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Graham's Law Calculator

Applies Graham's law, rate1/rate2 = √(M2/M1), to two gases given by formula or molar mass, returning the effusion-rate ratio, the time ratio and the root-mean-square molecular speeds; alternatively solves the unknown molar mass of gas 2 from a measured rate ratio.

Gas 1 formula (optional)Formula of gas 1, e.g. He, H2, O2, CO2, UF6 (case-sensitive symbols). Alternative to gas_1_molar_mass_g_mol. Gas 1 molar mass (g/mol) (optional)Molar mass of gas 1, if no formula is given. Gas 2 formula (optional)Formula of gas 2. Leave both gas 2 inputs empty and give rate_ratio_1_to_2 to solve its molar mass. Gas 2 molar mass (g/mol) (optional)Molar mass of gas 2, if no formula is given. Measured rate ratio rate1 / rate2 (optional)Measured effusion-rate ratio of gas 1 to gas 2 (equal to the time ratio t2/t1); used only to solve the molar mass of an unknown gas 2. Temperature (optional)Temperature for the root-mean-square speeds, in temperature_unit. Temperature unit (optional)Unit of the temperature inputs; the calculation uses kelvin (K = °C + 273.15).

When to use

You need how much faster one gas effuses or diffuses than another, the relative effusion times, or the molar mass of an unknown gas from its effusion rate relative to a known gas.

Do not use when: You need the absolute effusion rate through a specific orifice (depends on geometry and pressure), or the gases are at high pressure where mean free paths make diffusion non-ideal.

Formula

rate_1 / rate_2 = √(M2 / M1); t_1 / t_2 = √(M1 / M2); M2 = M1 × (rate_1 / rate_2)²; v_rms = √(3RT / M) with M in kg/mol and R = 8.314462618 J/(mol·K)

Graham's law follows from equal average kinetic energy of ideal gases at the same temperature; it applies to effusion through a small hole and approximately to diffusion. Molar masses use IUPAC 2021 abridged atomic weights.

Inputs

Parameter	Type	Unit	Required	Description
`gas_1_formula`	string		no	Formula of gas 1, e.g. He, H2, O2, CO2, UF6 (case-sensitive symbols). Alternative to gas_1_molar_mass_g_mol.
`gas_1_molar_mass_g_mol`	number	g/mol	no	Molar mass of gas 1, if no formula is given. Range: > 0, ≤ 100000
`gas_2_formula`	string		no	Formula of gas 2. Leave both gas 2 inputs empty and give rate_ratio_1_to_2 to solve its molar mass.
`gas_2_molar_mass_g_mol`	number	g/mol	no	Molar mass of gas 2, if no formula is given. Range: > 0, ≤ 100000
`rate_ratio_1_to_2`	number		no	Measured effusion-rate ratio of gas 1 to gas 2 (equal to the time ratio t2/t1); used only to solve the molar mass of an unknown gas 2. Range: > 0
`temperature`	number		default 25	Temperature for the root-mean-square speeds, in temperature_unit.
`temperature_unit`	enum: celsius \| kelvin		default celsius	Unit of the temperature inputs; the calculation uses kelvin (K = °C + 273.15).

Outputs

Output	Type	Unit	Description
`molar_mass_1_g_mol`	number	g/mol	M1 from the formula or as given.
`molar_mass_2_g_mol`	number	g/mol	M2 from the formula, as given, or solved from the rate ratio.
`rate_ratio_1_to_2`	number		√(M2 / M1): how many times faster gas 1 effuses than gas 2.
`time_ratio_1_to_2`	number		√(M1 / M2): time for gas 1 to effuse a given amount relative to gas 2.
`faster_gas`	string		The lighter gas effuses faster.
`rms_speed_1_m_s`	number	m/s	√(3RT/M1) at the given temperature.
`rms_speed_2_m_s`	number	m/s	√(3RT/M2) at the given temperature.
`temperature_k`	number	K	Temperature used for the speeds.
`solved_for`	string		rate_ratio_1_to_2 or molar_mass_2_g_mol.

Example

Helium vs oxygen at 25 °C: {"gas_1_formula":"He","gas_2_formula":"O2"} → {"molar_mass_1_g_mol":4.003,"molar_mass_2_g_mol":31.998,"rate_ratio_1_to_2":2.8274,"time_ratio_1_to_2":0.3537,"faster_gas":"He","rms_speed_1_m_s":1363.1,"rms_speed_2_m_s":482.1,"solved_for":"rate_ratio_1_to_2"}

Unknown gas effusing 4 times slower than H2: {"gas_1_formula":"H2","rate_ratio_1_to_2":4} → {"molar_mass_2_g_mol":32.256,"rate_ratio_1_to_2":4,"time_ratio_1_to_2":0.25,"solved_for":"molar_mass_2_g_mol"}

GET https://tttkmbb.com/api/v1/calculate/grahams-law?gas_1_formula=He&gas_2_formula=O2

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/grahams-law (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/grahams-law · Markdown: https://tttkmbb.com/chemistry/grahams-law.md · JSON definition: https://tttkmbb.com/chemistry/grahams-law.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="grahams-law"
OpenAPI operationId: calculate_grahams_law
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

FAQ

Is the ratio the same for diffusion?

Approximately. Graham's law is exact for effusion through a pinhole into vacuum; for diffusion through another gas it gives the right trend but collisions make real rates lower.

I measured times, not rates. What do I enter?

A gas that takes 3 times longer effuses 3 times slower, so rate_ratio_1_to_2 = t2 / t1. The output time_ratio_1_to_2 is the inverse of the rate ratio.

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