Critical constants Tc, Pc, Vc; law of corresponding states

2.4.9 · Chemistry › States of Matter (Quantitative)

1. Critical constants kya hain?

Meniscus kyun disappear hoti hai: $T_{c}$ ke neeche, ek $P$ – $V$ isotherm par ek flat "liquefaction plateau" dikhta hai (gas + liquid coexist karte hain). Jaise $T \to T_{c}$ , yeh plateau ek single point of inflection tak shrink ho jaata hai — wahan horizontal tangent aur curvature dono vanish ho jaate hain.

Figure — Critical constants Tc, Pc, Vc; law of corresponding states

2. Van der Waals se $T_{c}, P_{c}, V_{c}$ derive karna (scratch se)

Hum shuru karte hain 1 mole ke liye van der Waals equation se:

$(P + \frac{a}{V ^{2}}) (V - b) = R T$

$P$ ke liye solve karo:

$P = \frac{R T}{V - b} - \frac{a}{V ^{2}}$

Critical point ko mathematically kya define karta hai? Yeh isotherm ka inflection point hai, isliye $V$ ke saath $P$ ki dono pehli aur doosri derivatives vanish hoti hain:

$(\frac{\partial P}{\partial V})_{T} = 0, (\frac{\partial P}{\partial V ^{2}})_{T}^{2} = 0$

Step 1 — pehli derivative. Kyun? Plateau ke collapse par tangent horizontal hota hai. $\frac{\partial P}{\partial V} = - \frac{R T}{( V - b ) ^{2}} + \frac{2 a}{V ^{3}} = 0 \Rightarrow \frac{R T}{( V - b ) ^{2}} = \frac{2 a}{V ^{3}} (1)$

Step 2 — doosri derivative. Kyun? Ek inflection ke liye zero curvature bhi chahiye. $\frac{\partial ^{2} P}{\partial V ^{2}} = \frac{2 R T}{( V - b ) ^{3}} - \frac{6 a}{V ^{4}} = 0 \Rightarrow \frac{2 R T}{( V - b ) ^{3}} = \frac{6 a}{V ^{4}} (2)$

Step 3 — (2) ko (1) se divide karo. Kyun? Yeh $R$ , $T$ , $a$ ko khatam kar deta hai aur pure geometry bacha leta hai. $\frac{2/ ( V - b )}{1} = \frac{6 a / V ^{4}}{2 a / V ^{3}} \cdot \frac{( V - b ) ^{2}}{1} \Rightarrow \frac{2}{V - b} = \frac{3}{V}$ $2 V = 3 (V - b) \Rightarrow V_{c} = 3 b$

Step 4 — (1) se $T_{c}$ nikalo. $V_{c} = 3 b$ substitute karo, toh $V_{c} - b = 2 b$ : $\frac{R T _{c}}{( 2 b ) ^{2}} = \frac{2 a}{( 3 b ) ^{3}} = \frac{2 a}{27 b ^{3}} \Rightarrow R T_{c} = \frac{2 a}{27 b ^{3}} \cdot 4 b^{2} = \frac{8 a}{27 b}$ $T_{c} = \frac{8 a}{27 R b}$

Step 5 — $P_{c}$ nikalo $V_{c}$ aur $T_{c}$ ko vdW equation mein plug karke: $P_{c} = \frac{R T _{c}}{V _{c} - b} - \frac{a}{V _{c}^{2}} = \frac{R \cdot \frac{8 a}{27 R b}}{2 b} - \frac{a}{9 b ^{2}} = \frac{8 a}{54 b ^{2}} - \frac{a}{9 b ^{2}} = \frac{4 a}{27 b ^{2}} - \frac{3 a}{27 b ^{2}}$ $P_{c} = \frac{a}{27 b ^{2}}$

3. Critical Compressibility Factor (ek universal number)

Ise compute kyun karein? $Z_{c} = \frac{P _{c} V _{c}}{R T _{c}}$ banao. Ideal gas ke liye $Z = 1$ ; yahan ki value van der Waals ko reality ke against test karti hai.

$Z_{c} = \frac{P _{c} V _{c}}{R T _{c}} = \frac{( \frac{a}{27 b ^{2}} ) ( 3 b )}{R \cdot \frac{8 a}{27 R b}} = \frac{\frac{3 a}{27 b}}{\frac{8 a}{27 b}} = \frac{3}{8} = 0.375$

4. Law of Corresponding States (scratch se)

Idea (KAISE): reduced variables define karo — har property ko uski critical value se divide karke: $P_{r} = \frac{P}{P _{c}}, V_{r} = \frac{V}{V _{c}}, T_{r} = \frac{T}{T _{c}}$

$P = P_{r} P_{c}$ , $V = V_{r} V_{c} = 3 b V_{r}$ , $T = T_{r} T_{c}$ ko vdW equation mein substitute karo:

$(P_{r} \frac{a}{27 b ^{2}} + \frac{a}{( 3 b V _{r} ) ^{2}}) (3 b V_{r} - b) = R T_{r} \frac{8 a}{27 R b}$

Cleverly factor karo. Kyun? Hum chahte hain ki $a$ aur $b$ completely cancel ho jaayein. $\frac{a}{27 b ^{2}} (P_{r} + \frac{3}{V _{r}^{2}}) \cdot b (3 V_{r} - 1) = \frac{8 a}{27 b} T_{r}$

Dono sides par $\frac{a}{27 b}$ cancel ho jaata hai, aur bacha rehta hai:

Yeh kyun matter karta hai: tum ek unknown gas ka behaviour ek generalized compressibility chart ( $Z$ vs $P_{r}$ for various $T_{r}$ ) use karke predict kar sakte ho jo har gas ke liye kaam karta hai — ek chart, saari gases.

5. Worked Examples

6. Common Mistakes (Steel-manned)

7. Flashcards

Critical temperature

T_{c}

kya hai?

Woh highest temperature jis par ek gas ko sirf pressure se liquefy kiya ja sakta hai; uske upar koi liquid phase nahi banta.

Critical point define karne wali do mathematical conditions?

(\partial P / \partial V)_{T} = 0

aur

(\partial^{2} P / \partial V^{2})_{T} = 0

(horizontal tangent ke saath inflection).

Van der Waals

b

ke terms mein

V_{c}

V_{c} = 3 b

a, b, R

ke terms mein

T_{c}

T_{c} = \frac{8 a}{27 R b}

a, b

ke terms mein

P_{c}

P_{c} = \frac{a}{27 b ^{2}}

vdW se predict kiya gaya critical compressibility factor

Z_{c}

Z_{c} = P_{c} V_{c} / R T_{c} = 3/8 = 0.375

(sabhi gases ke liye same).

Sirf

T_{c}

aur

P_{c}

b

kaise nikaalein?

b = R T_{c} / (8 P_{c})

(

T_{c} / P_{c}

divide karo taki

a

cancel ho jaaye).

Reduced variables define karo.

P_{r} = P / P_{c}, V_{r} = V / V_{c}, T_{r} = T / T_{c}

Van der Waals equation ka reduced form?

(P_{r} + 3/ V_{r}^{2}) (3 V_{r} - 1) = 8 T_{r}

Law of Corresponding States state karo.

Same

T_{r}

aur

P_{r}

par gases ka same

V_{r}

(aur same

Z

) hota hai; sabhi gases ek universal reduced equation obey karti hain.

T_{c}

par meniscus kyun vanish hoti hai?

Isotherm par liquefaction plateau ek single inflection point tak shrink ho jaata hai; liquid aur gas densities equal ho jaati hain.

Recall Feynman: 12-saal ke bachche ko explain karo

Steam aur paani ki imagine karo. Normally agar steam ko cool karo ya squeeze karo, toh woh paani ban jaata hai — tum unke beech surface line dekh sakte ho. Lekin agar steam bahut zyada garm ho, toh chahe kitna bhi push karo, woh bas ek thick foggy cheez ban jaata hai, kabhi real paani nahi banta. Woh exact hottest temperature jahan paani ab bhi ban sakta hai, wahi "critical temperature" hai. Cool fact: agar tum har gas ko uske apne special critical numbers ke fraction ke roop mein measure karo, toh sabhi gases identical twins ki tarah behave karne lagti hain — yahi hai Law of Corresponding States.

Connections

Van der Waals Equation of State — woh parent equation jise humne differentiate kiya.
Real Gases and Compressibility Factor Z — $Z_{c}$ aur generalized charts.
Liquefaction of Gases and Andrews Isotherms — $P$ – $V$ isotherms jo plateau dikhate hain.
Ideal Gas Equation — woh $a = b = 0$ limit jahan $Z = 1$ hamesha hota hai.
Intermolecular Forces — $a$ term ka physical origin (attractions jo $T_{c}$ set karte hain).
Supercritical Fluids — critical point se aage pahunchi state.