2.4.2 · D5 · HinglishThermodynamics & Statistical Mechanics (Advanced)
Question bank — Legendre transforms connecting them
2.4.2 · D5· Physics › Thermodynamics & Statistical Mechanics (Advanced) › Legendre transforms connecting them
Prerequisites jo khulle rakhne chahiye: Internal Energy and the First Law, Maxwell Relations, Thermodynamic Stability and Convexity.
0. Woh toolkit jin par ye traps based hain (pehle padho)
Is bank mein paanch objects use hote hain. Yahan sab scratch se define kiye hain, taaki neeche koi bhi symbol mystery na rahe.
True or false — justify karo
Ek Legendre transform koi information nahi khoता kyunki woh ek involution hai (do baar karne par original wapas milta hai).
True. Slope relation se, ko dobara transform karne par slope milti hai aur rebuild ho jaata hai; kyunki tum hamesha wapas aa sakte ho, original data ka har bit ke andar maujood hai — sirf (slope, intercept) ke roop mein re-encode hua hai.
enthalpy ki sahi definition hai.
False. mein ka slope hai (from ), toh Legendre subtract karta hai, matlab hum add karte hain : .
substitute karke ko ka function likhna ek valid fundamental relation deta hai.
False. Fundamental matlab apne natural variables mein exact ho; woh substitution ki value rakhta hai lekin intercept bhool jaata hai, toh tum recover nahi kar sakte bina integrate kiye. Sirf sab kuch rakhta hai, ke zariye.
Gibbs free energy "lab favourite" hai kyunki uske dono natural variables aur aisa knobs hain jo ek experimenter control kar sakta hai.
True. Ek bench par rakha beaker fixed (room/bath) aur (atmosphere) par hota hai; kyunki , yahi exactly ke natural variables hain, jis se equilibrium ek saaf -minimization ban jaata hai.
Har Legendre-generated potential automatically ek Maxwell relation deta hai.
True. Har ek exact differential hai, toh mixed-partial test ke zariye uske do second partials barabar hone chahiye; par woh test lagaane se milta hai, aur mein se har ek ek Maxwell relation deta hai.
Legendre transform ke liye function ka convex hona zaroori hai, jo thermodynamics ke liye ek serious restriction hai.
False ("serious restriction" wala part). ki invertibility ke liye convexity chahiye, aur stability usse supply karti hai: jaise ( mein upar ki taraf khulta hai). Toh stable equilibria ke liye transform hamesha well-defined hota hai — ek limitation nahi, balki ek feature hai.
aur mein exactly ek hi physical information hai.
True. ek Legendre transform hai aur isliye ek involution, toh invertible hai; aur ek hi fundamental surface ke do encodings hain, bas alag independent variables se parametrize kiye gaye hain.
Kyunki mein minus sign hai, yahan entropy negative ho sakti hai.
False. Minus sign slope relation ki khasiyat hai (yahan , ); iska matlab negative nahi hota. abhi bhi hold karta hai; bas fixed par badhne se ghatata hai.
Error dhundho
" ko se paane ke liye, hum ko se swap karte hain."
Galat pair. conjugate pair ko swap karta hai, ko nahi. swap karne se milta hai.
" par tangent line ka intercept bas hai."
Nahi — tangent ki value par hai, par nahi. -intercept hai , yahi toh construction ka poora point hai (upar figure dekho).
"Kyunki aur ka slope tha, ka bhi slope hai."
ka slope hai, nahi. differentiate karne par term vanish ho jaati hai aur sirf bachta hai.
" hume bataata hai ki ke natural variables aur hain."
Ulta hai. Isme aane wale differentials batate hain ki natural variables extensive hain; aur conjugate slopes hain, natural variables nahi.
"Hum add karte hain banane ke liye kyunki pressure positive hota hai."
ka sign rule se irrelevant hai. Hum add karte hain kyunki slope negative hai, aur Legendre slope×variable subtract karta hai: .
" do swaps ke dono minus signs collect karta hai."
