Third law of thermodynamics — S → 0 as T → 0: Foundations

1.7.25 · D1 · Physics › Thermodynamics › Third law of thermodynamics — S → 0 as T → 0

Is page par yeh assume kiya gaya hai ki tum kuch nahi jaante. Third Law se milne se pehle, tumhe — ek-ek karke — har letter, ratio, aur squiggle diya jaayega jo woh use karta hai. Hum inhe ek aisi order mein build karte hain jahan har naya idea sirf un ideas par tikta hai jo pehle se khade hain.

1. Temperature $T$ — "particles kitna jig kar rahe hain"

Ek tray of marbles ka picture karo. Isse dheere hilao toh marbles thodi si drift karte hain; zyada zyada hilao toh woh bounce karte hain. Tumhari shaking ki strength temperature hai.

Figure — Third law of thermodynamics — S → 0 as T → 0

Is topic ko iske liye kyun chahiye. Poora Third Law ek story hai jo batata hai kya hota hai jab $T$ slide karke $0$ par aata hai. Agar tum nahi jaante $T$ ka matlab kya hai, toh " $T \to 0$ " sirf symbols hain.

2. Microstate, macrostate, aur count $W$

Yeh is page par ek sabse important idea hai, aur isko ek picture chahiye.

Chaar coins ki table par socho. "Do heads dikhna" ek macrostate hai. Lekin woh macrostate ho sakta hai HHTT, HTHT, HTTH, THHT, THTH, TTHH ke roop mein — chhe alag detailed arrangements. Toh us macrostate ke liye, $W = 6$ .

Is topic ko iske liye kyun chahiye. Entropy $W$ se bani hai. Third Law ki har baat — "ek arrangement," "residual entropy," "perfect crystal" — $W$ ki value ke baare mein ek statement hai.

Recall

$W$ par quick check Teen coins ke liye, kitne microstates macrostate "all heads" dete hain? Sirf ek: HHH, toh $W = 1$ .

3. Logarithm $ln$ — ek bade count ko friendly number mein badalna

Entropy $W$ use karne se pehle, tumhe ek mathematical tool chahiye: natural logarithm, likha jaata hai $ln$ .

Sirf do facts hi chahiye, aur dono seedha definition se aate hain:

Yeh tool kyun, plain $W$ kyun nahi? Kyunki real $W$ values astronomically large hoti hain ( $W$ ek gas ke liye $1 0^{1 0^{23}}$ ho sakta hai). Hum ek aisi quantity chahte hain jo (a) manageable size ki ho aur (b) jud jaaye jab tum do systems ko ek saath jodoge. Agar system A ke $W_{A}$ arrangements hain aur system B ke $W_{B}$ , toh combined system ke $W_{A} \times W_{B}$ arrangements hain (har A-way har B-way ke saath pair hoti hai). Logarithm us awkward multiplication ko simple addition mein convert karta hai: $ln (W_{A} W_{B}) = ln W_{A} + ln W_{B}$ . Woh "addition" property exactly wahi hai jo hum entropy mein chahte hain.

Figure par notice karo: jaise $W$ shrink hokar $1$ ki taraf jaata hai, curve $ln W$ slide karke $0$ par aa jaata hai. Yeh single feature poore Third Law ka seed hai.

4. Boltzmann's constant $k_{B}$ — unit converter

$k_{B}$ ko ek exchange rate socho: $ln W$ ek abstract sense mein "messiness ki matra" hai, aur $k_{B}$ isse physical bookkeeping units mein convert karta hai jo baaki thermodynamics use karta hai.

Ek cousin jo tum miloge: puri ek mole ke liye ( $N_{A} = 6.022 \times 1 0^{23}$ particles) hum likhte hain $R = N_{A} k_{B} = 8.314 J mol^{- 1} K^{- 1}$ , gas constant. Jab bhi tum $R$ dekhte ho, isse " $k_{B}$ ek mole bhar ke particles ke liye" padho.

5. Entropy $S$ — messiness measured

Ab hum is show ke star ko assemble kar sakte hain.

