For a closed system, the differential of the Helmholtz free energy is:

d F \leq - P d V - T d S

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d F \leq P d V - T d S

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d F \geq - P d V - T d S

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d F \geq P d V - T d S

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Solution

The correct option is A $d F \leq - P d V - T d S$
Helmholtz free energy is given by $F = U - T S$
We know $d U = T d S - P d V$
Taking differential of Helmholtz free energy we get
$d F = d U - T d S - S d T d F = T d S - P d V - T d S - S d T d F = - P d V - S d T$
At constant temperature and volume $d W = 0$ (Closed System)
$d U = δ q$
Also $d S \geq \frac{δ q}{T}$ So we have $d U \leq T d S$
Hence $d F \leq - P d V - T d S$

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