Why light gases win the race: Graham’s law with moving particles

How to use this simulation

1. The simulation opens on the tube experiment with ammonia (NH₃, 17 g/mol) entering from the left and hydrogen chloride (HCl, 36.5 g/mol) from the right. Press play. Blue and orange particles wander in from the ends, and wherever a pair meets it turns into a white dot, the ammonium chloride that makes the ring.
2. Watch the dashed line and the green line. The dashed one is Graham’s prediction, 59.4 cm from the ammonia end of the 100 cm tube. The green one is where the simulated meetings actually happen. It starts jumpy and settles near the dashed line after a few simulated seconds. The arrows at the bottom show the mean speeds in m/s: the lighter gas has the longer arrow.
3. Use "Gas pair" to try hydrogen with oxygen, helium with methane, carbon dioxide with oxygen, or choose "Set your own" and drag the two molar mass sliders. The Temperature slider changes every speed by the square root of the temperature ratio. Reset restarts with the same seed, so the same inputs give the same run.
4. Switch "Experiment" to Effusion (pinhole). The two gases now share a box with a small hole in the right wall. Each particle that finds the hole flies into the vacuum and is replaced, so the mixture stays constant. Read the live ratio of escapes against √(M₂ / M₁). Early on the ratio wobbles, because few particles have escaped; give it half a minute.

A perfume bottle, a quiet room and a nose at the far wall

Someone opens a bottle of perfume at one end of a classroom. A few seconds later the students nearby smell it. A minute later the whole room does. Nobody switched on a fan, and nothing pushed the molecules along. They spread out on their own.

That spreading is called diffusion. Now picture something different: a car tyre with a pinprick. Air hisses out through the tiny hole, and a tyre filled with a lighter gas would go flat sooner. That is effusion. Both are about gas molecules on the move, and both follow a rule found by the Scottish chemist Thomas Graham in the nineteenth century.

The rule says lighter molecules go faster. At room temperature a hydrogen molecule moves at about 1,782 m/s on average and an oxygen molecule at about 446 m/s. That is 4.00 times slower, which is exactly √(32 ÷ 2). The simulation above turns that gap into a white ring in a tube and into a race through a pinhole.

Below we start from zero and finish with solved exam problems. Keep the simulation open beside you and test every claim as you read.

From zero: gas particles that never stop moving

A gas is mostly empty space with tiny molecules flying in straight lines. They change direction only when they hit something: another molecule or a wall. Between hits, a molecule flies freely.

In air at normal pressure a molecule travels less than a ten-thousandth of a millimetre before its next collision, and it collides billions of times every second. So one perfume molecule does not shoot across the room. It zig-zags in a random walk, and the crowd of them slowly spreads from where there are many to where there are few.

That is the whole idea of diffusion: random motion plus a crowd that starts bunched up. No force pushes it. In the tube simulation each end is topped up like a wet cotton plug and every particle scatters after a fixed mean free path, so you see the spreading fronts move in from both sides.

Effusion is different. The hole is so small that molecules do not squeeze through together. They go one at a time, whenever one happens to hit the hole. The more often a molecule hits the wall, the more often it can escape, and a fast molecule hits the wall more often. So speed decides the rate.

Speed depends on two things you control: temperature and molar mass. Heat the gas and every molecule speeds up (v ∝ √T). Swap to a heavier gas and at the same temperature the molecules are slower (v ∝ 1/√M). Try the Temperature slider and the custom pair, then watch the "Mean speed" reading change.

Key terms for diffusion and effusion

Examination answers lose marks on loose wording. Learn these definitions as written.

Diffusion and effusion side by side

Students mix the two words up all the time. Both are about moving gas molecules and both obey Graham’s law, but the situations differ.

There is another topic with a similar name on this site: Diffusion and osmosis in biology. That one is about dissolved particles and water crossing a cell membrane. This page is the gas-phase chemistry law: how fast different gases spread, and why. The two meet in one idea, that particles move randomly and spread from crowded places to sparse ones.

Graham’s law, built from ½mv²

Nothing here has to be memorised blindly. The law comes from one idea about temperature.

The ammonia and hydrogen chloride ring

This is the classic school experiment, and the one the simulation opens with.

Try it in the simulation: six experiments

Run these in order. Press Reset between them so every run starts from the same seed.

