Entropy — Clausius definition dS = dQ_rev - T: Foundations

1.7.22 · D1 · Physics › Thermodynamics › Entropy — Clausius definition dS = dQ_rev - T

Yeh page parent note ke har letter, ratio, aur squiggle ko build karta hai — "state" kya hoti hai wahan se shuru karke — taaki jab aap $d S = d Q_{r e v} / T$ se milein toh aapke paas already har piece ho.

0. Stage: ek "state" aur ek "system"

Kisi bhi symbol se pehle, us cheez ki picture banao jo hum study karte hain.

Figure — Entropy — Clausius definition dS = dQ_rev - T

Figure dekho. Diagram par har point ek state hai. Do points ko join karne wali ek curve ek process hai — ek specific route jo gas state $A$ se state $B$ tak leti hai. Poora topic ek sawal hai: kaunsi quantities sirf do dots ki parwah karti hain, aur kaunsi quantities puri curve ki parwah karti hain?

1. $V$ — Volume

Figure mein piston ko bahar slide karte hue picture karo: box bada hota hai, $V$ badhta hai. Volume ek state quantity hai — aap gas ki taraf point karke keh sakte ho "yeh abhi $V$ occupy kar raha hai" bina uski history jaane. Hume yeh chahiye kyunki gases apni boundary ko baahaar dhakka dekar kaam karti hain, aur woh dhakka $V$ ko change karta hai.

2. $P$ — Pressure

Gas molecules ko piston face par drumming karte hue picture karo. Zyada frequent, zyada hard hits ⇒ zyada $P$ . Pressure bhi ek state quantity hai. Hume yeh chahiye kyunki gas jo kaam karti hai woh is baat par depend karta hai ki wall move hote waqt woh kitni takkat se dhakka deti hai.

3. $T$ — Absolute temperature

Temperature parent note ki water analogy mein "crowding" number hai: heat ko kisi thandi (chhoti $T$ ) cheez mein daalo toh $d Q / T$ ek bada jump hai; kisi garam (badi $T$ ) cheez mein, ek chhota jump.

4. $U$ — Internal energy

Ek ideal gas ke liye, $U$ sirf $T$ par depend karta hai (tezi se jiggle = zyada energy), ek fact jo parent note Example 1 mein use karta hai.

5. $Q$ aur $d Q$ — Heat (woh slippery wali)

Yahi core baat hai: $Q$ ek path quantity hai. Ek hi do dots ke beech alag-alag routes alag-alag total $Q$ deliver karte hain. Hume heat chahiye kyunki yeh do tareekon mein se ek hai (doosra work hai) jisse energy gas mein enter karti hai.

6. $W$ aur $d W = P d V$ — Work

$d W = P d V$ kyun? Piston par force hai (pressure) × (area) $= P A$ . Isse tiny distance $d x$ baahaar dhakko toh kaam hai force × distance $= P A d x$ . Lekin $A d x$ exactly nayi sliver of volume $d V$ hai. Toh $d W = P d V$ . Figure mein shaded strip dekho: $P$ – $V$ curve ke neeche uska area hi work hai. Kyunki curve ki shape woh area set karti hai, work — heat ki tarah — ek path quantity hai.

7. $d$ vs $Δ$ vs $\int$ vs $\oint$ — calculus notation

8. Exact vs inexact differentials — mathematical core

Figure do paths (violet, orange) ek hi do dots ke beech dikhata hai. Ek exact quantity ke liye dono paths ek hi total dete hain (ek pahadi par chadhi gayi unchai: sirf endpoints matter karte hain). Ek inexact quantity ke liye — heat $d Q$ , work $d W$ — dono totals alag hote hain (pahadi detour par jala hua fuel: route matter karta hai).

9. Reversible — woh word jo saara kaam karta hai

10. Carnot ratio $\frac{Q _{H}}{T _{H}} = \frac{Q _{C}}{T _{C}}$

Parent note ko yeh kyun chahiye: yeh ek equation kehti hai ki heat/temperature ratio ek loop ke around balance karta hai. Kisi bhi reversible loop ko tiny Carnot cycles se tile karo aur balance har loop tak failta hai, deta hai $\oint d Q_{r e v} / T = 0$ — $S$ ke existence ka proof. Dekho Carnot Cycle and Efficiency aur Clausius Inequality.

Ek state quantity vs ek path quantity — dono mein se ek-ek do. ::: State: $U, P, V, T, S$ (sirf dot par depend karte hain). Path: $Q, W$ (curve par depend karte hain).
$d Q / T$ ke liye temperature kelvin mein kyun honi chahiye? ::: Kelvin absolute zero se shuru hota hai aur hamesha positive hota hai, isliye $1/ T$ kabhi zero se divide nahi karta aur na hi sign flip karta hai.
$d W = P d V$ kahan se aata hai? ::: Force $= P A$ , distance $= d x$ , work $= P A d x = P d V$ kyunki $A d x = d V$ .
Ek inexact differential kya satisfy karne mein fail hota hai? ::: Path-independence — $\int_{A}^{B}$ route par depend karta hai aur $\oint \neq = 0$ .
Integrating factor kya hai aur heat ke liye kaunsa kaam karta hai? ::: Ek multiplier jo ek inexact differential ko exact banata hai; $d Q_{r e v}$ ke liye yeh $1/ T$ hai.
$\oint d X = 0$ aapko $X$ ke baare mein kya bataata hai? ::: $X$ ek state function hai (kisi bhi round trip ke baad apni value par wapas aata hai).
$d S = d Q_{r e v} / T$ mein reversible word kyun zaroori hai? ::: Sirf tab $T$ well-defined hota hai aur system aur surroundings ke liye barabar hota hai, jo $d Q / T$ ko exact banata hai.
Carnot ratio $Q_{H} / T_{H} = Q_{C} / T_{C}$ se hum kya conclude kar sakte hain? ::: Heat/temperature ratio ek loop ke around balance karta hai, jo har reversible cycle ke liye $\oint d Q_{r e v} / T = 0$ tak extend hota hai.

Connections

Parent: Clausius definition of entropy
First Law of Thermodynamics — $d Q = d U + P d V$ supply karta hai
Exact and Inexact Differentials — kyun $1/ T$ ek integrating factor hai
Reversible vs Irreversible Processes — "rev" ka matlab
Carnot Cycle and Efficiency — ratio $Q_{H} / T_{H} = Q_{C} / T_{C}$
Clausius Inequality — jahan loop integral $\leq 0$ ban jaata hai
Second Law of Thermodynamics — jahan entropy ja rahi hai
Statistical Entropy — Boltzmann S = k ln W — $S$ ka microscopic meaning

Foundations — Entropy — Clausius definition dS = dQ_rev - T