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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

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

we can rewrite Equation (6.82) in the form

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

where

$\displaystyle H = E + p V$ (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

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

which implies that

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

Comparison of Equations (6.91) and (6.94) yields

$\displaystyle \left(\frac{\partial H}{\partial S}\right)_p$ $\displaystyle = T,$ (6.95)
$\displaystyle \left(\frac{\partial H}{\partial p}\right)_S$ $\displaystyle = V.$ (6.96)

We also know that

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

or

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

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

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


next up previous
Next: Helmholtz Free Energy Up: Classical Thermodynamics Previous: Internal Energy
Richard Fitzpatrick 2016-01-25