Rate of diffusion of a gas is (a) directly proportional to its density (b) directly proportional to its molecular weight (c) directly proportional to its square root of its molecular weight (d) inversely proportional to the square root of its molecular weight

Rate of diffusion of a gas is (a) directly proportional to its density (b) directly proportional to its molecu

Choose the Best Answer · Q2

Q. Rate of diffusion of a gas is

(a) directly proportional to its density

(b) directly proportional to its molecular weight

(d) inversely proportional to the square root of its molecular weight

Puducherry TnboardTextbookSubjectiveImportance★★★★★

4% · 3/72 Questions

✓ Free question

Concept understanding — Graham's Law of Diffusion

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:

\frac{r_A}{r_B} = \sqrt{\frac{M_B}{M_A}} = \sqrt{\frac{\rho_B}{\rho_A}}

where:

$r$ = rate of diffusion/effusion (volume or moles per unit time)
$M$ = molar mass
$\rho$ = density (at same T and P)

\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}

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:

\frac{1}{2} m v^2 = \text{constant}

Here $m$ is the mass of one molecule and $v$ is its speed. Rearranging:

v \propto \frac{1}{\sqrt{m}}

Since molar mass $M$ is proportional to molecular mass $m$ , the average molecular speed is inversely proportional to $\sqrt{M}$ . 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 ( $M_A < M_B$ ), then $r_A > r_B$ . Check: $\sqrt{M_B/M_A} > 1$ , so $r_A/r_B > 1$ — correct. Always put the lighter gas in the numerator if you want a ratio > 1.

Worked Example

Problem: Hydrogen ( $M = 2\ \text{g/mol}$ ) and oxygen ( $M = 32\ \text{g/mol}$ ) are allowed to effuse through identical pinholes. How much faster does hydrogen effuse?

Solution:

\frac{r_{H_2}}{r_{O_2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4

Hydrogen effuses 4 times faster than oxygen.

Tip

For quick comparisons, remember that the ratio of rates is the square root of the inverse ratio of molar masses. Lighter = faster, by the square root factor.

Key Points for Exams

Conditions matter: Graham's Law holds strictly only at constant temperature and pressure, and for gases behaving ideally.
Rate can be measured as volume per unit time, moles per unit time, or distance travelled per unit time in a diffusion tube.
Density form: Since density $\rho \propto M$ at fixed T and P, you can use $\sqrt{\rho_2/\rho_1}$ directly.
For mixtures: Graham's Law is used in uranium enrichment — separating $^{235}\text{UF}_6$ from $^{238}\text{UF}_6$ by effusion, though the mass difference is tiny.

Important

Graham's Law is a direct consequence of the fact that all gases have the same average kinetic energy at the same temperature. This is the single most important idea to remember — the rest is just algebra.

Graham's law of diffusion is a standard NCERT/CBSE Class 11 Chemistry topic, and "Graham's law of diffusion formula and numericals" is a commonly searched query during board exam preparation. It's also a frequent JEE Main and NEET important-question type, often used to compare effusion rates of two gases.

Step 1. Graham's law of diffusion states $\text{rate}\propto\dfrac{1}{\sqrt{M}}$ , where M is the molar (molecular) mass of the gas.

Step 2. This directly rules out (a) and (b), which claim a direct proportionality to density or molecular weight, and (c), which claims direct proportionality to the square root of molecular weight (the correct relationship is inverse, not direct).

✓Final answer

(d) inversely proportional to the square root of its molecular weight

←Previous question · evaluationQ1 — Gases deviate from ideal behavior at high pressure. Which of the follo…→Next question · evaluationQ3 — Which of the following is the correct expression for the equation of s…

Q.Rate of diffusion of a gas is

🎓Unlock everything free for 14 days

Unlock everything free for 14 days