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GATE 2022 PH – Question 17

Thermodynamics and Statistical Mechanics · Laws of thermodynamics · 1 mark · Multiple choice

Which relation between internal energy U and Helmholtz free energy F is true?

  1. $U=-T^2[\partial(F/T)/\partial T]_V$
  2. $U=T^2[\partial(F/T)/\partial T]_V$
  3. $U=T(\partial F/\partial T)_V$
  4. $U=-T(\partial F/\partial T)_V$

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Correct answer: (A) $U=-T^2[\partial(F/T)/\partial T]_V$

Explanation

**Definitions.** The Helmholtz free energy is $F=U-TS$, and its differential is $dF=-S\,dT-P\,dV$, so the entropy is
$$S=-\left(\frac{\partial F}{\partial T}\right)_V .$$

**Internal energy:**
$$U=F+TS=F-T\left(\frac{\partial F}{\partial T}\right)_V .$$

**Rewrite** using the quotient rule: $\dfrac{\partial}{\partial T}\!\left(\dfrac FT\right)=\dfrac1T\dfrac{\partial F}{\partial T}-\dfrac{F}{T^2}$, so
$$-T^2\frac{\partial}{\partial T}\left(\frac FT\right)=F-T\frac{\partial F}{\partial T}=U .$$

$$U=-T^2\left[\frac{\partial(F/T)}{\partial T}\right]_V\quad(\text{option A}).$$