van der Waals equation (P + a - V²)(V − b) = RT — physical meaning of a, b

2.4.8 · Chemistry › States of Matter (Quantitative)

WHY correction ki zaroorat hi kyun hai?

WHAT ideal gas assume karta hai (KTG se yaad karo):

Molecules ka apna volume = 0.
Molecules ke beech koi forces nahi (sirf elastic collisions ke alawa).

WHY yeh fail hota hai: High pressure aur low temperature par molecules ek doosre ke karib squeeze ho jaate hain. Ab unki finite size aur mutual attraction ko ignore nahi kiya ja sakta. Isliye $P V / n R T \neq = 1$ (yeh ratio ==compressibility factor $Z$ == hai).

Volume correction ( $b$ ) derive karna

HOW hum ise scratch se nikalte hain:

Ideal gas: molecules poore container volume $V$ mein freely move karte hain. Real gas: har molecule ek hard sphere hai jiska radius $r$ hai. Do molecule-centres kabhi $2 r$ se closer nahi ja sakte.

Do molecules lo. Har ek ke around ek sphere of radius $2 r$ hai jisme koi doosra centre enter nahi kar sakta. Uss "forbidden" sphere ka volume hai $V_{forbidden(pair)} = \frac{4}{3} π (2 r)^{3} = 8 \cdot \frac{4}{3} π r^{3} = 8 v_{mol}$ Yeh step kyun? Kyunki $(2 r)^{3} = 8 r^{3}$ , ek pair ka excluded region ek molecule ke volume ka 8 guna hota hai.

Ise do molecules mein baanto → excluded volume per molecule $= 4 v_{mol}$ .

Ek mole ke liye ( $N_{A}$ molecules): $b = 4 N_{A} \cdot \frac{4}{3} π r^{3} = 4 \times (actual volume of a mole of molecules)$

The correction: actually available space $(V - nb)$ hai, $V$ nahi. Isliye $V \to (V - nb)$ replace karo.

Pressure correction ( $a$ ) derive karna

HOW hum ise scratch se nikalte hain:

Gas ke andar deep koi molecule sabhi directions mein equally pull hota hai → net force zero. Jo molecule wall se takraane wala hota hai, woh peeche ke molecules dwara backward pull hota hai → woh wall se reduced momentum ke saath takraata hai. Isliye observed pressure $P$ , "internal" ideal pressure se kam hoti hai.

$P_{ideal} = P_{observed} + Δ P$

$Δ P$ kitna bada hai? Wall-molecule par inward pull ∝ (usse pull karne wale molecules ki density). Wall-molecules ki jo pull ho rahi hai unki sankhya ∝ (phir se density). Isliye: $Δ P \propto (density)^{2} \propto (\frac{n}{V})^{2}$ Squared kyun? Kyunki attraction ek pair effect hai — jo molecule pull ho raha hai aur jo molecules pull kar rahe hain, dono density ke saath scale karte hain.

Proportionality constant ko $a$ likho: $Δ P = \frac{a n ^{2}}{V ^{2}}$

The correction: true (ideal-jaise) pressure $(P + \frac{a n ^{2}}{V ^{2}})$ hai.

Figure — van der Waals equation (P + a - V²)(V − b) = RT — physical meaning of a, b

Real gases mein $a$ aur $b$ padhna

Compressibility factor $Z$ — fingerprint

$Z = \frac{P V _{m}}{R T}$

$Z = 1$ : ideal.
$Z < 1$ : attractions dominate ( $a$ effect) — gas ideal se zyada compressible hai. Moderate pressures par hota hai.
$Z > 1$ : size/repulsion dominate karta hai ( $b$ effect) — gas kam compressible. Bahut high pressure par hota hai.

Kyun: Low P par molecules door hote hain, attraction $\to$ soft pressure $\to Z < 1$ . High P par molecules jam jaate hain, finite size $\to$ compress karna mushkil $\to Z > 1$ .

Common mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho ek room mein bouncy balls hain jo thodi magnetic bhi hain.

Kyunki har ball moti hai, woh poora room use nahi kar sakti — kuch space hamesha block rehta hai. Woh blocked space $b$ hai.
Kyunki woh magnetic hain, wall ki taraf jaati hui ball apne doston dwara peeche khicch jaati hai, isliye woh zyada halke se hit karti hai jitna expect tha. Woh halkapan $a$ ka pull hai. van der Waals ne bas likh diya "real room = full room minus moti-space" aur "real punch = ideal punch minus magnet-tug." Ho gaya.

Flashcards

Constant

a

physically kya represent karta hai?

Intermolecular attractive forces ki magnitude; bada

a

⇒ zyada strong attractions ⇒ aasaani se liquefy hoga.

Constant

b

physically kya represent karta hai?

Finite molecular size ki wajah se excluded (co-)volume per mole; bada

b

⇒ bade molecules.

Pressure correction

(n / V)^{2}

ke proportional kyun hai?

Attraction ek pair effect hai: jo molecule pull ho raha hai aur jo pull kar rahe hain, dono density ke saath scale karte hain, isliye product ∝ density² hai.

n

moles ke liye van der Waals equation likho.

(P + a n^{2} / V^{2}) (V - nb) = n R T

b

aur actual molecular volume ka kya relation hai?

b = 4 \times

ek mole molecules ka actual volume (ek pair ka excluded volume =

8 v

, jo

4 v

/molecule mein split hota hai).

a

aur

b

ki units kya hain?

a

: atm L² mol⁻²;

b

: L mol⁻¹.

Z < 1

physically kya matlab hai?

Attractions dominate karte hain; gas ideal se zyada compressible hai (moderate pressures par).

Z > 1

physically kya matlab hai?

Molecular size/repulsion dominate karta hai; gas ideal se kam compressible hai (bahut high pressure par).

Low pressure par real vs ideal pressure kya hoti hai?

Real pressure ideal se kam hoti hai kyunki attractions wall collisions ko soft bana dete hain.

Bahut high pressure par kaun sa correction dominate karta hai?

Volume term

b

(finite size), jo

Z > 1

banata hai.

Connections

Kinetic Theory of Gases — woh ideal assumptions jo vdW repair karta hai
Compressibility Factor Z — $a$ aur $b$ $Z$ mein kaise dikhte hain
Critical Constants and Liquefaction — $T_{c}, P_{c}, V_{c}$ ko $a, b$ ke terms mein
Intermolecular Forces — $a$ ka origin (dispersion, dipole, H-bond)
Ideal Gas Law — $a \to 0, b \to 0$ limit
Boyle Temperature — jahan $a$ aur $b$ effects cancel hote hain, $Z \approx 1$