Foundations — Thermodynamic potentials — U, H, F, G (preview)

1.7.26 · D1 · Physics › Thermodynamics › Thermodynamic potentials — U, H, F, G (preview)

Is page par kuch bhi assumed nahi hai. Hum har letter, har $d$, har $\partial$ ko ground up se banate hain, usi order mein jisme parent note ko chahiye. Agar tum thermometer padh sakte ho aur ek piston imagine kar sakte ho, tum is page ko finish kar sakte ho.

0. Story ke paanch characters

Koi bhi formula se pehle, physical quantities se milo. Do describe karte hain kitna stuff hai aur energy kitni spread out hai; do describe karte hain system par kitna push ho raha hai.

Exactly yahi chaar kyun? Kyunki ye do partner pairs mein aate hain:

• $(T,S)$ — thermal pair. Inhe multiply karo aur $TS$ ke units energy ke hain.
• $(p,V)$ — mechanical pair. Inhe multiply karo aur $pV$ ke bhi units energy ke hain.

Har pair ek "cheez hai jis par push kar sakte ho" aur ek "cheez jo respond karti hai." Yahi pairing chaaon potentials ke peeche ka secret skeleton hai, aur hum iska picture §4 mein banate hain.

1. Chhota $d$ kya matlab hai: ek tiny change

Parent note $dU$, $dS$, $dT$ jaise cheezein se bhara hai. Symbol $d$ ka matlab kabhi "d se multiply karo" nahi hota. Iska matlab hai "mein ek tiny change."

Physicists bade changes ki jagah tiny changes ke saath kyun kaam karte hain? Kyunki rules ($TdS$, $pdV$) sirf tabhi exactly sahi hote hain jab step itna chhota ho ki us step ke dauran $T$ aur $p$ na badlein. Ek tiny honest step lo, phir unhe millions ki sankhya mein jodo (yahi integration hai) ek bade change ko cover karne ke liye. Figure mein staircase dekho: har tread ek $dX$ hai; poori chadhaai $\Delta X$ hai.

2. Energy aur First Law: $dU=\delta Q-\delta W$

Tum us wallet ko exactly do tareekon se change kar sakte ho: heat daalo, ya gas ko kaam karne do.

First Law bas in tiny bits ke liye likha gaya energy conservation hai: $dU=\delta Q-\delta W.$

3. Do substitutions jo $\delta Q,\delta W$ ko $TdS$ aur $pdV$ mein badal dete hain

Parent note seedha $dU=TdS-pdV$ par jump karta hai. Yahan se ye pieces aate hain.

Gas ka piston par kaam. Piston ko pressure $p$ ke against tiny volume $dV$ bahar push karo: $\delta W=pdV.$ Kyun? Force = pressure × area, move ki gayi distance × area = swept volume, toh force × distance = $pdV$. Picture: piston thoda sa bahar slide hota hai, gas ne usse dhakelte hue $pdV$ energy kharchi.

Heat aur entropy. Ek reversible (idealised, infinitely gently kiya gaya) step ke liye, entropy ki definition exactly yahi hai $\delta Q=TdS.$ Hum yeh kyun keh sakte hain? Yeh literally entropy ki definition hai — dekho Entropy and the Second Law। Ise ulta padho: entropy change $dS=\delta Q/T$ hai "dali gayi heat, divided by tum kitne already hot ho."

Dono ko First Law mein substitute karo: $dU=\delta Q-\delta W=TdS-pdV.$

Woh single equation — §0 ke do energy pairs ko connect karti hui — poore parent note ka seed hai. Sab kuch ($H,F,G$, Maxwell relations, equilibrium) usi se ugta hai। Dekho First Law of Thermodynamics।

4. Partner-pair square: swap kyun possible hai

Phir se do pairs dekho. $dU=TdS-pdV$ ke andar:

• $T$, $dS$ ke saamne baitha hai → hum kehte hain $T$, $S$ ka conjugate ("partner") hai।
• $p$, $dV$ ke saamne baitha hai → $p$, $V$ ka conjugate hai।

Parent note ko iski parwaah kyun hai? Kyunki lab mein tum ek pair ka ek member aasaani se control kar sakte ho aur doosre ko sirf mushkil se:

Toh hum energy ko rewrite karna chahenge taaki easy-to-control partner "input variable" ban jaaye. Woh tool jo ek partner ko uske conjugate ke saath swap karta hai woh Legendre transform hai — aage introduce kiya gaya, aur Legendre Transform mein poori tarah se cover kiya gaya।

5. Slope symbol ${\left(\frac{\partial U}{\partial S}\right)}_V=T$

Parent note $T$ aur $-p$ ko partial derivatives ke roop mein read karta hai. Yahan woh notation zero se hai।

Topic ke liye yeh kyun important hai? Kyunki $dU=TdS-pdV$ ko us generic rule ke against pattern-match kiya ja sakta hai ki ek do-variable function kaise change hoti hai: $dU={\left(\frac{\partial U}{\partial S}\right)}_{V}dS+{\left(\frac{\partial U}{\partial V}\right)}_{S}dV.$ Dono sides par $dS$ aur $dV$ ke coefficients line up karo: ${\left(\frac{\partial U}{\partial S}\right)}_V=T,{\left(\frac{\partial U}{\partial V}\right)}_S=-p.$ Yeh "coefficient padhne" wala move parent note ke har worked example mein use hota hai।

6. Baaki teen wallets banana

Ab har symbol earn ho gaya, parent ki definitions clearly padhti hain:

• $pV$ add karo → tum us room ka account kar rahe ho jo system ne pressure ke against banaya → enthalpy $H$।
• $TS$ ghataao → tum woh "entropy tax" alag kar rahe ho jo second law temperature $T$ par charge karta hai → Helmholtz free energy $F$।
• Dono karo → Gibbs free energy $G$ (chemist ka wallet — dekho Gibbs Free Energy and Chemical Equilibrium)।

Exactly inhi products ko kyun jodtey/ghatatey hain? Kyunki har ek §0 se ek energy chunk hai, aur use jodna precisely woh Legendre swap hai jo ek aasaan lab variable ko driver's seat mein le aata hai।

7. Prerequisite map

Ise top-down padho: tiny change aur chaar state variables ke ideas First Law ko feed karte hain; heat aur work substitutions use master equation mein badal dete hain; conjugate pairs Legendre swap ko possible banate hain; woh chaar potentials produce karta hai; aur partial derivatives tumhe baaki sab kuch extract karne dete hain।

Equipment checklist

Khud ko test karo — right side cover karo aur zor se jawab do।