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

WHAT is γ?

WHY does $C_p > C_v$ ? (Derive Mayer's relation)

HOW — from the First Law. For one mole, $dQ = dU + dW = dU + P\,dV$ .

Constant volume: $dV = 0 \Rightarrow dQ = dU$ . So $C_v = \left(\frac{dQ}{dT}\right)_V = \frac{dU}{dT}.$ Why this step? At fixed volume no work happens, so every joule of heat raises internal energy directly.

Constant pressure: $dQ = dU + P\,dV$ . For an ideal gas $PV = RT$ , so at constant $P$ , $P\,dV = R\,dT$ . Thus $C_p = \frac{dU}{dT} + R = C_v + R.$ Why this step? The piston must expand to keep $P$ fixed; that expansion does work $R\,dT$ , which you must supply as extra heat.

WHY does γ depend on degrees of freedom?

Equipartition theorem: each active quadratic degree of freedom holds $\tfrac12 k_B T$ per molecule, i.e. $\tfrac12 R T$ per mole. With $f$ degrees of freedom: $U = \frac{f}{2}RT \quad\Rightarrow\quad C_v = \frac{dU}{dT} = \frac{f}{2}R.$

Then $C_p = C_v + R = \left(\frac f2 + 1\right)R$ , and

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

The three cases

Gas type	Translational	Rotational	$f$	$C_v$	$C_p$	$\gamma=1+2/f$
Monatomic (He, Ne, Ar)	3	0	3	$\tfrac32 R$	$\tfrac52 R$	$5/3\approx1.67$
Diatomic (O₂, N₂, H₂)	3	2	5	$\tfrac52 R$	$\tfrac72 R$	$7/5=1.40$
Polyatomic (CO₂ linear*, H₂O, CH₄)	3	3	6	$3R$	$4R$	$4/3\approx1.33$

Worked examples

Common mistakes

Recall Feynman: explain to a 12-year-old

Imagine a toy that can wiggle. A marble (monatomic) can only roll around — one kind of motion. A pencil (diatomic) can roll AND spin end-over-end — more ways to move. A spinning ball of dough (polyatomic) can tumble in every direction. Heat is like energy you pour into the toy. The toy with MORE ways to move shares the heat among all its motions, so its "temperature" (rolling speed) goes up slower. γ is just a score: the simplest toy (marble) gets the biggest γ = 1.67, the fanciest toy gets the smallest γ ≈ 1.33.

Active-recall flashcards

#flashcards/physics

Define γ

The ratio

C_p/C_v

of molar heat capacity at constant pressure to that at constant volume; always > 1.

Why is

C_p>C_v

At constant P the gas also does expansion work (

R\,dT

per mole) which needs extra heat.

State Mayer's relation

C_p - C_v = R

(for an ideal gas).

Formula for γ from degrees of freedom

\gamma = 1 + 2/f

where

f

is active degrees of freedom.

γ for monatomic gas

5/3 \approx 1.67

(f = 3).

γ for diatomic gas (room T)

7/5 = 1.40

(f = 5).

γ for polyatomic (non-linear)

4/3 \approx 1.33

(f = 6).

C_v

of diatomic gas

\tfrac52 R \approx 20.8

J/mol·K.

Given γ, how to find f?

f = 2/(\gamma-1)

Why is f = 5 (not 7) for diatomic at room T?

Vibrational DOF are quantum-frozen at ordinary temperatures.

Equipartition energy per DOF per mole

\tfrac12 RT

Why does diatomic have only 2 rotational DOF?

Rotation about the bond axis has negligible moment of inertia, so it's frozen out.

Connections

First Law of Thermodynamics — source of $dQ = dU + P\,dV$
Degrees of Freedom & Equipartition Theorem — gives $C_v = \tfrac f2 R$
Adiabatic Process — uses $PV^\gamma = \text{const}$ , γ defined here
Speed of Sound in Gas — $v=\sqrt{\gamma RT/M}$ depends directly on γ
Internal Energy of Ideal Gas — $U = \tfrac f2 RT$
Mayer's Relation

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, γ = Cp/Cv ka matlab simple hai: gas ko garam karne ke do tareeke hain. Agar volume fixed rakho (Cv), to saari heat sidha internal energy badhane mein jaati hai. Agar pressure fixed rakho (Cp), to gas ko expand bhi karna padta hai, piston ko dhakka dena padta hai, isliye thodi extra heat chahiye — tabhi Cp hamesha Cv se bada hota hai. Yehi reason hai ki γ hamesha 1 se zyada hota hai. Ye extra heat exactly R ke barabar hoti hai, isko Mayer's relation kehte hain: $C_p - C_v = R$ .

Ab main point: γ batata hai ki molecule ke paas kitne "degrees of freedom" (f) hain — yaani energy store karne ke kitne tareeke. Formula yaad rakho: $\gamma = 1 + 2/f$ . Monatomic gas (jaise Helium, Argon) sirf idhar-udhar bhaag sakti hai (3 translational), to f=3, γ=5/3=1.67. Diatomic (O₂, N₂) ek dumbbell ki tarah ghoom bhi sakti hai (2 rotation extra), to f=5, γ=1.40. Polyatomic (paani, methane) har direction mein tumble karti hai (3 rotation), f=6, γ=1.33.

Notice karo — molecule jitna fancy, utne zyada boxes mein energy bat jaati hai, isliye temperature dheere badhti hai, aur γ utna hi 1 ke kareeb aata hai. Exam mein agar γ diya ho to ulta f nikaal sakte ho: $f = 2/(\gamma-1)$ . Yeh γ aage adiabatic process ( $PV^\gamma$ = const) aur sound ki speed ( $v=\sqrt{\gamma RT/M}$ ) mein kaam aata hai, isliye yeh chhota sa number bahut powerful hai. Bas ek cheez yaad rakhna: room temperature pe vibration "frozen" hoti hai, isliye diatomic ke liye f=5 lete hain, 7 nahi.

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

WHAT is γ?

WHY does $C_p > C_v$ ? (Derive Mayer's relation)

WHY does γ depend on degrees of freedom?

The three cases

Worked examples

Common mistakes

Active-recall flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Thermodynamics

Connections