Isse left to right plain words mein padho: "Entropy exchange-rate $k_{B}$ times the log of secret arrangements ki sankhya ke barabar hai." Poori story ke liye Boltzmann entropy S = k ln W dekhein.

Is topic ko iske liye kyun chahiye. Yahi woh bridge hai jo parent note baar baar cross karta hai. Third Law is formula ko $T \to 0$ par evaluate karne se zyada kuch nahi hai, jahan $W$ ghar kar $1$ ho jaata hai.

6. Ground state aur degeneracy $g$

$T \to 0$ par relevant count exactly yahi hai: $W = g$ . Boltzmann mein plug karo:

$S (0) = k_{B} ln g .$

Perfect crystal: $g = 1$ → $S (0) = k_{B} ln 1 = 0$ . (Third Law ki headline — exactly woh statement jo is page ke top par callout mein hai.)
Frozen-in disorder (jaise CO): $g > 1$ → $S (0) = k_{B} ln g > 0$ . Yeh leftover residual entropy kehlata hai; Residual entropy of ice and CO dekhein.

Is topic ko iske liye kyun chahiye. $g$ woh hinge hai jo "entropy exactly zero hai" aur "entropy ek leftover constant hai" ke beech hai. Parent note ka poora "perfect crystal" clause actually " $g = 1$ " clause hai.

7. Reversible heat $δ Q_{r e v}$ aur entropy ratio $\frac{δ Q _{r e v}}{T}$

Parent entropy ko ek doosre tarike se bhi compute karta hai — substance ko warm karte waqt heat add karke. Iske liye do aur symbols chahiye — aur pehle, ek word un do tarah ke "small change" ke baare mein jo $d$ aur $δ$ letters se likhe jaate hain.

$T$ se divide kyun, sabse pehle? Kyunki wahi sip of heat zyada extra disorder cause karta hai jab system thanda aur orderly ho, bajaaye jab woh pehle se hi hot aur messy ho. $T$ se divide karna exactly wahi encode karta hai: chhote $T$ par ratio bada hai, bade $T$ par chhota. Yeh recipe Second law of thermodynamics se aati hai.

8. Heat capacity $C_{p}$ aur integral $\int_{0}^{T}$

Recipe mein substitute karne par, warming ka ek sip contribute karta hai $d S = \frac{C _{p} d T}{T}$ . Absolute zero se $T$ tak built-up total entropy paane ke liye, hume saare sips add up karne padte hain. Woh "infinitely many tiny pieces ko add karna" exactly wahi hai jo integral sign ka matlab hai:

Is topic ko iske liye kyun chahiye, aur Third Law isse kyun kaam karata hai. Kyunki Third Law starting value $S (0) = 0$ ko perfect crystal ke liye pin karta hai, integral ek absolute entropy deta hai, sirf ek difference nahi — woh payoff Absolute entropy and standard molar entropy mein explore kiya gaya hai. Aur yeh tabhi converge hota hai (finite rehta hai) kyunki $C_{p} \to 0$ jaise substance thanda hota hai, ek fact Heat capacity and Debye T-cubed law se. Third Law ke bina $C_{p} \to 0$ force kiye, fraction $C_{p} / T$ bottom par blow up ho jaata.

Prerequisite map

Yeh diagram sirf dikhata hai kaun sa idea feed into karta hai kismein — har arrow ko "ke liye zaroori hai" padho. (Agar boxes unfamiliar lagte hain, syntax ignore karo aur arrows ko stepping stones ki tarah follow karo.)

Equipment checklist

Khud ko test karo — tum parent note ke liye ready ho jab har reveal obvious lage.

Third Law ko ek sentence mein state karo.

Ek perfect crystalline substance ki entropy zero ke paas jaati hai jaise temperature absolute zero ke paas jaata hai.

T \to 0

" ka plain words mein kya matlab hai?

Temperature ko absolute zero ke jitna paas lao jitna chaaho (kabhi bilkul wahan pahunche bina).

Unattainability principle kya hai?

