2.4.3
Ek thermodynamic potential ek aisi function hoti hai jiske mixed second partial derivatives differentiation ke order ki parwah nahi karte . Calculus ka woh ek simple fact, jab har potential par apply hota hai, nikaal deta hai chaar powerful relations jo measurable quantities (jaise ∂ V / ∂ T ∂ S / ∂ P Maxwell relations sirf thermodynamic potentials ke mixed partials ki equality hain .
Definition Exact differential aur mixed partials ki equalityAgar z = z ( x , y ) d z = ( ∂ x ∂ z ) y d x + ( ∂ y ∂ z ) x d y ≡ M d x + N d y ( ∂ y ∂ M ) x = ( ∂ x ∂ N ) y = ∂ x ∂ y ∂ 2 z reciprocity / Schwarz / Euler condition hai. Har Maxwell relation is ek identity ka hi disguised roop hai.
HUM KYA exploit karte hain: Thermodynamic potentials (U , H , F , G
HUM ISSE KAISE use karte hain: Har potential ke natural differential se M N
Potential
Definition
Differential
Natural vars
Internal energy U
—
d U = T d S − P d V ( S , V )
Enthalpy H
H = U + P V d H = T d S + V d P ( S , P )
Helmholtz F
F = U − T S d F = − S d T − P d V ( T , V )
Gibbs G
G = U − T S + P V d G = − S d T + V d P ( T , P )
Legendre transform KYUN? Yeh change karne ke liye ki kaun sa variable "independent" hai. Example: H = U + P V d H = d U + P d V + V d P = ( T d S − P d V ) + P d V + V d P = T d S + V d P . − P d V d P V Yeh step KYUN? P V + P d V + V d P P d V
Har differential ki form hai d z = M d x + N d y ( ∂ M / ∂ y ) x = ( ∂ N / ∂ x ) y
U d U = T d S − P d V x = S , y = V , M = T , N = − P ( ∂ V ∂ T ) S = − ( ∂ S ∂ P ) V Yeh step KYUN? M = ( ∂ U / ∂ S ) V = T N = ( ∂ U / ∂ V ) S = − P ∂ 2 U / ∂ V ∂ S
H d H = T d S + V d P x = S , y = P , M = T , N = V ( ∂ P ∂ T ) S = ( ∂ S ∂ V ) P Yeh step KYUN? Minus sign nahi hai kyunki d P + V d V − P
F d F = − S d T − P d V x = T , y = V , M = − S , N = − P ( ∂ V ∂ S ) T = ( ∂ T ∂ P ) V Yeh step KYUN? ∂ ( − S ) / ∂ V = ∂ ( − P ) / ∂ T ⇒ sabse useful relation hai: RHS easily measurable hai (pressure T
G d G = − S d T + V d P x = T , y = P , M = − S , N = V ( ∂ P ∂ S ) T = − ( ∂ T ∂ V ) P Yeh step KYUN? RHS thermal expansion coefficient α = V 1 ( ∂ V / ∂ T ) P
Sign pattern (KYUN): Minus sign tab aata hai jab differential mein − P d V V U F conjugate mein minus kahan hai ). Practical rule: U H independent variable hai) left side par T F G S measurable ones hain.
Worked example "Energy equation" — Maxwell relations in actionStart: d U = T d S − P d V T d V ( ∂ V ∂ U ) T = T ( ∂ V ∂ S ) T − P Yeh step KYUN? Hum T d S → ( ∂ S / ∂ V ) T d V
Ab Helmholtz Maxwell relation ( ∂ S / ∂ V ) T = ( ∂ P / ∂ T ) V ( ∂ V ∂ U ) T = T ( ∂ T ∂ P ) V − P Yeh KYUN matter karta hai: Ideal gas ke liye P = n R T / V T ( ∂ P / ∂ T ) V = T ⋅ n R / V = P ( ∂ U / ∂ V ) T = 0 sirf thermodynamics se prove kar diya ki ideal gas ki internal energy volume-independent hoti hai — Joule's law derived, assumed nahi .
