Maxwell relations — derivation from each potential: Foundations

2.4.3 · D1 · Physics › Thermodynamics & Statistical Mechanics (Advanced) › Maxwell relations — derivation from each potential

Is page ke liye assume kiya gaya hai ki tumne parent page ka koi bhi notation nahi dekha. Hum har symbol ko zero se banate hain, use ek picture se anchor karte hain, aur tabhi use karte hain. Upar se neeche padho; yahan koi bhi cheez uske neeche likhi kisi cheez par depend nahi karti.

1. Do variables ki ek function — "landscape"

Kisi bhi physics se pehle, hume ek aisi quantity ka idea chahiye jo do knobs par ek saath depend karti ho.

Figure — Maxwell relations — derivation from each potential

Poora topic isi tarah ki pictures par rehta hai. Baad mein, "height" $z$ ek energy hogi aur floor-axes $(x, y)$ physical quantities honge. Lekin geometry bilkul yahi hill wali hai.

Recall Hume ek ki jagah

do variables kyun chahiye? Kyunki ek gas ko apni state pin down karne ke liye do numbers chahiye ::: jaise uska volume aur uska temperature. Ek number kaafi nahi hai yeh batane ke liye ki gas kya kar rahi hai.

2. Ek direction mein slope — partial derivative

3. Total change — differential $d z$

Ab: agar main ek chhota sa step lun jo thoda East aur thoda North dono ek saath ho, toh height mein total kitna change hoga?

Hum do slopes ko short names dete hain, $M$ aur $N$ : $d z = M d x + N d y, M \equiv (\frac{\partial z}{\partial x})_{y}, N \equiv (\frac{\partial z}{\partial y})_{x} .$

4. Crossed-slopes-agree fact (Schwarz / equality of mixed partials)

Yahi poore topic ka engine hai.

Yeh ek line hi har Maxwell relation hai. Jab ek baar hume pata ho ki thermodynamics mein $z, x, y, M, N$ ki jagah kya hai, hum bas plug in kar dete hain.

Recall Koi function ke kaam karne ke liye kaunsi ek condition honi chahiye?

Use smooth hona chahiye (ek genuine height function / state function) ::: taaki uski value sirf is par depend kare ki tum kahaan ho, na ki us path par jis se tum wahan pahunche.

5. State function vs. path — energy kyun qualify karti hai

Chaar thermodynamic potentials $U, H, F, G$ state functions (landscapes par heights) hain, toh unke differentials exact hain aur Section 4 har ek par apply hota hai. Heat $δ Q$ aur work $δ W$ nahi hain, yahi wajah hai ki parent note ko unhe $T d S$ aur $P d V$ ke zariye convert karna hota hai aage badhne se pehle.

6. Physical axes — $T, S, P, V$ se miliye

Ab hum abstract $(x, y, z)$ ko real physical quantities se replace karte hain. Char symbols poore subject ko carry karte hain.

$S$ aur $V$ measure karne mein opposite tarike se aasaan-ya-mushkil hain: tum $P, V, T$ ko gauges se easily padh sakte ho, lekin tum nahi ek probe andar dalke $S$ padh sakte. Yahi asymmetry poori wajah hai ki Maxwell relations useful hain — woh ek unmeasurable slope (jaise $\partial S / \partial P$ ) ko ek measurable slope se (jaise $\partial V / \partial T$ ) trade karti hain.

7. Master differential $d U = T d S - P d V$

Ab abstract " $d z = M d x + N d y$ " physical ban jaata hai.

Kyunki $U$ ek state function hai (Section 5), crossed-slopes fact (Section 4) us par apply hota hai — aur $H, F, G$ par bhi jab hum unhe banate hain.

8. Legendre transform — kaunse axis par tum khade ho, yeh badalna

Parent note $H, F, G$ ko $U$ se " $P V$ add karke" ya " $T S$ subtract karke" banata hai. Yahan picture hai.

Definition "Natural variables"

Woh pair of axes jiske liye ek potential ka differential clean coefficients ke saath aata hai ( $T, S, P, V$ aur kuch messy nahi). $U$ ke liye woh $(S, V)$ hain. Transforms dete hain:

potential	$U$ hai, saath mein...	natural axes
$U$	—	$(S, V)$
$H = U + P V$	$V$ ko $P$ se trade kiya	$(S, P)$
$F = U - T S$	$S$ ko $T$ se trade kiya	$(T, V)$
$G = U - T S + P V$	dono trade kiye	$(T, P)$

Poori machinery Thermodynamic potentials & Legendre transforms mein hai. Is page ke liye tumhe bas itna chahiye: har potential ek aur smooth hill hai, ek alag pair of axes par drawn, aur Section 4 har ek par apply hota hai.

Foundations topic ko kaise feed karte hain

Ise aise padho: calculus (left branch) engine deta hai; physics (right branch) hills supply karta hai; dono milke chaar Maxwell relations produce karte hain.

Equipment checklist

Apne aap ko test karo — agar koi reveal tumhe surprise kare, toh main note se pehle woh section dobara padho.

Curly $\partial$ tumhe kya warn karta hai, jo straight $d$ nahi karta? ::: Ki doosre variables held fixed hain; yeh ek frozen direction mein slope hai.
$(\partial z / \partial x)_{y}$ mein subscript tumhe kya batata hai? ::: Exactly kaunsa variable constant rakha gaya hai — aur yeh quantity ka hissa hai, optional nahi.
$d z = M d x + N d y$ mein $M$ kya hai? ::: $x$ -direction mein height ka slope, $(\partial z / \partial x)_{y}$ — woh number jo $d x$ ke saath multiply ho raha hai.
Equality of mixed partials ek line mein state karo. ::: $(\partial M / \partial y)_{x} = (\partial N / \partial x)_{y}$ — crossed slopes agree karte hain.
Yeh equality $U, H, F, G$ ke liye kyun hold karti hai lekin heat $δ Q$ ke liye nahi? ::: $U, H, F, G$ state functions (heights) hain; heat ek path function hai, kisi bhi hill ka differential nahi.
Do conjugate pairs ke naam batao. ::: $(T, S)$ aur $(P, V)$ .
Term $- P d V$ negative kyun hai? ::: Expansion surroundings par kaam karta hai, internal energy drain karta hai, toh $U$ drop karta hai jab $V$ badhta hai.
Legendre transform kya change karta hai, aur kya preserve karta hai? ::: Yeh swap karta hai ki kaunsa variable natural axis hai (jaise $V \to P$ ); yeh saari physical information preserve karta hai.

Connections

Parent topic — jahan ye foundations use hoti hain
Equality of mixed partial derivatives (Schwarz theorem) — engine (Section 4)
First and Second Laws of Thermodynamics — $d U = T d S - P d V$ ka source (Section 7)
Thermodynamic potentials & Legendre transforms — $H, F, G$ banana (Section 8)