Adiabatic relations — PV^γ = const, TV^(γ−1) = const (derivation): Foundations

1.7.16 · D1 · Physics › Thermodynamics › Adiabatic relations — PV^γ = const, TV^(γ−1) = const (deriva

Neeche hum is topic ka poora alphabet build karenge, ek symbol ek time pe. Har naye symbol ko sirf wahi symbols use karne ki permission hai jo pehle define ho chuke hain. Agar parent page ne assume kiya tha ki tum pehle se jaante the, toh hum yahan ruk ke usse draw karenge.

1. Gas khud: $P$ , $V$ , $T$

Ek box imagine karo jisme ek sliding lid (piston) laga hai. Andar countless tiny gas molecules hain jo bounce kar rahi hain.

Figure — Adiabatic relations — PV^γ = const, TV^(γ−1) = const (derivation)

Upar figure padho: yellow bar piston hai (jo $V$ set karta hai); blue dots chhote arrows ke saath woh molecules hain jinki speed $T$ hai aur jinki walls par drumming $P$ hai. Yeh ek box picture woh stage hai jis par baaki sab kuch hota hai — is page ke end tak ise yaad rakho.

Is topic ko teeno ki zaroorat kyun hai: ek adiabatic curve ek $P$ – $V$ diagram par trace hua path hai, aur $T$ woh hai jo silently us par change hoti hai. Tum $P V^{γ} =$ const nahi keh sakte jab tak pehle $P$ aur $V$ na ho; tum "expansion cools" explain nahi kar sakte bina $T$ ke.

2. Gas ko count karna: $n$ aur $R$

Is topic ko iski zaroorat kyun hai: poori derivation ek game hai "mere paas teen variables $P, V, T$ hain lekin main sirf do chahta hoon." $P V = n R T$ woh tool hai jo kisi bhi variable ko marzi se eliminate karta hai — $P$ ko $n R T / V$ se swap karne ke liye use hota hai, aur baad mein $V$ ko $n R T / P$ se.

3. Stored warmth: internal energy $U$

4. Change ki grammar: $d$ , $δ$ , $Δ$

Isse pehle ki hum energy flowing ke baare mein baat karein, hume "thodi si matra" ke notation ki zaroorat hai. Yeh nayi physics nahi hai — yeh change ki grammar hai — isliye hum inhe first use se pehle yahan milte hain.

5. Energy in transit: heat $Q$ aur work $W$ (sign conventions ke saath)

Energy gas ke bank ( $U$ ) mein exactly do tareekon se andar ya bahar ja sakti hai. Ab jab hum $d / δ$ symbols ke saath hain, hum inhe likh sakte hain.

Upar figure padho: blue curve ek possible $P$ – $V$ path hai. Yellow strip ki height $P$ hai aur width $d V$ hai, isliye iska area exactly $δ W = P d V$ hai — woh energy jo gas us whisker mein wide hone ke liye spend karti hai. Ek path ke saath aise saare strips ko add karna total work deta hai; isliye work "area under the $P$ – $V$ curve" hai.

6. First Law — ek line mein energy balance

Ab hum woh law state kar sakte hain jo $Q$ , $U$ , aur $W$ ko ek saath tie karta hai — woh beating heart jis par poori derivation khadi hai.

Do pieces jo hum pehle build kar chuke hain ( $d U = n C_{V} d T$ Section 3 se, aur $δ W = P d V$ Section 5 se) ko adiabatic first law $0 = d U + δ W$ mein substitute karne se parent ki starting equation milti hai: $0 = n C_{V} d T + P d V .$

7. Heat capacities: $C_{V}$ , $C_{P}$ , aur Mayer's bridge

Is topic ko iski zaroorat kyun hai: derivation ke Step 4 mein ugly $R / C_{V}$ ko clean $γ - 1$ se replace kiya jaata hai. Woh replacement hi Mayer's relation disguise mein hai.

8. Show ka star: $γ$

Upar figure padho: ek shared yellow point se hum do curves draw karte hain. Green isotherm ( $P V =$ const, $T$ fixed) gently girta hai; red adiabat ( $P V^{γ} =$ const) steeper girta hai kyunki $γ > 1$ . Woh extra steepness gas ke expand hote hue cool hone ka visual signature hai — pressure zyada tez girta hai jab koi heat energy refill nahi karta.

9. Undo-tool: logarithm $ln$

10. Sab kuch ek saath jodna — derivation, step by step

Ab har symbol define hai. Dekho kaise teen tools ( $P V = n R T$ , Mayer's relation, $ln$ ) first law ko adiabatic relations mein badal dete hain. Har step mein kya kiya aur kyun bhi hai.

Start. Section 6 se adiabatic first law: $0 = n C_{V} d T + P d V .$ Kyun: $Q = 0$ income hata deta hai, isliye work poori tarah internal energy se pay hoti hai.

Step A — $P$ eliminate karo. Hamare paas teen variables $P, V, T$ hain lekin sirf $T$ aur $V$ mein relation chahiye. Ideal gas law deta hai $P = \frac{n R T}{V}$ ; substitute karo: $0 = n C_{V} d T + \frac{n R T}{V} d V .$ Kyun: $P V = n R T$ exactly ek variable trade karne ka tool hai; hum $T, V$ rakhte hain kyunki hum $T V$ relation ki taraf ja rahe hain.

Step B — variables separate karo. Har term ko $n C_{V} T$ se divide karo taaki har variable apne differential ke saath baithe: $\frac{d T}{T} + \frac{R}{C _{V}} \frac{d V}{V} = 0.$ Kyun: integrate karne ke liye, har side mein sirf ek variable hona chahiye; yeh "separation" dono terms ko independently integrable banata hai.

Step C — $γ$ inject karo. Mayer's relation deta hai $\frac{R}{C _{V}} = \frac{C _{P} - C _{V}}{C _{V}} = γ - 1$ : $\frac{d T}{T} + (γ - 1) \frac{d V}{V} = 0.$ Kyun: yeh woh moment hai jab $γ$ — gas ka fingerprint — maths mein enter karta hai.

Step D — $ln$ ke saath integrate karo. Har $\frac{d x}{x}$ integrate hokar $ln x$ deta hai: $ln T + (γ - 1) ln V = const .$ Kyun: logarithm $1/ x$ ka natural antiderivative hai; ek constant ke barabar logs ka sum matlab product constant hai.

Step E — result padho. $T V^{γ - 1} = const, then using T = \frac{P V}{n R} : P V^{γ} = const .$ Kyun: $T$ ko $V^{γ - 1}$ se multiply karne se $V$ -powers add hote hain, $V^{1 + (γ - 1)} = V^{γ}$ , aur constant $n R$ naye constant mein absorb ho jaata hai.

Yeh woh full skeleton hai jise parent page flesh out karta hai — ab har step mein sirf wahi symbols use hote hain jo humne zero se define kiye.

Yeh topic ko kaise feed karta hai

Ise top-down padho: paanch gas symbols ideal gas law build karte hain; $T$ aur $C_{V}$ $d U$ build karte hain; $Q = 0$ aur work term first law build karte hain; ideal gas law $P$ eliminate karta hai; Mayer's relation $γ - 1$ deliver karta hai; aur $ln$ kaam khatam karta hai.

Equipment checklist

Apne aap test karo — har line ka right side cover karo reveal karne se pehle.

P

physically kya measure karta hai?

Walls par molecular collisions se unit area par force (in

Pa

atm

T

kya measure karta hai, aur yahan konsi unit honi chahiye?

Average molecular kinetic energy; hamesha kelvin (

K

) mein.

Ideal gas law state karo aur derivation mein iska kaam batao.

P V = n R T

; yeh

P, V, T

mein se kisi ek ko eliminate karne deta hai.

Ek ideal gas ke liye,

U

sirf kise depend karta hai?

Sirf temperature

T

par.

Hum

d U = n C_{V} d T

kyun likh sakte hain jab volume change ho?

Kyunki ideal gas ke liye

U = U (T)

hai, isliye

C_{V}

par

V

-subscript sirf iske measure karne ke tarike ke baare mein hai.

First Law state karo aur har term ka sign meaning batao.

δ Q = d U + δ W

;

Q

= gas mein add ki gayi heat,

W

= gas dwara kiya gaya work.

"Adiabatic" ka matlab ek symbol mein kya hai, aur First Law kya banta hai?

Q = 0

; First Law banta hai

0 = d U + δ W

δ Q

aur

δ W

kyun likhte hain

d Q

d W

ki jagah?

Heat aur work stored quantities nahi hain; yeh sirf transit mein exist karte hain.

Small-expansion work likho aur expansion ke liye iska sign batao.

δ W = P d V

; positive jab gas expand kare (

d V > 0

C_{V}

C_{P}

kya hai?

Constant volume par vs constant pressure par heat per mole per kelvin;

C_{P} > C_{V}

Mayer's relation state karo.

C_{P} - C_{V} = R

γ

define karo aur iska range batao.

γ = C_{P} / C_{V}

, aur

γ > 1

R / C_{V}

kis cheez mein simplify hota hai?

γ - 1

Integrate karte waqt

ln

kya karta hai jise hum exploit karte hain?

Products ko sums mein badalta hai (aur

\int d V / V = ln V

), isliye logs ka sum = const matlab product = const.

Connections

Parent: the full derivation
First Law of Thermodynamics — energy balance $δ Q = d U + δ W$ provide karta hai.
Isothermal Process — contrasting case jahan $T$ (aur isliye $U$ ) fixed rakha jaata hai.
Mayer's Relation — $C_{P} - C_{V} = R$ kahan se aata hai.
Degrees of Freedom and $\gamma$ — $γ$ $5/3$ , $7/5$ ... values kyun leta hai.
Speed of Sound in Gases — adiabatic bulk modulus $γ P$ use karta hai.
Work Done in Thermodynamic Processes — $δ W = P d V$ ka matlab.