Graham's Law of Diffusion – From Intuition to Precision
Imagine you're in a room where someone opens a bottle of perfume at one end. You don't smell it instantly — it takes time for the perfume molecules to wander across the room. Now imagine the same experiment with a bottle of ammonia. You'd smell the ammonia much faster. Why? The ammonia molecules are lighter.
That's the core physical idea: lighter gas molecules move faster, on average, than heavier ones at the same temperature. Since diffusion and effusion are processes driven by molecular motion, a lighter gas will spread out (diffuse) or escape through a tiny hole (effuse) more quickly than a heavier gas.
Note
Diffusion is the mixing of gases due to random molecular motion. Effusion is the escape of a gas through a tiny hole into a vacuum. Graham's Law applies to both.
The Precise Statement
Graham's Law of Diffusion/Effusion states:
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).
Mathematically, for two gases A and B:
rBrA=MAMB=ρAρB
where:
r = rate of diffusion/effusion (volume or moles per unit time)
M = molar mass
ρ = density (at same T and P)
r2r1=M1M2
Why the Square Root? (The Physics)
The reason comes from kinetic molecular theory. At a given temperature, the average kinetic energy of gas molecules is the same for all gases:
21mv2=constant
Here m is the mass of one molecule and v is its speed. Rearranging:
v∝m1
Since molar mass M is proportional to molecular mass m, the average molecular speed is inversely proportional to M. And since the rate of diffusion/effusion is directly proportional to this average speed, you get Graham's Law.
Watch out
A common mistake is to invert the ratio. If gas A is lighter (MA<MB), then rA>rB. Check: MB/MA>1, so rA/rB>1 — correct. Always put the lighter gas in the numerator if you want a ratio > 1.
Worked Example
Problem: Hydrogen (M=2g/mol) and oxygen (M=32g/mol) are allowed to effuse through identical pinholes. How much faster does hydrogen effuse?
Graham's law: at the same T and P, the rate of diffusion of a gas is inversely proportional to the square root of its density (or molar mass), so lighter gases diffuse faster.
Diffusion is the tendency of a gas to spread throughout the available space and to mix with another gas.
Statement of Graham's law of diffusion:
At constant temperature and pressure, the rate of diffusion (or effusion) of a gas is inversely proportional to the square root of its density (or its molar mass).
Mathematically:
rate of diffusion is proportional to 1 / sqrt(d)
or rate of diffusion is proportional to 1 / sqrt(M)
(where d = density and M = molar mass of the gas.)
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
CBSE 2023Set ANNUAL1 markMCQ
Q.An unknown gas diffuses at a rate of 0.5 times that of Nitrogen at the same temperature and pressure. The molar mass of the unknown gas is ________.
(a) 114 g mol^-1
(b) 112 g mol^-1
(c) 120 g mol^-1
(d) 110 g mol^-1
›Reveal solutionSolution
Graham's law (rate is proportional to 1/sqrt(M)) applied to the unknown gas vs. nitrogen gives a molar mass of 112 g/mol.
Graham's law of diffusion states that, at the same temperature and pressure, the rate of diffusion of a gas is inversely proportional to the square root of its molar mass:
Q.A bottle of ammonia and a bottle of HCl connected through a long tube are opened simultaneously at both ends. The white ammonium chloride ring will be first formed:
(a) near the ammonia bottle
(b) at the centre of the tube
(c) throughout the length of the tube
(d) near the hydrogen chloride bottle
›Reveal solutionSolution
Graham's law says a lighter gas diffuses faster than a heavier one (rate is inversely proportional to the square root of molar mass). Since NH3 is lighter than HCl, it travels further along the tube in the same time, so the white NH4Cl smoke ring forms closer to the HCl bottle.
When NH3 gas meets HCl gas, they react to form a white solid: NH3(g) + HCl(g) -> NH4Cl(s), visible as a white ring/smoke where the two gases meet.
Graham's law of diffusion: rate of diffusion is inversely proportional to the square root of the molar mass, rate ∝ 1/√M.
Molar mass of NH3 = 17 g/mol; molar mass of HCl = 36.5 g/mol.
Q.The molecules of a gas A travels four times faster than the molecules of gas B at the same temperature. The ratio of molecular weight (MA/MB) will be ________.
(a) 1/4
(b) 1/16
(c) 16
(d) 4
›Reveal solutionSolution
Using Graham's law, uA/uB = 4 gives MA/MB = 1/16.
Graham's law of diffusion/effusion states that the rate (or root-mean-square speed) of gas molecules is inversely proportional to the square root of their molar mass: