γ = Cp - Cv — for monatomic, diatomic, polyatomic

1.7.17 · Physics › Thermodynamics

WHAT is γ?

WHY does $C_{p} > C_{v}$ ? (Derive Mayer's relation)

HOW — First Law se. Ek mole ke liye, $d Q = d U + d W = d U + P d V$ .

Constant volume: $d V = 0 \Rightarrow d Q = d U$ . To $C_{v} = (\frac{d Q}{d T})_{V} = \frac{d U}{d T} .$ Ye step kyun? Fixed volume pe koi work nahi hota, isliye heat ka har joule seedha internal energy badhata hai.

Constant pressure: $d Q = d U + P d V$ . Ideal gas ke liye $P V = R T$ , to constant $P$ pe, $P d V = R d T$ . Is tarah $C_{p} = \frac{d U}{d T} + R = C_{v} + R .$ Ye step kyun? $P$ fixed rakhne ke liye piston ko expand karna padta hai; wo expansion work $R d T$ karta hai, jo tumhe extra heat ke roop mein supply karni padti hai.

WHY does γ depend on degrees of freedom?

Equipartition theorem: har active quadratic degree of freedom $\frac{1}{2} k_{B} T$ energy store karta hai per molecule, yaani $\frac{1}{2} R T$ per mole. $f$ degrees of freedom ke saath: $U = \frac{f}{2} R T \Rightarrow C_{v} = \frac{d U}{d T} = \frac{f}{2} R .$

Phir $C_{p} = C_{v} + R = (\frac{f}{2} + 1) R$ , aur

Figure — γ = Cp - Cv — for monatomic, diatomic, polyatomic

The three cases

Gas type	Translational	Rotational	$f$	$C_{v}$	$C_{p}$	$γ = 1 + 2/ f$
Monatomic (He, Ne, Ar)	3	0	3	$\frac{3}{2} R$	$\frac{5}{2} R$	$5/3 \approx 1.67$
Diatomic (O₂, N₂, H₂)	3	2	5	$\frac{5}{2} R$	$\frac{7}{2} R$	$7/5 = 1.40$
Polyatomic (CO₂ linear*, H₂O, CH₄)	3	3	6	$3 R$	$4 R$	$4/3 \approx 1.33$

Worked examples

Common mistakes

Recall Feynman: 12-saal ke bachhe ko samjhao

Socho ek aisa khilona jo hil-jul sakta hai. Ek marble (monatomic) sirf idhar-udhar roll kar sakta hai — ek hi tarah ki motion. Ek pencil (diatomic) roll bhi kar sakti hai AUR end-over-end spin bhi — zyada tareekey hain hilne ke. Ek ghoomti hui atta ki loi (polyatomic) har direction mein tumble kar sakti hai. Heat ek tarah ki energy hai jo tum khilone mein dalte ho. Jis khilone ke paas zyada hilne ke tareekey hain, wo heat ko saari motions mein baant leta hai, isliye uska "temperature" (rolling speed) dheere badhta hai. γ bas ek score hai: sabse simple khilone (marble) ko sabse bada γ = 1.67 milta hai, sabse fancy khilone ko sabse chhota γ ≈ 1.33 milta hai.

Active-recall flashcards

#flashcards/physics

γ define karo

Molar heat capacity at constant pressure aur constant volume ka ratio

C_{p} / C_{v}

; hamesha > 1 hota hai.

C_{p} > C_{v}

kyun hai?

Constant P pe gas expansion work bhi karti hai (

R d T

per mole) jiske liye extra heat chahiye.

Mayer's relation batao

C_{p} - C_{v} = R

(ideal gas ke liye).

Degrees of freedom se γ ka formula

γ = 1 + 2/ f

jahan

f

active degrees of freedom hain.

Monatomic gas ke liye γ

5/3 \approx 1.67

(f = 3).

Diatomic gas ke liye γ (room T)

7/5 = 1.40

(f = 5).

Polyatomic (non-linear) ke liye γ

4/3 \approx 1.33

(f = 6).

Diatomic gas ka

C_{v}

\frac{5}{2} R \approx 20.8

J/mol·K.

γ diya ho to f kaise nikaalein?

f = 2/ (γ - 1)

Diatomic ke liye room T pe f = 5 (7 nahi) kyun?

Ordinary temperatures pe vibrational DOF quantum-frozen hote hain.

Equipartition energy per DOF per mole

\frac{1}{2} R T

Diatomic ke sirf 2 rotational DOF kyun hain?

Bond axis ke around rotation ka moment of inertia negligible hai, isliye wo frozen out ho jaata hai.

Connections

First Law of Thermodynamics — $d Q = d U + P d V$ ka source
Degrees of Freedom & Equipartition Theorem — $C_{v} = \frac{f}{2} R$ deta hai
Adiabatic Process — $P V^{γ} = const$ use karta hai, γ yahan define hota hai
Speed of Sound in Gas — $v = γ R T / M$ seedha γ pe depend karta hai
Internal Energy of Ideal Gas — $U = \frac{f}{2} R T$
Mayer's Relation