Graham's Law of Diffusion

When a bottle of perfume is opened across a room, its scent reaches you within seconds — this everyday phenomenon is diffusion, the spontaneous mixing of gas molecules driven purely by their kinetic energy. A closely related process, effusion, describes how gas molecules escape through a tiny pinhole into a vacuum. Both processes obey a beautiful quantitative relationship discovered by the Scottish chemist Thomas Graham in 1848: the rate at which a gas diffuses or effuses is inversely proportional to the square root of its molar mass. Lighter molecules move faster and therefore spread more quickly than heavier ones. Graham's Law emerges directly from the kinetic molecular theory of gases. At a fixed temperature, all gas molecules share the same average kinetic energy (½mv²). Because heavier molecules must have lower average speeds to possess the same kinetic energy, they diffuse more slowly. This means two gases at the same temperature and pressure can be compared quantitatively: doubling the molar mass reduces the rate by a factor of √2 ≈ 1.41. The law applies rigorously to effusion and serves as an excellent approximation for diffusion in the absence of intermolecular collisions. Graham's Law has profound practical significance beyond classroom problems. During the Manhattan Project and in modern nuclear power, uranium hexafluoride gas ( $\ce{UF6}$ ) is passed repeatedly through porous barriers to separate the fissile isotope $\ce{^{235}U}$ from the more abundant $\ce{^{238}U}$ . Because the two isotopic forms differ by only three atomic mass units, thousands of sequential diffusion stages are required to achieve weapons-grade or reactor-grade enrichment — a direct consequence of the square-root relationship in Graham's Law.

Key takeaways

Kinetic Energy Foundation
Deriving the Speed Ratio
Graham's Law Statement
Time vs. Rate Relationship
Isotope Separation — A Real-World Case
Diffusion vs. Effusion — A Subtle Distinction

Kinetic Energy Foundation

The kinetic molecular theory states that for any two ideal gases A and B at the same temperature T, their average kinetic energies are equal: ½M_A v_A² = ½M_B v_B², where M is molar mass (kg mol⁻¹) and v is the root-mean-square speed. This equality is the thermodynamic engine behind Graham's Law.

Deriving the Speed Ratio

Rearranging the equal-kinetic-energy condition: v_A²/v_B² = M_B/M_A. Taking the square root of both sides gives v_A/v_B = √(M_B/M_A). Since the rate of effusion is proportional to the average molecular speed (rate ∝ v), the ratio of effusion rates equals the inverse square root of the ratio of molar masses.

Graham's Law Statement

For gases A and B at the same temperature and pressure: Rate_A / Rate_B = √(M_B / M_A). Equivalently, Rate ∝ 1/√M for a single gas compared to a standard. A lighter gas (smaller M) always has a higher rate. Note: rate can be measured as volume of gas effusing per unit time, or as the inverse of the time taken to effuse a fixed volume.

Time vs. Rate Relationship

Because rate and time are inversely related (if gas A effuses twice as fast, it takes half the time), an equivalent form is: t_A / t_B = √(M_A / M_B). This version is extremely useful in problems where experimental data gives the time taken for each gas to pass through a small orifice.

Isotope Separation — A Real-World Case

For uranium enrichment, the two gases are $\ce{^{235}UF6}$ (M = 235 + 6×19 = 349 g mol⁻¹) and $\ce{^{238}UF6}$ (M = 238 + 114 = 352 g mol⁻¹). The separation factor per stage = √(352/349) = √1.0086 ≈ 1.0043. This tiny factor means that to enrich natural uranium (0.72% $\ce{^{235}U}$ ) to ~3–5% for reactor fuel requires roughly 1,000–2,000 cascade stages, and ~100,000 stages for weapons-grade material (>90%).

Diffusion vs. Effusion — A Subtle Distinction

Effusion is the passage of gas through a single tiny orifice (diameter much smaller than the mean free path) into a vacuum. Graham's Law is exact here. Diffusion is the mixing of two gases; it is complicated by intermolecular collisions. Graham's Law still predicts the correct trend for diffusion but is only an approximation in practice because collision frequency and mean free path also depend on molecular size and intermolecular forces.

The Key Equation

\frac{\text{Rate}_A}{\text{Rate}_B} = \sqrt{\frac{M_B}{M_A}}

\text{Rate}_A — Effusion rate of gas A (mol s⁻¹ or L s⁻¹): Amount or volume of gas A passing through the orifice per unit time under constant conditions.
\text{Rate}_B — Effusion rate of gas B (mol s⁻¹ or L s⁻¹): Amount or volume of gas B passing through the same orifice per unit time at the same temperature and pressure.
M_A — Molar mass of gas A (g mol⁻¹): Mass of one mole of gas A molecules; determines average molecular speed at a given temperature.
M_B — Molar mass of gas B (g mol⁻¹): Mass of one mole of gas B molecules. Note that M_B appears in the numerator: the heavier gas B appears under the square root in the numerator, giving gas A a higher rate when M_A < M_B.

Worked Example

Problem. A sample of hydrogen gas ( $\ce{H2}$ , M = 2.016 g mol⁻¹) and an unknown gas X both effuse through a porous plug. It takes $\ce{H2}$ 45 seconds to effuse a fixed volume, and gas X takes 180 seconds to effuse the same volume. (a) Calculate the molar mass of gas X. (b) Identify gas X from: $\ce{O2}$ (32 g mol⁻¹), $\ce{CO2}$ (44 g mol⁻¹), $\ce{SO2}$ (64 g mol⁻¹).

M_X = \left(\frac{t_X}{t_{H_2}}\right)^2 \times M_{H_2} = \left(\frac{180}{45}\right)^2 \times 2.016 \approx 32.3 \text{ g mol}^{-1}

Answer. Molar mass of gas X ≈ 32.3 g mol⁻¹; Gas X is oxygen, O₂.

Figure

Diagram

Origins · 1 minThomas Graham and how this was discovered

Thomas Graham

1805–1869 · Scottish

Thomas Graham was a Scottish chemist and the founding father of colloid chemistry. In 1833 he published his law of effusion and diffusion, establishing the inverse square-root relationship between gas speed and molar mass. He also discovered dialysis — the passage of crystalloids but not colloids through semipermeable membranes — laying the groundwork for kidney dialysis machines used in medicine today. He served as Master of the Mint from 1855 until his death. Graham coined the term 'colloid' (from the Greek for 'glue') to describe substances like gelatin and starch that cannot pass through membranes — a discovery he made by observing that his diffusion apparatus filtered out certain substances entirely.

Kinetic Molecular Theory of Gases — Graham's Law is derived directly from the postulate of KMT that all gases have the same average kinetic energy at the same temperature, which forces lighter molecules to move faster.
The Ideal Gas Law (PV = nRT) — Graham's Law assumes ideal gas behaviour; understanding PV = nRT provides the thermodynamic context for why temperature and pressure must be held constant when comparing effusion rates.
Root-Mean-Square Speed of Gas Molecules — The RMS speed formula v_rms = √(3RT/M) is the quantitative expression of why heavier gases move slower and directly gives the mathematical basis for Graham's Law.
Isotope Separation and Uranium Enrichment — The most important industrial application of Graham's Law is the gaseous diffusion separation of ²³⁵UF₆ from ²³⁸UF₆ for nuclear fuel and weapons programmes.
Mean Free Path and Molecular Collisions — The distinction between ideal effusion (orifice smaller than mean free path) and bulk diffusion depends on mean free path; understanding this clarifies when Graham's Law applies exactly versus approximately.