Common mistake "Natural variables jo chahun woh hain."Kyun sahi lagta hai: U ( T , V ) Fix: Sirf jab U natural variables ( S , V ) d U = T d S − P d V ( T , V ) Potential ko uske natural variables ke saath pair karna zaroori hai.
Common mistake Minus signs galat karna.Kyun sahi lagta hai: Charon symmetric lagte hain, toh aap assume karte ho sab "= Fix: Sign directly differential ke sign se aata hai. Agar term − P d V N = − P Hamesha differential se re-derive karo, signs blindly mat memorize karo.
Common mistake Subscript (held-constant variable) confuse karna.Kyun sahi lagta hai: Log subscript ordinary calculus ki tarah drop karte hain. Fix: Thermodynamics mein ( ∂ T / ∂ V ) S = ( ∂ T / ∂ V ) P quantity ka part hai ; Maxwell relation sirf dikhaye gaye subscripts ke saath hold hoti hai (dusra natural variable fixed rakhte hain).
Recall Feynman: ek 12-saal ke bachche ko explain karoEk smooth pahaad imagine karo. Neeche se upar pahunchne ke liye, fark nahi padta agar pehle east phir north jao ya pehle north phir east — same total height climb hoti hai. "Height function" order ki parwah nahi karta. Thermodynamic potentials bhi us pahaad ki tarah hain: unki energy ek smooth function hai, toh jab tum measure karte ho ki woh do alag directions mein kitni steeply change hoti hai, answers ko ek special crossed tarike se match karna hoga. Woh matching hume free facts deti hai — jaise kuch hard-to-measure cheez (gas squeeze hone par kitni disorderly hoti hai) ko kuch easy cheez measure karke figure out karna (woh warm hone par kitna expand hoti hai). Same pahaad, do paths, equal climb.
Mnemonic Potentials aur signs yaad karne ka tarika"Good Physicists Have Studied Under Very Fine Teachers" ki zaroorat nahi — square use karo:
Square likhao U H F G with S V T P corners par. Natural-variable square (Born square) "Valid Facts and Theoretical Understanding Generate Solutions "... sabse simple: yaad rakho d U = T d S − P d V baaki teen ko Legendre transform se build karo (P V T S Derive karo, memorize mat karo.
Thermodynamic potentials & Legendre transforms — chaar potentials kahan se aate hainFirst and Second Laws of Thermodynamics — d U = T d S − P d V Equality of mixed partial derivatives (Schwarz theorem) — calculus engineJoule expansion and internal energy — energy equation ke zariye applicationHeat capacities $C_P - C_V$ — Maxwell relations use karke derive kiyaThermal expansion coefficient and isothermal compressibility — RHS par appear karte hain
Kaun sa ek calculus fact har Maxwell relation ke peeche hai? State function ke mixed second partial derivatives ki equality (exact differential): ∂ 2 z / ∂ x ∂ y = ∂ 2 z / ∂ y ∂ x
U ( S , V ) d U = T d S − P d V
G ( T , P ) d G = − S d T + V d P
Helmholtz free energy F ( ∂ S / ∂ V ) T = ( ∂ P / ∂ T ) V
Gibbs free energy G ( ∂ S / ∂ P ) T = − ( ∂ V / ∂ T ) P
U ( ∂ T / ∂ V ) S = − ( ∂ P / ∂ S ) V
H ( ∂ T / ∂ P ) S = ( ∂ V / ∂ S ) P
U H Legendre transform H = U + P V d H = T d S + V d P
Maxwell relation ke liye potential ko uske natural variables mein kyun hona chahiye? Sirf tab differential ke coefficients clean quantities (T , − P , ...
Energy equation batao aur ideal gas ke liye yeh kya prove karta hai. ( ∂ U / ∂ V ) T = T ( ∂ P / ∂ T ) V − P 0 U
Maxwell relation mein minus sign kahan se aata hai? Directly differential ke term ke sign se (jaise − P d V ⇒ N = − P
Exact differential dz eq M dx plus N dy
Equality of mixed partials
Potentials are state functions
Internal energy U vars S V