• Pick Diffusion with NH₃ and HCl and press play. Wait for the green line to settle near the dashed Graham line, about 59.4 cm from the ammonia end. Notice which end the ring is nearer to.
• Switch the pair to H₂ and O₂. The ring is replaced by a dotted first-meeting marker, because H₂ and O₂ do not react; Graham’s line sits at 80.0 cm. Hydrogen is 4 times faster, so it covers 4 times the distance of oxygen.
• Choose "Set your own" and drag both sliders to the same value. The marker (and the Graham line) sit at 50 cm, the middle. Equal masses mean equal rates.
• Raise the temperature from 300 K to 600 K. Every speed grows by √600 ÷ 300 = 1.41 times, yet the ring stays put and simply forms sooner, because both gases speed up equally.
• Switch to Effusion with H₂ and O₂ and let it run for half a minute. The live ratio drifts toward 4.00 but wobbles early on. A few dozen escapes are a small sample, and Graham’s law is about averages.
• Choose CO₂ and O₂ in the effusion box. The ratio falls below one, near 0.85, because heavier carbon dioxide escapes more slowly than oxygen.

Solved problems

Each answer uses the same numbers as the simulation. Write the formula first, then substitute.

Common mistakes

These cost marks every year. Check your own working against each one.

• Writing rate ∝ 1/M. The square root is the whole point: a gas 16 times heavier is only √16 = 4 times slower.
• Swapping the masses. For rate₁ / rate₂ the heavier gas’s molar mass goes on top: √(M₂ / M₁).
• Treating time like rate. Time is inversely related to rate, so the slower gas takes longer: t₁ / t₂ = √(M₁ / M₂).
• Using degrees Celsius in a speed formula. It needs kelvin: 27 °C is 300 K.
• Applying the law at different temperatures or pressures. Graham’s law compares gases under the same conditions.
• Expecting the ring in the middle. Only equal molar masses meet halfway, at 50 cm. The ring sits nearer the heavier gas’s end.

Where Graham’s law is used

Uranium enrichment. Natural uranium holds only about 0.72% of the fissile isotope uranium-235. To separate it, uranium is turned into the gas UF₆ and pushed through porous barriers. The molar masses are 349 g/mol for the light isotope and 352 g/mol for the heavy one, so the light one passes slightly faster, by a factor of √(352 / 349) = 0.9957. That is only -0.43% per stage, so a plant needs hundreds of stages in a row (an ideal cascade to reach 4% takes about -410).

Balloons. Helium has M = 4, air about 29. Helium escapes through the pores of a balloon √(29 / 4) = 2.69 times faster than air, which is why a helium balloon goes soft before an air-filled one.

Gas leaks. Methane (16 g/mol) leaks through a tiny crack 1.66 times faster than propane (44 g/mol) would. Cooking gas has no smell of its own, so a strong-smelling chemical is added: the smell then diffuses through the room to warn you of a leak.

Breathing. Oxygen and carbon dioxide cross the thin walls of the lungs by diffusion, and the biology topic Diffusion and osmosis picks that story up. The gas law here sets the speed; the membrane adds its own rules.

Exam corner (class 11 chemistry and A-level)

Graham’s law questions come in three shapes. The first asks for a rate or time ratio, the second for an unknown molar mass, the third for a distance in a tube. Problems 1 to 4 above cover all three.

• State Graham’s law: at constant temperature and pressure, the rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass (or density).
• Define effusion and diffusion in one line each and give an example of each.
• Derive Graham’s law from equal kinetic energy at the same temperature (Step 1 to Step 3 above).
• Calculate the ratio of rates for two gases, or the time one takes if you know the other.
• Find the molar mass of an unknown gas from effusion times.
• Explain why the ammonium chloride ring forms nearer the HCl end.

Revision summary

Read this list the night before. Each line is one mark in an examination.

• Diffusion is mixing of gases by random motion; effusion is escape through a tiny hole into a vacuum.
• At one temperature all molecules have the same average kinetic energy, so lighter ones are faster.
• Mean speed v̄ = √(8RT / πM): it grows with √T and falls with √M.
• Graham’s law: rate₁ / rate₂ = √(M₂ / M₁), and time goes the other way: t₁ / t₂ = √(M₁ / M₂).
• The NH₃ and HCl ring forms 59.4 cm from the ammonia end of a 100 cm tube, because the ratio is 1.47.
• Use kelvin, keep the same temperature and pressure, and take the square root.

Frequently asked questions