Enthalpy

The analysis in the previous section is based on the premise that $S$ and $V$ are the two independent parameters that specify the system. Suppose, however, that we choose $S$ and $p$ to be the two independent parameters. Because

$$p dV = d(p V)-V dp,$$ (6.90)

we can rewrite Equation (6.82) in the form

$$dH = T dS + V dp,$$ (6.91)

where

$$H = E + p V$$ (6.92)

is termed the enthalpy. The name is derived from the Greek enthalpein, which means to ``to warm in."

The analysis now proceeds in an analogous manner to that in the preceding section. First, we write

$$H = H(S,p),$$ (6.93)

which implies that

$$dH = \left(\frac{\partial H}{\partial S}\right)_p dS + \left(\frac{\partial H}{\partial p}\right)_S dp.$$ (6.94)

Comparison of Equations (6.91) and (6.94) yields

$$\left(\frac{\partial H}{\partial S}\right)_p = T,$$ (6.95)

$$\left(\frac{\partial H}{\partial p}\right)_S = V.$$ (6.96)

We also know that

$$\frac{\partial^{ 2} H}{\partial p \partial S} = \frac{\partial^{ 2}H}{\partial S \partial p},$$ (6.97)

or

$$\left(\frac{\partial}{\partial p}\right)_S\left(\frac{\partial H}... ...rac{\partial}{\partial S}\right)_p\left(\frac{\partial H}{\partial p}\right)_S.$$ (6.98)

Thus, it follows from Equations (6.95) and (6.96) that

$$\left(\frac{\partial T}{\partial p}\right)_S = \left(\frac{\partial V}{\partial S}\right)_p.$$ (6.99)