-swap subtract nahi, add karta hai. Sahi hai , jisse milta hai .
"Maxwell relations equation of state se aate hain."
Woh ki exactness se aate hain (mixed partials commute karte hain), kisi bhi specific equation of state se independent. Isliye woh har substance ke liye hold karte hain.
Why questions
Kyun mein minus sign thermodynamics ke liye transform ka "dil" hai?
Kyunki woh se driven differential ko se driven mein convert karta hai: naye potential ka independent variable ban jaata hai slope , exactly wahi lab-controllable quantity jo hum chahte the.
Kyun hum dono mein simultaneously natural ek potential nahi bana sakte?
aur ek conjugate pair hain — ek doosre ke saath potential ka slope hai. Ek fundamental relation har pair ka ek member use karta hai, dono ek saath nahi.
Kyun stability guarantee karta hai ki Legendre inversion single-valued hai?
Stability convexity force karti hai (), toh strictly monotonic hai; ek strictly increasing slope function one-to-one hai aur isliye har jagah invertible hai.
Kyun naive substitution fundamental nahi hai jabki hai?
Geometrically, kai curves ek hi slopes ka set share kar sakti hain agar tum intercepts bhool jaao. intercept ko term ke zariye store karta hai, toh unique curve pin ho jaati hai; substituted nahi karta.
Kyun bench-top reaction ke liye sabse clean equilibrium criterion deta hai?
Fixed par, , toh equilibrium sirf chemical potentials balance karne tak reduce ho jaata hai — dekho Chemical Potential and Phase Equilibrium. Koi doosra potential ko apna natural variable nahi banata.
Kyun ko "free" energy kehte hain?
term woh energy subtract karta hai jo disordered thermal motion mein bandhi hai, bacha rahe woh hissa jo constant par kaam karne ke liye "free" hai.
Kyun wahi Legendre machinery mechanics mein Hamiltonian ke roop mein milti hai?
Wahan woh velocity ko uske conjugate momentum se swap karta hai: — dekho Lagrangian to Hamiltonian (Legendre in Mechanics). Same (slope, intercept) re-encoding, alag naam.
Edge cases
Agar ho kisi point par (ek inflection, zero curvature), toh Legendre transform ka kya hoga?
Slope locally invertible rehna band ho jaata hai (uski derivative vanish ho jaati hai), toh transform wahan singular ho jaata hai — yahi exactly woh jagah hai jahan phase transition / loss of stability dikhti hai.
Ek first-order phase transition par ek curve mein kink hota hai (slope jump karta hai). Iska ke liye kya matlab hai?
discontinuously jump karta hai — do phases ke alag volumes hain. mein kink transition ka geometric signature hai; transform ki slope uss ek pressure par multi-valued hai.
Agar ka koi region concave ho convex ke bajaye (unstable), toh Legendre transform kya karta hai?
Concave matlab , toh slope one-to-one nahi rehti aur naive transform multi-valued / non-physical branch deta hai; physical fix hai convex-hull (common-tangent) construction, jo unstable branches ko flat coexistence lines se replace karta hai — dekho Thermodynamic Stability and Convexity.
limit mein kya ban jaata hai, aur woh consistent kyun hai?
jab , kyunki term vanish ho jaata hai; consistent hai kyunki absolute zero par koi thermal energy "free up" karne ke liye nahi hoti, toh free energy aur internal energy ek ho jaate hain.
ke liye ke saath, uska transform concave kyun hai (neeche khulta hai)?
convex hai (), aur ek convex hamesha ek concave mein transform hoti hai (yahan ); sign flip seedha slope relation se aata hai aur involution verify karta hai jab tum wapas transform karte ho.
Recall Jaane se pehle ek-line self-test
Dhako aur zor se jawab do. Har Legendre step ek variable ke saath teen kya kaam karta hai, bolo. ::: Ek extensive variable ko uske conjugate intensive slope se swap karta hai, term add karta hai intercept preserve karne ke liye, aur ek naya potential deta hai jo us slope mein natural ho.