We start from the definition of Gibbs energy and the fundamental relation.
Step 1 — Definition.G=H−TSWhy this step? This is the definition of G; everything must follow from it.
Step 2 — Get the entropy as a derivative of G.
From the differential dG=−SdT+VdP (which itself comes from dG=dH−TdS−SdT and dH=TdS+VdP cancelling the TdS terms), at constant P:
(∂T∂G)P=−SWhy this step? We need to replace S by something involving G, so H is the only unknown left.
Step 3 — Substitute S into the definition.G=H−T[−(∂T∂G)P]=H+T(∂T∂G)P
Rearrange:
H = G - T\left(\frac{\partial G}{\partial T}\right)_P \tag{$\star$}Why this step? This already gives H in terms of G and its slope — but it's ugly. The G/T trick cleans it up.
Step 4 — Differentiate G/T using the quotient rule.∂T∂(TG)P=T2T(∂T∂G)P−G⋅1=T1(∂T∂G)P−T2GWhy this step? We chose G/T precisely so that the numerator is T∂TG−G, which is −[G−T∂TG]=−H by (⋆).
Step 5 — Insert (⋆).
Multiply the numerator by −1: numerator =−(G−T∂TG)=−H. Therefore
(∂T∂(G/T))P=−T2HWhy this step? The whole point: the S disappeared, H survives.
Why differentiate G/T instead of G? → because T∂TG−G=−H, so entropy cancels.
What is (∂G/∂T)P? → −S.
State GH in 1/T form. → ∂(ΔG/T)/∂(1/T)=ΔH.
What constant-volume equation parallels it? → ∂(A/T)/∂(1/T)=U.
Gibbs–Helmholtz equation (T form)
(∂T∂(G/T))P=−T2H
Gibbs–Helmholtz equation (1/T form)
(∂(1/T)∂(G/T))P=H
What does (∂G/∂T)P equal?
−S (entropy), NOT H
Why divide by T before differentiating?
So the −TS entropy term cancels, leaving only H
Helmholtz (constant-V) analogue
(∂(1/T)∂(A/T))V=U
Two-point integrated form (const ΔH)
T2ΔG2−T1ΔG1=ΔH(T21−T11)
GH applied to ΔG∘=−RTlnK gives?
van 't Hoff: d(1/T)dlnK=−RΔH∘
Plot to extract ΔH
ΔG/T vs 1/T; slope =ΔH
Starting relation for derivation
G=H−TS and dG=−SdT+VdP
Recall Feynman: explain to a 12-year-old
Imagine your "happiness budget" G is made of real money H minus a "fun tax" TS that grows with the temperature T. If I just watch how the budget changes when it gets hotter, I'm really only seeing the tax change — that hides the real money. But if I look at "budget per degree" (G/T) and watch that, the tax part cancels out and I can finally see how much real money H I actually have. Gibbs–Helmholtz is just that smart way of looking.
Dekho, Gibbs free energy G=H−TS hota hai — usme do cheezein mixed hain: enthalpy H (asli heat content) aur entropy term TS. Problem yeh hai ki agar tum seedha G ko temperature ke saath differentiate karoge, toh tumhe sirf entropy milti hai: (∂G/∂T)P=−S. Yani H chhup jaata hai. Gibbs–Helmholtz ka jugaad yeh hai ki pehle G ko T se divide karo, phir differentiate karo — tab TS wala term magically cancel ho jaata hai aur sirf H bachta hai: ∂(G/T)/∂T=−H/T2.
Sabse useful form yeh hai: G/T ko 1/T ke against plot karo, toh uska slope hi ΔH ban jaata hai (koi minus sign tension nahi). Yahi reason hai ki van 't Hoff equation isi se nikalti hai — bas ΔG∘=−RTlnK daalo aur dlnK/d(1/T)=−ΔH∘/R aa jaata hai.
Practically iska faayda kya? Tum reaction ka ΔG alag-alag temperatures pe measure kar sakte ho (equilibrium constant K se), aur uske slope se directly heat of reaction ΔH nikaal sakte ho — bina calorimeter ke. Bas ek dhyan rakhna: yeh ΔH ko constant maan ke chalta hai, toh bahut bade temperature range pe Kirchhoff ki correction lagani padti hai. Mantra yaad rakho: "T se divide karo, entropy bhaag jaayegi, H reh jaayega."
