Thermodynamic potentials — U (internal), H (enthalpy), F (Helmholtz), G (Gibbs)

2.4.1 · Physics › Thermodynamics & Statistical Mechanics (Advanced)

---

1. Master equation (sab kuch yahan se shuru hota hai)

YE form KYUN? First law: $dU=\delta Q-\delta W$. Ek reversible path ke liye $\delta Q=TdS$ (entropy ki definition) aur $\delta W=pdV$. Kyunki $U$ ek state function hai, $dU$ do states ke beech kisi bhi path ke liye same hoga — isliye yeh exact differential hamesha hold karta hai, chahe humne ise reversibly derive kiya ho.

Key reading: $U=U(S,V)$. Variables $S$ aur $V$ ko $U$ ke natural variables kehte hain kyunki $dU$ naturally $dS$ aur $dV$ ke terms mein likha jaata hai. $dU=TdS-pdV$ se hum directly padhte hain: $T={\left(\frac{\partial U}{\partial S}\right)}_V,-p={\left(\frac{\partial U}{\partial V}\right)}_S$

---

2. Baaki teeno ko Legendre transforms se banana

Enthalpy $H$ — $V\to p$ swap karo

$H\equiv U+pV$ Differential KAISE nikalte hain: $dH=dU+pdV+Vdp=(TdS-pdV)+pdV+Vdp=TdS+Vdp$ Toh $H=H(S,p)$, jahan $T=(\partial H/\partial S{)}_p$ aur $V=(\partial H/\partial p{)}_S$.

Helmholtz free energy $F$ — $S\to T$ swap karo

$F\equiv U-TS$ $dF=dU-TdS-SdT=-SdT-pdV$ Toh $F=F(T,V)$, jahan $-S=(\partial F/\partial T{)}_V$ aur $-p=(\partial F/\partial V{)}_T$.

Gibbs free energy $G$ — dono swap karo

$G\equiv U-TS+pV=H-TS=F+pV$ $dG=-SdT+Vdp$ Toh $G=G(T,p)$, jahan $-S=(\partial G/\partial T{)}_p$ aur $V=(\partial G/\partial p{)}_T$.

---

3. Har potential "useful" KYUN hai

---

4. Maxwell relations (exactness se free bonus)

$dF=-SdT-pdV$ se: $\frac{\partial }{\partial V}{|}_T(-S)=\frac{\partial }{\partial T}{|}_V(-p)\Rightarrow \boxed{{\left(\frac{\partial S}{\partial V}\right)}_T={\left(\frac{\partial p}{\partial T}\right)}_V}$ Baaki teeno ($U,H,G$ se):

