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

2.4.1 · D1 · Physics › Thermodynamics & Statistical Mechanics (Advanced) › Thermodynamic potentials — U (internal), H (enthalpy), F (He

Is page par kuch bhi assume nahi kiya gaya. Agar parent note mein koi symbol likha tha, toh hum use yahan pehle ek picture se build karte hain. Upar se neeche padho — har block sirf wohi use karta hai jo pehle aaya.

0. System: hum dekh kya rahe hain?

Yeh picture kyun chahiye. Neeche har symbol ek number hai jo tum is ek snapshot se padh sakte ho. Thermodynamics ka study yahi hai ki yeh snapshot-numbers aapas mein kaise relate karte hain aur kaise change hote hain jab tum piston push karte ho ya gas ko heat karte ho.

1. Volume $V$ — kitni jagah hai

Picture: cylinder ki length times uski cross-section — upar figure mein shaded region. Bas itna hi. $V$ poore subject ka sabse concrete symbol hai: tum ise dekh sakte ho.

Topic ko yeh kyun chahiye. $V$ ek cheez hai jise tum physically clamp kar sakte ho (piston lock karo) ya float hone do. Tum konsa clamp karte ho yeh decide karta hai ki konsa potential useful hai — isliye $V$ pehle samajhna zaroori hai.

2. Pressure $p$ — gas kitni zor se push karta hai

Picture: chhote gas molecules piston se bounce karte hue. Har bounce ek chhoti si shove hai. Saari shoves ko piston ki area par add karo aur us area se divide karo — woh $p$ hai (orange arrows).

Topic ko yeh kyun chahiye. $p$ woh doosri cheez hai jise tum clamp kar sakte ho (khula beaker = fixed atmospheric pressure). $V$ aur $p$ do "mechanical" handles hain.

3. Temperature $T$ — jiggling kitni vigorous hai

Picture: wahi bouncing molecules jaise pehle the, lekin ab unke motion arrows ki length woh hai jo $T$ track karta hai — lambe arrows = hot, chhote arrows = cold.

Topic ko yeh kyun chahiye. $T$ woh handle hai jo thermostat control karta hai. Zyatsar real experiments mein $T$ fixed hoti hai (ek beaker jo room mein rakha ho), aur yahi wajah hai ki $F$ aur $G$ jaise potentials — jo $T$ ko natural variable ki tarah use karne ke liye bane hain — exist karte hain.

4. Internal energy $U$ — andar ki total energy

Picture: imagine karo har single molecule ki energy sum karo — woh grand total $U$ hai.

Topic ko yeh kyun chahiye. $U$ root potential hai. Baaki teen ( $H$ , $F$ , $G$ ) sab sirf $U$ mein kuch pieces add ya subtract karne se bante hain.

5. Heat $δ Q$ aur work $δ W$ — energy boundary cross karti hai

Picture: §0 ki figure mein box wall cross karte do arrows — ek red "heat in" arrow aur ek blue "work out" arrow.

Topic ko yeh kyun chahiye. Yahi equation hai jahan se parent note shuru karta hai. Sab kuch — chaaron potentials — yahi law hai, rewrite kiya hua.

6. Entropy $S$ — "spread-out-ness" ka tax

Picture: ek box mein kuch marbles. Agar sirf ek tidy arrangement hai (sab ek corner mein), toh $S$ low hai. Agar woh kaheen bhi ho sakti hain (bahut saare arrangements), toh $S$ high hai. Figure mein "few ways" (left, low $S$ ) versus "many ways" (right, high $S$ ) dikhaya gaya hai.

Topic ko yeh kyun chahiye. $δ Q_{rev} = T d S$ aur $δ W = p d V$ ko First Law mein substitute karne par messy $δ$ -law clean master equation ban jaati hai.

7. Master equation assemble karna

Ise picture ki tarah padho. $d U$ piggy bank mein total change hai. $+ T d S$ = heat se deposit ki gayi energy. $- p d V$ = piston push karne mein kharch ki gayi energy. Parent note ka har potential isi ek line ko algebraically rearrange karke bana hai.

8. Derivative $(\frac{\partial U}{\partial S})_{V}$ ka matlab kya hai

Picture: energy-landscape $U (S, V)$ par khade ho. Sirf $S$ -direction mein ek kadam chalo (kabhi $V$ mein sideways nahi). Us kadam ki steepness partial derivative hai.

Topic ko yeh kyun chahiye. $d U = T d S - p d V$ se, term-by-term match karne par milta hai $T = (\partial U / \partial S)_{V}$ aur $- p = (\partial U / \partial V)_{S}$ . Yahi "equations of state for free" hain jo parent note ne promise kiye the — aur yeh sirf slope ki readings hain.

V

ek picture mein kya hai?

Woh shaded space jo gas fill karta hai; piston andar = kam, bahar = zyada.

p

physically kya hai?

Force per unit area molecules ke wall se bounce karne se.

p

aur

V

hamesha product

p V

ke roop mein kyun aate hain?

Force-per-area × area-swept = work/energy; yeh conjugate partners hain.

T

microscopically kya measure karta hai?

Molecules ki average jiggle (kinetic) energy.

U

kya hai?

Andar store ki gayi total energy — "piggy bank mein paisa," ek state function.

First Law state karo.

d U = δ Q - δ W

(andar change = heat in − work out).

Entropy change

d S = δ Q_{rev} / T

kyun hai (kyun

T

se divide karte hain)?

Wahi heat ek cold system ko ek hot system se zyada disorder karta hai, isliye disorder gain = heat over temperature scale hoti hai.

Konse pairs conjugate hain?

T

with

S

, aur

p

with

V

— har pair multiply karke ek energy deta hai.

Master equation likho.

d U = T d S - p d V

(\partial U / \partial S)_{V}

mein subscript tumhe kya batata hai?

V

fixed rakho jabki

S

slide karo;

U

ki resulting slope measure karo.

Mixed second partials equal kyun hote hain?

U

ek state function hai, isliye do nudges ka order change nahi karta ki tum kahan land karte ho.

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