Heat capacities Cp, Cv; relationship Cp − Cv = nR (ideal gas)

2.5.6 · Chemistry › Thermodynamics (Chemical)

Derivation of Cp − Cv = nR (Ideal Gas)

Step 1: First Law Foundation

First law of thermodynamics se shuru karo: $d U = δ Q - δ W = δ Q - P d V$

Ek ideal gas ke liye, internal energy sirf temperature par depend karti hai: $d U = n C_{v} d T$

KYUN? Ideal gases mein molecules interact nahi karte (unke beech koi potential energy nahi), isliye $U$ sirf kinetic energy par → sirf $T$ par depend karta hai.

Step 2: Enthalpy Definition

Enthalpy ko is tarah define kiya jaata hai: $H = U + P V$

Differential lo: $d H = d U + P d V + V d P$

Substitute karo $d U = n C_{v} d T$ aur rearrange karo: $d H = n C_{v} d T + P d V + V d P$

Step 3: Apply Constant Pressure Condition

Constant pressure par ( $d P = 0$ ): $d H = n C_{v} d T + P d V$

Lekin definition ke anusaar, constant pressure par: $d H = n C_{p} d T$

In dono ko equate karo: $n C_{p} d T = n C_{v} d T + P d V$

Step 4: Use Ideal Gas Law

Ideal gas ke liye constant pressure par, $P V = n R T$ ko differentiate karo: $P d V = n R d T$

KYUN? Kyunki $P$ constant hai, $d (P V) = P d V = d (n R T) = n R d T$ .

Step 5: Substitute and Simplify

Step 3 mein $P d V$ replace karo: $n C_{p} d T = n C_{v} d T + n R d T$

$n d T$ se divide karo:

Physical Interpretation

nR term woh work represent karta hai jo gas constant pressure par heat hone ke dauran expansion mein karta hai.

Constant V par: Koi expansion nahi → $Q = Δ U = n C_{v} Δ T$
Constant P par: Expansion hota hai → $Q = Δ H = n C_{p} Δ T = n C_{v} Δ T + n R Δ T$

Extra $n R Δ T$ woh $P Δ V$ work energy hai jo external pressure ke against push karne mein "bahar nikal jaati hai".

Heat Capacity Ratio γ (Gamma)

Adiabatic index ya heat capacity ratio: $γ = \frac{C _{p}}{C _{v}} = \frac{C _{v} + R}{C _{v}} = 1 + \frac{R}{C _{v}}$

Ideal gases ke liye values:

Monatomic (He, Ar): $γ = 5/3 \approx 1.67$
Diatomic (N₂, O₂): $γ = 7/5 = 1.40$
Polyatomic (CO₂, CH₄): $γ \approx 1.29 - 1.33$

γ KYUN GHATTA HAI? Zyada degrees of freedom (rotational, vibrational) $C_{v}$ badhate hain, isliye ratio $C_{p} / C_{v}$ ghatta hai.

Memory Aids

Recall Feynman Technique: Ek 12-Saal Ke Bachche Ko Samjhao

Socho tumhare paas hawa se bhara ek balloon hai. Tum andar ki hawa ko garam karna chahte ho.

Method 1: Balloon ko tight pakad lo taaki woh expand na ho sake. Jab tum heat add karte ho, saari energy hawa ke molecules ko tez chalane (garam hone) mein jaati hai.

Method 2: Heat karte waqt balloon ko freely expand hone do. Ab tumhari kuch heat energy balloon ko bada push karne mein "waste" hoti hai sirf molecules ko tez chalane ki jagah. Isliye tumhe utna hi temperature badhane ke liye ZYADA heat chahiye!

Method 2 mein jo "extra heat" chahiye woh hamesha same amount hoti hai per molecule ke liye: isse $R$ (gas constant) kehte hain. Isliye scientists kehte hain $C_{p} - C_{v} = R$ . Yeh "expansion tax" hai—gas ko garam karte waqt bada hone dene ki jo price tum pay karte ho!

Connections

First Law of Thermodynamics — $C_{p} - C_{v}$ derive karne ki foundation
Enthalpy — Kyun $C_{p}$ mein enthalpy aata hai; relationship $H = U + P V$
Ideal Gas Law — $P d V$ ko $n R d T$ se relate karne ke liye use hota hai
Adiabatic Process — Equation $P V^{γ} = const$ mein $γ = C_{p} / C_{v}$ use hota hai
Degrees of Freedom — $C_{v}$ ki values determine karta hai (equipartition theorem)
Internal Energy — Ideal gases ke liye, $d U = n C_{v} d T$ hamesha
Isothermal vs Adiabatic Processes — Heat capacities process behavior determine karte hain
Real Gases (van der Waals) — $C_{p} - C_{v} = R$ se deviations

#flashcards/chemistry

Cv (molar heat capacity at constant volume) kya hota hai? :: Woh heat jo constant volume par 1 mole substance ka temperature 1 K badhane ke liye chahiye; equals (∂U/∂T)_V. Ideal gases ke liye, saari heat internal energy ban jaati hai kyunki koi work nahi hota.

Cp (molar heat capacity at constant pressure) kya hota hai? :: Woh heat jo constant pressure par 1 mole substance ka temperature 1 K badhane ke liye chahiye; equals (∂H/∂T)_P. Cv se zyada heat chahiye kyunki gas expansion work karta hai.

Ideal gases ke liye Mayer's relation batao.

Cp − Cv = R (1 mole ke liye) ya Cp − Cv = nR (n moles ke liye). Difference R expansion work per mole per kelvin represent karta hai.

Gases ke liye Cp hamesha Cv se zyada kyun hota hai?

Constant pressure par, gas heating ke dauran expand hota hai, work karta hai W = PΔV = nRΔT. Yeh work energy internal energy increase se "extra" hoti hai, isliye zyada heat chahiye: Cp = Cv + R.

Ideal gas ke liye Cv ki terms mein ΔU kya hota hai?

ΔU = nCvΔT, kisi bhi process ke liye valid hai (sirf constant volume ke liye nahi), kyunki ideal gases ke liye U sirf T par depend karta hai.

Heat capacity ratio γ kya hota hai?

γ = Cp/Cv; monatomic gases ke liye γ = 5/3, diatomic ke liye γ = 7/5. Adiabatic process equations mein use hota hai: PV^γ = constant.

Kya Cp − Cv = R liquids aur solids ke liye valid hai?

Nahi, yeh sirf ideal gases ke liye exact hai. Condensed phases ke liye, Cp − Cv = TVα²/κ_T (usually negligible), isliye Cp ≈ Cv.

2 moles He (Cv = 3R/2) ke liye, constant P par ΔT = 50 K ke liye heat kya hogi?

Pehle Cp = Cv + R = 5R/2 nikalo. Phir Q = nCpΔT = 2(5R/2)(50) = 250R = 2078 J.

Qp − Qv mein nRΔT term physically kya represent karta hai?

Expansion work W = PΔV = nRΔT jo gas constant pressure par heat hone par karta hai.