperature, so they escape through a tiny hole or spread through another gas more quickly. Graham’s law quantifies this effect. It was used to separate uranium isotopes and it is still the basis for many gas separation methods. This article explains the law, solves several problems and shows the difference between diffusion and effusion.
What are diffusion and effusion?
Diffusion is the gradual mixing of gases as their molecules move and collide. A perfume molecule released in a room diffusing through the air is an example. Effusion is the escape of a gas through a very small hole into a vacuum, without collisions at the opening. A leaking balloon is a loose example of this process. Graham’s law applies exactly to effusion, and approximately to diffusion.
What is Graham’s law?
At the same temperature, gases have the same average kinetic energy, ½mv2. A lighter molecule must therefore move faster than a heavier one, and the ratio of their rms speeds is √(M2/M1). Since effusion rates are proportional to molecular speeds, we obtain:
Rate1 / Rate2 = √(M2 / M1)
The same equation can be used with times, because the time needed to effuse a given amount of gas is inversely proportional to the rate: t1/t2 = √(M1/M2). Note that the heavier gas is on the top in the equation for the rates, and on the bottom in the equation for the times.
Relative effusion rates of common gases
| Gas | Molar mass (g/mol) | Relative rate (O2 = 1.00) |
|---|---|---|
| H2 | 2.016 | 3.98 |
| He | 4.003 | 2.83 |
| CH4 | 16.04 | 1.41 |
| N2 | 28.014 | 1.07 |
| O2 | 31.998 | 1.00 |
| CO2 | 44.01 | 0.853 |
| UF6 | 352.02 | 0.302 |
How do you solve Graham’s law problems?
Worked example 1: comparing two gases
How many times faster does helium effuse than nitrogen?
Rate(He) / Rate(N2) = √(28.014 / 4.003) = √6.998 = 2.65. Helium escapes about 2.6 times faster, which is the reason why helium balloons deflate faster than air-filled ones.
Worked example 2: finding the molar mass of an unknown gas
An unknown gas takes 2.50 times as long to effuse through a small hole as the same amount of oxygen. Find its molar mass.
- t(X) / t(O2) = √(MX / M(O2)) = 2.50.
- Square both sides: MX / 31.998 = 6.25.
- MX = 31.998 × 6.25 = 200 g/mol.
This value is close to the molar mass of mercury, 200.59 g/mol.
Worked example 3: ammonia and hydrogen chloride
Cotton wads soaked in ammonia (17.03 g/mol) and hydrochloric acid (HCl, 36.46 g/mol) are placed at the two ends of a glass tube 100 cm long. A white ring of NH4Cl forms where the gases meet. How far from the ammonia end does it form?
- The ratio of the distances is equal to the ratio of the rates: d(NH3) / d(HCl) = √(36.46 / 17.03) = 1.463.
- d(NH3) = 100 × 1.463 / (1 + 1.463) = 59.4 cm.
The ring forms 59.4 cm from the ammonia end, closer to the HCl end, because the lighter gas travels farther.
Worked example 4: separating uranium isotopes
The two isotopes in UF6 have molar masses 349.03 g/mol (235UF6) and 352.04 g/mol (238UF6). The ratio of their rates is √(352.04 / 349.03) = 1.0043. One effusion step enriches the mixture by only about 0.4 %, so thousands of stages were needed to reach useful concentrations.
Why is real diffusion slower than the law predicts?
Molecules in air at normal pressure collide with other molecules about 1010 times per second and travel only some 70 nm between collisions. Their paths are zigzag lines and not straight lines. Graham’s law gives the correct order of the relative rates but not their absolute values. Diffusion in a liquid is many orders of magnitude slower than in gas.
Who was Thomas Graham?
Thomas Graham (1805–1869), a Scottish chemist, published his findings on the diffusion of gases in 1831 and later on effusion. He later studied diffusion in liquids and introduced the terms colloid and dialysis. He served as Master of the Mint in London from 1855. The law was explained theoretically by kinetic theory a few decades afterwards.
Where is the law used?
- Isotope separation: gaseous diffusion plants enriched uranium for reactors and weapons during and after World War II. Modern plants mostly use centrifuges instead.
- Gas leak detection: light gases, such as helium, are used in leak testing because they pass through tiny gaps quickly.
- Mass spectrometry and vacuum technology: the flow of gases through small holes affects instrument design.
Which mistakes are common?
- Inverting the ratio: in the rates equation the heavier gas is in the numerator under the square root.
- Forgetting the square root: the rate does not depend on the molar mass in direct proportion.
- Mixing rates and times: a faster gas has a shorter time.
- Different temperatures: the law assumes that both gases are at the same temperature and pressure.
- Using atomic masses: use the molar mass of the actual molecule, such as 31.998 for O2.