Graham's Law of Diffusion Calculator
Compare the diffusion or effusion rates of two gases, or solve for one unknown molar mass, using Graham's square-root relationship.
Workbook ready for the demonstration values.
Gas comparison inputs
Live result
Gas 1 effuses 4.0000 times as fast as Gas 2.
Current comparison
| Gas | Process | Rate | Molar mass (g/mol) | Rate relative to Gas 2 |
|---|---|---|---|---|
| Gas 1 | Effusion | 4.0000 | 2.0160 | 4.0000 |
| Gas 2 | Effusion | 1.0000 | 32.2560 | 1.0000 |
The table uses the same canonical values as the live result and the downloaded workbook. Rates may be any consistent unit, such as mol/s, molecules/s, or a dimensionless relative rate.
How to use this Graham's law calculator
What this calculator does
This calculator applies Graham's law to two gases under comparable temperature and pressure conditions. It estimates the ratio of their diffusion or effusion rates from their molar masses and checks whether entered rates are consistent with that theoretical relationship. It is useful for textbook problems, laboratory planning, gas-selection comparisons, and quick checks of an unknown molar mass. It does not model real diffusion coefficients in liquids, turbulent mixing, membrane-specific resistance, intermolecular reactions, or large pressure differences.
When to use it
Use it when comparing how quickly two gases escape through a tiny opening, when estimating which gas spreads faster under similar conditions, when checking a measured rate ratio against known molar masses, or when rearranging Graham's law to estimate one missing molar mass. The underlying relationship is described in the OpenStax discussion of gas effusion and diffusion.
How to calculate
- The calculator opens with a complete demonstration: hydrogen-like Gas 1 has a molar mass of 2.016 g/mol and a relative rate of 4, while Gas 2 has a molar mass of 32.256 g/mol and a relative rate of 1. The example workbook is available immediately.
- Choose Effusion or Diffusion in the Process menu. This changes the wording, not the idealized square-root ratio.
- Replace the four demonstration values with positive decimal values. Rates must use the same unit for both gases. Molar masses must be in grams per mole.
- Read the primary rate ratio, mass ratio, consistency difference, formula substitution, and comparison table. A zero consistency difference means your entered rates exactly agree with the masses at the displayed precision.
- Select Download Excel to export the current validated model. Select Reset to clear the demonstration and results; export remains unavailable until all four required values are valid again.
Input guide
Process is a required choice between effusion and diffusion. Effusion is escape through a very small opening; diffusion is spreading and mixing. Rate of Gas 1 and Rate of Gas 2 are required positive decimals in any identical rate unit. Examples are 4 and 1. Increasing Gas 1's rate raises the entered rate ratio; entering different units for the two rates is a common mistake. Molar mass of Gas 1 and Molar mass of Gas 2 are required positive decimals in g/mol. Examples are 2.016 and 32.256. Increasing a gas's molar mass lowers its predicted rate relative to the other gas. Do not enter molecular mass in kilograms per mole unless you convert both values to g/mol first. Plain decimal notation is accepted; scientific notation, commas used as decimal separators, units typed into the field, zero, and negative values are rejected.
Output guide
Rate ratio, Gas 1 ÷ Gas 2 is the entered relative rate and is dimensionless because identical rate units cancel. A value above 1 means Gas 1 is faster; below 1 means Gas 2 is faster; exactly 1 means equal rates. Mass ratio, Gas 2 ÷ Gas 1 is also dimensionless. Its square root is the Graham's-law predicted rate ratio. Consistency difference is the absolute percentage difference between the entered rate ratio and the ratio predicted by the molar masses. Zero is exact agreement; a larger value indicates that the four inputs do not describe one ideal Graham's-law comparison. The formula line shows the current substitution, while the Current comparison table reports each gas's process, rate, molar mass, and rate relative to Gas 2. These are calculated identities and comparisons, not experimental guarantees.
Worked example
With Gas 1 at 2.016 g/mol and Gas 2 at 32.256 g/mol, the mass ratio is 32.256 ÷ 2.016 = 16. Graham's law takes the square root: √16 = 4. Therefore, a Gas 1 rate of 4 and a Gas 2 rate of 1 are perfectly consistent, the displayed rate ratio is 4.0000, and the consistency difference is 0.0000%. In practical terms, Gas 1 is predicted to effuse four times as fast as Gas 2 under comparable ideal conditions.
Formula and assumptions
Graham's law is written as r₁/r₂ = √(M₂/M₁), where r is a diffusion or effusion rate and M is molar mass. The inverse square-root dependence reflects the kinetic-theory result that, at the same temperature, lighter molecules have higher characteristic speeds. The LibreTexts physical chemistry treatment of Graham's law explains this kinetic-energy connection. For reliable use, compare gases at the same temperature and similar pressure and geometry. Real diffusion through another gas can deviate from the simple ideal relationship because collisions, concentration gradients, and transport properties also matter.
Common mistakes and interpretation
Keep the rate units identical, place each mass under the matching gas, and remember that the masses appear in reverse order inside the square root. A gas four times as massive is not four times slower; it is two times slower because the relationship uses a square root. The equation predicts a ratio, not an absolute rate, unless one absolute rate is already known. A nonzero consistency difference is not automatically an experimental error: it may reflect non-ideal conditions or simply an intentionally hypothetical set of inputs.