Helmholtz Free Energy

Suppose that $T$ and $V$ are the are the two independent parameters that specify the system. Because

$$T dS = d(T S)-S dT,$$ (6.100)

we can rewrite Equation (6.82) in the form

$$dF = -S dT - p dV,$$ (6.101)

where

$$F = E - T S$$ (6.102)

is termed the Helmholtz free energy.

Proceeding as before, we write

$$F= F(T,V),$$ (6.103)

which implies that

$$dF= \left(\frac{\partial F}{\partial T}\right)_V dT + \left(\frac{\partial F}{\partial V}\right)_T dV.$$ (6.104)

Comparison of Equations (6.101) and (6.104) yields

$$\left(\frac{\partial F}{\partial T}\right)_V = -S,$$ (6.105)

$$\left(\frac{\partial F}{\partial V}\right)_T = -p.$$ (6.106)

We also know that

$$\frac{\partial^{ 2} F}{\partial V \partial T} = \frac{\partial^{ 2}F}{\partial T \partial V},$$ (6.107)

or

$$\left(\frac{\partial}{\partial V}\right)_T\left(\frac{\partial F}... ...rac{\partial}{\partial T}\right)_V\left(\frac{\partial F}{\partial V}\right)_T.$$ (6.108)

Thus, it follows from Equations (6.105) and (6.106) that

$$\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V.$$ (6.109)