\left(\frac{\partial T}{\partial p}\right)_S = \left(\frac{\partial V}{\partial S}\right)_p,\quad \left(\frac{\partial S}{\partial p}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_p$$ --- ## 5. Worked examples > [!example] (a) Ideal gas ki Enthalpy sirf $T$ par depend karti hai > Dikhao ki $(\partial H/\partial p)_T = 0$ ideal gas ke liye. > **Step 1:** $G$ se Maxwell use karo: $(\partial H/\partial p)_T = V - T(\partial V/\partial T)_p$. > *Yeh step KYUN?* Humein ek measurable form chahiye; $dH=TdS+Vdp \Rightarrow (\partial H/\partial p)_T = T(\partial S/\partial p)_T + V$ se shuru karo, phir Maxwell $(\partial S/\partial p)_T=-(\partial V/\partial T)_p$ substitute karo. > **Step 2:** Ideal gas $pV=nRT \Rightarrow (\partial V/\partial T)_p = nR/p = V/T$. > *KYUN?* Fixed $p$ par equation of state ko differentiate karo. > **Step 3:** $(\partial H/\partial p)_T = V - T\cdot(V/T) = 0$. ✓ Toh $H=H(T)$ sirf. > [!example] (b) Ideal gas ki Gibbs energy pressure ke saath (isothermal) > Fixed $T$ par $G(T,p)-G(T,p_0)$ nikalo. > **Step 1:** $dG = -SdT + Vdp$; constant $T$ par, $dG = Vdp$. *KYUN?* $dT=0$ pehle term ko khatam kar deta hai. > **Step 2:** $V = nRT/p$, toh $\Delta G = \int_{p_0}^{p} \frac{nRT}{p}dp = nRT\ln(p/p_0)$. > *Yeh step KYUN?* Equation of state substitute karo aur integrate karo. Yahi $RT\ln(p/p_0)$ chemical potential $\mu(T,p)=\mu^\circ + RT\ln(p/p^\circ)$ ke peeche ka workhorse hai. > [!example] (c) Potentials se $C_p - C_V$ > **Step 1:** $C_V = T(\partial S/\partial T)_V$, $C_p = T(\partial S/\partial T)_p$. *KYUN?* $\delta Q = TdS$ const $V$ ya $p$ par. > **Step 2:** $S(T,V)$ expand karke aur Maxwell use karke: > $$C_p - C_V = T\left(\frac{\partial p}{\partial T}\right)_V\left(\frac{\partial V}{\partial T}\right)_p$$ > Ideal gas ke liye: $(\partial p/\partial T)_V = nR/V$, $(\partial V/\partial T)_p = nR/p$, product $\times T = nR$. Toh $C_p-C_V = nR$. ✓ --- ## 6. Common mistakes (Steel-manned) > [!mistake] "$F$ aur $G$ same hain — dono 'free energy' hain." > **Sahi kyun lagta hai:** dono mein $TS$ minus hota hai, dono equilibrium par minimize hote hain, dono ko "free energy" kehte hain. **Fix:** dono $pV$ se differ karte hain ($G=F+pV$) aur inke *different natural variables* hain. $F$ use karo jab **volume** fixed ho (rigid box), $G$ use karo jab **pressure** fixed ho (open beaker). Galat choose karna matlab tumhara "equilibrium par minimize" criterion teri constraints ke liye bilkul galat hai. > [!mistake] "Legendre transform ka matlab randomly $pV$ *add* karna hai." > **Sahi kyun lagta hai:** $H=U+pV$ add karta hai, $F=U-TS$ subtract karta hai — inconsistent lagta hai. **Fix:** rule hai *variable ko hataane ke liye conjugate-pair product subtract karo, slope lane ke liye add karo.* $H=U-(-p)V$: hum $V$ ko $p$ ke saath swap kar rahe hain. Sign is baat se aata hai ki $dU=TdS-pdV$ mein kaunsa variable minus carry karta hai. Signs hamesha $dU$ se track karo, kabhi memorize mat karo. > [!mistake] "Maxwell relation $(\partial S/\partial V)_T=(\partial p/\partial T)_V$ — main signs guess kar lunga." > **Sahi kyun lagta hai:** chaar relations hain, do mein minus signs hain, mix karna easy hai. **Fix:** har ek ko matching potential ke $d\Phi$ se mixed partials ki equality use karke derive karo. Signs $d\Phi$ ke signs se *forced* hain — kabhi guess nahi karna. --- ## 7. Active recall > [!recall]- Quick self-test (answers cover karo) > - $G$ ke natural variables kya hain? → $T,p$. > - $dF$ likho. → $-SdT - pdV$. > - Constant $T,p$ par kaun sa potential minimize hota hai? → $G$. > - Constant $T$ par $-\Delta F$ kisko equal hai? → maximum obtainable work. > - $G$ se Maxwell relation bolo. → $(\partial S/\partial p)_T = -(\partial V/\partial T)_p$. > [!recall]- Feynman: 