Graham's law of effusion - diffusion (rate ∝ 1 - √M)

WHY is it $1/\sqrt{M}$ ? — Derivation from first principles

We derive it, we never memorize it blind.

Figure — Graham's law of effusion - diffusion (rate ∝ 1 - √M)

HOW to use it — rate can be measured many ways

Rate can be expressed as volume/time, moles/time, or distance/time (in a diffusion tube). If two gases effuse for the same time, the amounts are in the ratio of rates:

$\frac{r_1}{r_2}=\frac{n_1/t}{n_2/t}=\frac{n_1}{n_2}=\frac{V_1}{V_2}=\frac{d_1}{d_2}=\sqrt{\frac{M_2}{M_1}}$

Also, since at same $P,T$ density $\rho \propto M$ : $\frac{r_1}{r_2}=\sqrt{\frac{\rho_2}{\rho_1}}$

Common Mistakes (Steel-manned)

Active Recall

Recall Predict before revealing (Forecast-then-Verify)

If gas X effuses 3× faster than gas Y, how do their molar masses compare? → $M_Y/M_X = 3^2 = 9$ , so Y is 9× heavier.
Two gases, same $T$ ; does higher $M$ mean higher or lower $v_{rms}$ ? → Lower.
Rewrite Graham's law using densities. → $\frac{r_1}{r_2}=\sqrt{\rho_2/\rho_1}$ .

Recall Feynman: explain to a 12-year-old

Imagine a room full of ping-pong balls and heavy bowling balls, all kicked with the same energy. The light ping-pong balls zoom around super fast; the heavy bowling balls lumber slowly. Now poke a tiny hole in the wall. The fast ping-pong balls find and slip through the hole way more often, so they leak out first. That's effusion — light gas escapes faster. And because energy links to speed squared, making a ball 4 times heavier only makes it 2 times slower (not 4), which is where the square root comes from.

Flashcards

State Graham's law in words

At constant T and P, the rate of effusion/diffusion of a gas is inversely proportional to the square root of its molar mass.

Graham's law ratio form

r₁/r₂ = √(M₂/M₁)

Why does 1/√M appear (not 1/M)?

Because KE = ½mv² is equal for all gases at same T, so v ∝ 1/√M and rate ∝ v.

Which is faster to effuse, H₂ or O₂, and by how much?

H₂, by a factor of √(32/2) = 4.

Rate in terms of density

r₁/r₂ = √(ρ₂/ρ₁), since ρ ∝ M at same T,P.

A gas effuses at half the rate of O₂ (M=32). Its molar mass?

128 g/mol (0.5 = √(32/M) → M = 128).

Difference between effusion and diffusion

Effusion = escape through a tiny hole into vacuum; diffusion = mixing of gases via random motion. Both obey 1/√M.

If gas A is 4× slower than gas B, mass ratio?

M_A/M_B = 4² = 16, so A is 16× heavier.

Condition required for Graham's law simple form

Same temperature and same pressure for both gases.

Formula for v_rms

v_rms = √(3RT/M).

Connections

Kinetic Theory of Gases — the origin of $v_{rms}$ and equal-KE principle.
Root Mean Square Speed — direct parent formula.
Maxwell-Boltzmann Distribution — why molecules have a spread of speeds.
Ideal Gas Equation — supplies $PV=nRT$ and density $\rho \propto M$ .
Isotope Separation (Uranium Hexafluoride) — real-world use of tiny mass differences.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Graham's law ka core idea bahut simple hai: same temperature par har gas ke molecules ki average kinetic energy same hoti hai (kyunki temperature hi kinetic energy ka measure hai). Ab agar energy same hai aur $KE = \tfrac12 mv^2$ , toh jo molecule halka (kam mass wala) hoga, woh tez bhagega. Isliye halki gas ek chhote se hole se jaldi bahar nikal jaati hai — isko effusion kehte hain.

Formula banta hai $\text{rate} \propto 1/\sqrt{M}$ . Yahan square root important hai — log aksar bhool jaate hain. Reason: energy speed ke square se juda hai, toh mass ko square root ke through speed se link karna padta hai. Isiliye O₂ (mass 32) aur H₂ (mass 2) mein rate ka ratio $\sqrt{32/2} = 4$ aata hai, na ki 16.

Do gases compare karte time formula flip hota hai: $r_1/r_2 = \sqrt{M_2/M_1}$ . Yaad rakhne ka trick — "dusri gas ka mass upar". Sanity check hamesha karo: halki gas ka rate zyada hona chahiye. Ye law real life mein bhi kaam aata hai, jaise uranium isotopes separate karne mein (UF₆ gas ke halke aur bhaari version ko alag karna).

Bas ye dhyan rakho: formula sirf tab simple form mein lagta hai jab temperature aur pressure dono same ho, warna full $v_{rms}=\sqrt{3RT/M}$ use karna padega. Isko ratke mat, derive karke samjho — exam mein confidence rahega.

Graham's law of effusion - diffusion (rate ∝ 1 - √M)

WHAT is happening?

WHY is it $1/\sqrt{M}$ ? — Derivation from first principles

HOW to use it — rate can be measured many ways

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — States of Matter (Quantitative)

Connections