2.4.4
Intuition Ek-line pictureHalke gas molecules average par zyada fast hote hain (same temperature = same average kinetic energy), isliye woh ek tiny hole se escape karte hain ya zyada jaldi spread out karte hain. Rate of effusion inversely proportional to the square root of molar mass hoti hai.
Effusion ::= gas molecules ka tiny pinhole ke through vacuum mein escape karna, ek ek karke, bina hole mein collisions ke.Diffusion ::= ek gas ka doosri gas mein mixing/spreading — random molecular motion ki wajah se.Rate ::= unit time mein kitni gas pass hoti hai (moles/s, volume/s, ya distance/time).
Graham's law (1846): same temperature aur pressure par,
Rate ∝ M 1 M
Hum isse derive karte hain, blindly memorize nahi karte.
Rate ko volume/time , moles/time , ya distance/time (diffusion tube mein) ke roop mein express kar sakte hain. Agar do gases same time ke liye effuse karein, toh amounts, rates ke ratio mein hongi:
r 2 r 1 = n 2 / t n 1 / t = n 2 n 1 = V 2 V 1 = d 2 d 1 = M 1 M 2
Aur, same P , T ρ ∝ M r 2 r 1 = ρ 1 ρ 2
Worked example Example 1 — H₂ vs O₂H₂, O₂ se kitni zyada fast effuse karta hai?
r O 2 r H 2 = M H 2 M O 2 = 2 32 = 16 = 4 Yeh step kyun? Doosri gas ka mass upar rakho. H₂ 4× faster effuse karta hai. Sahi lagta hai: H₂, 16× lighter hai, aur 16 = 4
Worked example Example 2 — Effuse hone ka timeGas A (M = 16 M = 64 r B r A = 16 64 = 2 V r = V / t t ∝ 1/ r t B = 2 × t A = 40 Yeh step kyun? Slow gas ⇒ zyada time. Time aur rate inverse hain.
Worked example Example 3 — Unknown molar mass nikalnaEk unknown gas, O 2 M = 32 M r O 2 r u nk = 0.5 = M 32 ⟹ 0.25 = M 32 ⟹ M = 128 g/mol Yeh step kyun? Root hatane ke liye dono sides ko square karo, phir solve karo. Jo gas 4× slow ho woh 16× heavier hogi — check: 32 × ... 0.5 32 × 4 = 128
Worked example Example 4 — Diffusion tube (NH₃ vs HCl)Ek tube mein, NH₃ (M = 17 M = 36.5 d H C l d N H 3 = 17 36.5 = 2.15 ≈ 1.47 HCl end ke paas banti hai.
Yeh step kyun? Halka NH₃ faster move karta hai, isliye same time mein zyada distance cover karta hai.
Common mistake Same gas ka mass upar rakhnaGalat idea: r 2 r 1 = M 2 M 1 Yeh sahi kyun lagta hai: Hum instinctively subscript 1 ko subscript 1 ke saath rakhte hain.
Fix: Rate, mass ka inverse hai, isliye masses swap hone chahiye: r 2 r 1 = M 1 M 2
Common mistake Square root bhool jaanaGalat idea: rate ∝ 1/ M Yeh sahi kyun lagta hai: "heavier = slower" linear lagta hai.
Fix: KE ∝ v 2 speed squared se linked hai; root lene par v ∝ 1/ M
Common mistake "Same T and P" ignore karnaGalat idea: Graham's law ko alag temperatures wali gases par apply karna.
Yeh sahi kyun lagta hai: formula mein sirf masses dikhte hain.
Fix: 3 R T tabhi cancel hota hai jab T . Alag T v r m s
Common mistake Effusion aur diffusion speeds in air ko confuse karnaFix: Dono same 1/ M ratio phir bhi sahi rehta hai, actual speed nahi.
Recall Reveal se pehle predict karo (Forecast-then-Verify)
Agar gas X, gas Y se 3× faster effuse kare, toh unke molar masses ka comparison kya hoga?
→ M Y / M X = 3 2 = 9 9× heavier hai.
Do gases, same T M v r m s Lower .
Graham's law ko densities use karke likho. → r 2 r 1 = ρ 2 / ρ 1
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho ek kamre mein ping-pong balls aur bhaari bowling balls hain, sabko same energy se kick kiya gaya. Halki ping-pong balls bahut fast ghoomti hain; bhaari bowling balls dheere dheere chalti hain. Ab wall mein ek tiny hole karo. Fast ping-pong balls us hole ko zyada baar dhundhti aur slip through karti hain, isliye woh pehle leak out karti hain. Yahi effusion hai — halki gas zyada fast escape karti hai. Aur kyunki energy, speed squared se linked hai, ek ball ko 4 guna heavier banana use sirf 2 guna slow karta hai (4 guna nahi), yahi woh jagah hai jahan se square root aata hai.
"Light and lively, heavy and lazy."
Flipped formula ke liye: "Rate 1 over Root of the other one's Mass." Odd-one-out gas ka mass upar jaata hai.
Graham's law ko words mein batao Constant T aur P par, kisi gas ki rate of effusion/diffusion, uski molar mass ke square root ke inversely proportional hoti hai.
Graham's law ratio form r₁/r₂ = √(M₂/M₁)
1/√M kyun aata hai (1/M kyun nahi)? Kyunki KE = ½mv² sabhi gases ke liye same T par equal hoti hai, isliye v ∝ 1/√M aur rate ∝ v.
H₂ ya O₂ — kaun faster effuse karta hai, aur kitna? H₂, √(32/2) = 4 ke factor se.
Rate in terms of density r₁/r₂ = √(ρ₂/ρ₁), kyunki same T,P par ρ ∝ M.
Ek gas, O₂ (M=32) ki half rate se effuse karti hai. Uski molar mass? 128 g/mol (0.5 = √(32/M) → M = 128).
Effusion aur diffusion mein farq Effusion = vacuum mein tiny hole ke through escape; diffusion = random motion se gases ka mixing. Dono 1/√M follow karte hain.
Agar gas A, gas B se 4× slow ho, toh mass ratio kya hai? M_A/M_B = 4² = 16, toh A, 16× heavier hai.
Graham's law ke simple form ke liye kya condition chahiye? Dono gases ka same temperature aur same pressure hona zaroori hai.
v_rms ka formula v_rms = √(3RT/M).
Kinetic Theory of Gases — v r m s Root Mean Square Speed — direct parent formula.Maxwell-Boltzmann Distribution — molecules mein speeds ka spread kyun hota hai.Ideal Gas Equation — P V = n R T ρ ∝ M Isotope Separation (Uranium Hexafluoride) — tiny mass differences ka real-world use.
Rate proportional to v_rms
Graham's law rate ∝ 1 / sqrt M
Measured as volume, moles or distance per time
H2 effuses 4x faster than O2
