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

1.7.17 · D1 · Physics › Thermodynamics › γ = Cp - Cv — for monatomic, diatomic, polyatomic

Is page pe kuch bhi assumed nahi hai. Parent note γ = Cp/Cv padhne se pehle, tumhe har woh letter fluently samajhna hoga jo wahan use hua hai. Hum unhe ek ek karke build karte hain, har ek pichle pe top karke.

1. Heat, $Q$ — transit mein energy

Picture. Socho ek gas ek cylinder mein band hai jiske neeche ek flame hai. Flame energy ko metal wall se guzaarke gas mein dhakelta hai. Woh flowing energy hi $Q$ hai.

Topic ko iske kya zarurat hai. Har heat-capacity idea is sawaal se shuru hota hai: "agar main itna heat $dQ$ daalu, toh kya hoga?" Toh $Q$ woh input hai jise hum track karte hain.

2. Temperature $T$ — average jiggle

Picture. Upar wali figure mein, tez arrows = zyada garam gas = zyada $T$. Temperature total energy nahi hai — yeh har molecule ki average speediness hai.

Topic ko iske kya zarurat hai. Heat capacity poochti hai "kitni heat temperature ko ek degree badhati hai?" — toh sawal pose karne ke liye bhi humein dono $Q$ aur $T$ chahiye.

3. Volume $V$, Pressure $P$, aur piston

Picture. Cylinder mein ek lid hai jo slide kar sakti hai — ek piston. Agar piston bolt se band hai, toh $V$ change nahi ho sakta ($dV=0$). Agar piston free hai aur uske upar ek fixed weight rakha hai, toh gas ek fixed push $P$ maintain karta hai par expand karne ke liye free hai, toh $V$ badhta hai.

Topic ko iske kya zarurat hai. Poori vajah ki $C_p\ne C_v$ yahi do setups hain — locked piston (constant $V$) versus loaded piston (constant $P$). Mayer's relation samjhe bina dono ko picture karna impossible hai.

4. Work $W$ — gas jo energy push karne mein kharach karta hai

Topic ko iske kya zarurat hai. Constant pressure pe, kuch heat is pushing-work ke roop mein leak ho jaati hai gas ko warm karne ki jagah — yahi "leak" poori vajah hai ki $C_p>C_v$.

5. Internal energy $U$ — andar stored energy

Picture. Agar $T$ har molecule ki average speed hai, toh $U$ har molecule aur har tarah ki motion ka grand total hai.

Topic ko iske kya zarurat hai. First Law (aage) incoming heat ko "goes into $U$" aur "goes into work $W$" mein split karta hai. Sab kuch $U$ pe depend karta hai.

6. First Law — energy bookkeeping

Yeh sirf energy accounting hai: jo bhi joule tum daalo, uska hisaab dena hoga — ya toh andar rehta hai ($U$ badhata hai) ya piston ko push karne mein kharach hota hai ($PdV$). Fully built in First Law of Thermodynamics.

Topic ko iske kya zarurat hai. Yeh single equation, do baar apply karne pe (ek baar $dV=0$ ke saath, ek baar $P$ fixed ke saath), $C_v$ aur $C_p$ dono produce karta hai.

7. Moles $n$ aur gas constant $R$

Topic ko iske kya zarurat hai. Constant pressure pe, $PV=RT$ differentiate karne se $PdV=RdT$ milta hai — work-leak ka exact size. Aur Mayer's relation $C_p-C_v=R$ per mole state ki gayi hai, toh $n$ aur $R$ crystal clear hone chahiye. Dekho Mayer's Relation.

8. Heat capacities $C_v$ aur $C_p$

Notation padhna. Bada fraction $\frac{dQ}{dT}$ matlab "chota heat divided by us chote temperature-rise se jo iske wajah se hua" — ek rate. Chhota subscript $V$ ya $P$ kehta hai "measure karte waqt is cheez ko constant rakho."

Topic ko iske kya zarurat hai. Yahi do numbers IS topic hain. $\gamma$ inका ratio hai.

9. Degrees of freedom $f$ aur equipartition

Full build in Degrees of Freedom & Equipartition Theorem aur Internal Energy of Ideal Gas.

Topic ko iske kya zarurat hai. Yahi cheez $\gamma$ ko ek abstract ratio se ek molecule-shape detector banati hai: $\gamma =1+2/f$.

10. Ratio $\gamma$ khud

Jab ek baar $C_v=\frac{f}{2}R$ aur $C_p=C_v+R$ mil jaaye, toh ratio collapse hoke $\gamma =1+2/f$ ban jaata hai. Aage, $\gamma$ adiabatic $P{V}^{\gamma }=\text{const}$ aur sound ki speed $v=\sqrt{\gamma RT/M}$ ko power deta hai.

Yeh sab topic ko kaise feed karte hain

Equipment checklist

Connections

• First Law of Thermodynamics — jahan se $dQ=dU+PdV$ aata hai
• Degrees of Freedom & Equipartition Theorem — $C_v=\frac{f}{2}R$ deta hai
• Internal Energy of Ideal Gas — $U=\frac{f}{2}RT$
• Mayer's Relation — $C_p-C_v=R$
• Adiabatic Process — downstream $\gamma$ use karta hai
• Speed of Sound in Gas — $v=\sqrt{\gamma RT/M}$