Third Law ka ek companion form: absolute zero ko finite number of cooling steps mein reach nahi kiya ja sakta.

Microstate vs macrostate kya hai?

Microstate har particle ki ek detailed arrangement hai; macrostate outside-measurable summary hai (jaise total energy) jo kai microstates share karte hain.

Kis setting mein

S = k_{B} ln W

apne clean form mein exactly valid hai?

Fixed total energy par ek isolated system (microcanonical picture).

Kya us picture mein temperature ek primary macrostate label hai?

Nahi — yeh is baat se derived hai ki

S

energy ke saath kaise badalta hai (

\partial S / \partial E

);

T

aur

S

intertwined hain, independent inputs nahi.

Symbol

W

kya count karta hai, aur kya yeh probability hai?

Ek macrostate ke liye microstates ki sankhya — ek plain count, probability nahi (ek individual microstate ki probability

1/ W

hai).

ln 1

kya hai, aur kyun?

0

, kyunki

e^{0} = 1

isliye

1

banane ke liye zaroori power zero hai.

Entropy mein

W

directly use karne ki jagah

ln

kyun use karte hain?

Yeh huge counts ko shrink karta hai aur combined systems ke multiplication ko addition mein badal deta hai.

S = k_{B} ln W

mein

k_{B}

kya karta hai?

Unitless count

ln W

ko physical units (joules per kelvin) mein convert karta hai.

R

aur

k_{B}

mein kya sambandh hai?

R = N_{A} k_{B} = 8.314

J mol⁻¹K⁻¹, yaani ek mole ke liye

k_{B}

Ground state kya hai aur

T \to 0

hone par iske saath kya hota hai?

Single lowest-energy arrangement; koi thermal energy na bachne par, system isme settle ho jaata hai.

Degeneracy

g

kya measure karta hai, aur

S (0)

iske terms mein kya hai?

Lowest energy ke liye tied arrangements ki sankhya;

S (0) = k_{B} ln g

Ek perfect crystal ke liye

g

kya hai aur isliye

S (0)

kya hai?

g = 1

, toh

S (0) = k_{B} ln 1 = 0

d

aur

δ

mein kya fark hai?

d

ek state function ki infinitesimal change mark karta hai (jaise

d S

);

δ

ek path-dependent transfer ki infinitesimal matra mark karta hai (jaise

δ Q

δ Q_{r e v} = C_{p} d T

kin conditions mein hold karta hai?

Sirf ek reversible, constant-pressure process ke liye, jahan

C_{p} \equiv (\partial Q_{r e v} / \partial T)_{p}

d S = δ Q_{r e v} / T

kya kehta hai, aur

T

se divide kyun karte hain?

Heat ka ek gentle sip stored entropy ko heat over temperature se badhata hai;

T

se divide karna capture karta hai ki thande systems mein har sip par zyada disorder aata hai.

\int_{0}^{T} (\dots) d T^{'}

ka kya matlab hai?

Temperature

0

T

tak quantity ke tiny slivers ka continuous sum.

Absolute-entropy formula mein

S (0)

term kab zero hota hai?

Sirf ek perfect crystal ke liye (

g = 1

); agar

g > 1

toh tumhe

S (0) = k_{B} ln g > 0

rakhna padega.

Entropy integral

T = 0

par kyun converge karta hai?

Kyunki

C_{p} \to 0

(

T

se tez), toh

C_{p} / T

finite rehta hai.

Connections

Hinglish version of the parent
Boltzmann entropy S = k ln W — woh formula jismein yahan ke har symbol feed hote hain.
Second law of thermodynamics — $d S = δ Q_{r e v} / T$ recipe ka source.
Heat capacity and Debye T-cubed law — kyun $C_{p} \to 0$ taaki integral converge ho.
Absolute entropy and standard molar entropy — $S (0) = 0$ ka payoff.
Residual entropy of ice and CO — kya hota hai jab $g > 1$ .
Adiabatic demagnetization and reaching low temperatures — $T \to 0$ chase karna aur unattainability principle action mein.