12-saal ke bacche ko samjhao > Socho energy ek piggy bank mein paisa hai. $U$ andar ka total paisa hai. Lekin kabhi kabhi tum *sab* use nahi kar sakte — universe ek "messiness tax" ($TS$) charge karta hai jo tum kharch nahi kar sakte. Tax ke baad jo bacha woh **free** paisa hai jo tum actually kuch karne mein use kar sakte ho — yahi $F$ hai. Agar tum yeh game bahar khel rahe ho jahan hawa tumpar push karti hai, tumhe atmosphere se jagah rent karni padti hai ($pV$); usse add karne par $H$ milta hai, aur dono tax aur rent ke baad tumhe $G$ milta hai. Is baat par depend karta hai ki tumhara jar sealed hai (fixed volume) ya hawa mein khula hai (fixed pressure), alag "paisa" number batata hai ki tum settle down gaye ya nahi (equilibrium reach hua) — nature hamesha *sabse chhote* allowed wale ki taraf slide karti hai. > [!mnemonic] Potentials yaad karne ka tarika > **"Good Physicists Have Studied Under Very Fine Teachers"** — state functions ka square: > Corners $U, H, G, F$ with variables $S, V, T, p$ **Thermodynamic Square** par. > **Swaps** ke liye: *H = U **H**oists pV* ($pV$ add karo); *F = U **F**rees by losing TS* ($TS$ subtract karo); *G ko **dono** milte hain.* --- ## #flashcards/physics Definition of enthalpy $H$ ::: $H \equiv U + pV$ Definition of Helmholtz free energy $F$ ::: $F \equiv U - TS$ Definition of Gibbs free energy $G$ ::: $G \equiv U - TS + pV = H - TS$ Master differential of $U$ ::: $dU = TdS - pdV$ Differential of $H$ aur uske natural variables ::: $dH = TdS + Vdp$; natural vars $S,p$ Differential of $F$ aur uske natural variables ::: $dF = -SdT - pdV$; natural vars $T,V$ Differential of $G$ aur uske natural variables ::: $dG = -SdT + Vdp$; natural vars $T,p$ Legendre transform physically kya kar raha hai ::: ek variable ko uske conjugate slope ke saath swap karta hai (jaise $V\leftrightarrow p$) product subtract/add karke Constant $T,p$ par kaun sa potential minimize hota hai ::: Gibbs free energy $G$ Constant $S,V$ par kaun sa potential minimize hota hai ::: internal energy $U$ Constant $T$ par $-\Delta F$ ka physical meaning ::: maximum total work jo system kar sakta hai Constant $p$ par $\Delta H$ ka physical meaning ::: exchanged heat (heat of reaction) $F$ se Maxwell relation ::: $(\partial S/\partial V)_T = (\partial p/\partial T)_V$ $G$ se Maxwell relation ::: $(\partial S/\partial p)_T = -(\partial V/\partial T)_p$ General $C_p - C_V$ formula ::: $T(\partial p/\partial T)_V(\partial V/\partial T)_p$ Ideal gas ke isothermal pressure change par $G$ ka change ::: $\Delta G = nRT\ln(p/p_0)$ Maxwell relations sach KYUN hain ::: $d\Phi$ exact hai, isliye mixed second partials equal hote hain --- ## Connections - [[Legendre transforms]] — wo mathematical machine jo chaar potentials generate karti hai - [[Maxwell relations]] — $d\Phi$ ki exactness ka direct consequence - [[Entropy and the Second Law]] — $TdS$ term provide karta hai - [[Chemical potential and the grand potential]] — variable $N$ tak extend karta hai: $\mu\,dN$ add karo - [[Phase equilibria & Clausius–Clapeyron]] — $G$ minimization aur $F$-Maxwell relation use karta hai - [[Heat capacities Cp and Cv]] — Example (c) mein potentials ke zariye derive kiya gaya ## 🖼️ Concept Map ```mermaid flowchart TD U[Internal energy U] MASTER[dU = TdS - pdV] NAT[Natural vars S and V] LEG[Legendre transform] H[Enthalpy H] F[Helmholtz F] G[Gibbs G] EOS[Equations of state] EQ[Minimized at equilibrium] U -->|state function gives| MASTER MASTER -->|read off| NAT LEG -->|swaps var for slope| U U -->|add pV, swap V to p| H U -->|subtract TS, swap S to T| F U -->|subtract TS add pV| G H -->|dH = TdS + Vdp| EOS F -->|dF = -SdT - pdV| EOS G -->|dG = -SdT + Vdp| EOS F -->|at fixed T,V| EQ G -->|at fixed T,p| EQ ```