Derivation of $(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{p}}{)}_T=-T(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_p+V$

In the Exercise $10-1,$ we derived

$dH=TdS+Vdp$

in a closed system of constant composition, and in no additional

non$-$expansion work.

Step $1$ Derive an expression of $(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{p}}{)}_T$ using eq$.1$

Divide the both sides of eq$.1$ by $,$ and impose the constraint

of constant

Then, we obtain

$(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{p}}{)}_T=$

$=T(\frac{\text{d}\text{e}\text{l}\text{S}}{\text{d}\text{e}\text{l}\text{p}}{)}_T+V$

Step $2$ Apply a Maxwell relation to eq$.2$

Using a Maxwell relation of

$(\frac{\text{c}\text{d}\text{o}\text{t}\text{s}}{\text{v}\text{d}\text{o}\text{t}\text{s}\text{v}\text{d}\text{o}\text{t}\text{s}})=$

results in

$(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{p}}{)}_T=-T(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_p+V$

So$,$ we have successfully expressed $(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{p}}{)}_T$ only with $p,V,$ and $T.$