Which of the following relationship between the internal energy U and the Helmholtz's
free energy F is true?
Options:
(A) U = -T2
š (F
T)
šT
V
(B) U = +T2
š (F
T)
šT
V
(C) U = +T šF
šT2
V
(D) U = -T šF
šT2
V
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.17)**
**1. Thermodynamics Principles & Helmholtz Free Energy**:
- The Helmholtz free energy $F$ is defined via Legendre transform as:
$F = U - TS$
- Fundamental thermodynamic differential:
$dF = -S\,dT - P\,dV \implies S = -\left(\frac{\partial F}{\partial T}\right)_V$
**2. Step-by-Step Derivation**:
- Substitute entropy $S$ back into the expression for internal energy $U$:
$U = F + TS = F - T\left(\frac{\partial F}{\partial T}\right)_V$
- Now consider the derivative of $\frac{F}{T}$ with respect to $T$ at constant volume:
$\left(\frac{\partial}{\partial T}\left(\frac{F}{T}\right)\right)_V = \frac{T \left(\frac{\partial F}{\partial T}\right)_V - F}{T^2}$
- Multiplying both sides by $-T^2$:
$-T^2 \left(\frac{\partial}{\partial T}\left(\frac{F}{T}\right)\right)_V = F - T \left(\frac{\partial F}{\partial T}\right)_V = U$
- Thus, the exact thermodynamic relationship is:
$U = -T^2 \left[ \frac{\partial (F/T)}{\partial T} \right]_V$
**3. Official Verification**:
The verified official IIT Master Answer is **Option (A)**.
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