Part 1

1st law:

(1)

$du=\delta{Q}+\delta{W}=Tds-Pdv$

Enthalpy definition:

(2)

$h=u+Pv$

Enthalpy differential:

(3)

$dh=du+Pdv+vdP=Tds+vdP$

Maxwell equation from (2):

(4)

$(\frac{\partial{T}}{\partial{P}})_v=(\frac{\partial{v}}{\partial{s}})_T$

Definition of heat capacity and insertion (1) into definition:

(5)

$c_v=(\frac{\partial{u}}{\partial{T}})_v=T(\frac{\partial{s}}{\partial{T}})_v$

Write entropy as function of T and v:

(6)

$s=s(T,v)$

Derive expression for entropy differential and use (4) and (5):

(7)

$ds=(\frac{\partial{s}}{\partial{T}})_vdT+(\frac{\partial{s}}{\partial{v}})_T=\frac{c_v}{T}dT+(\frac{\partial{P}}{\partial{T}})_vdv$

Insert (7) into (1):

(8)

$du=\delta{Q}+\delta{W}=Tds-Pdv=T(\frac{c_v}{T}dT+(\frac{\partial{P}}{\partial{T}})_vdv)-Pdv=(T(\frac{\partial{P}}{\partial{T}})_v -P)+c_vdT$

QED

Part 2

van der Waals equation:

(9)

$(p+\frac{a}{v^2})(v-b)=kT$

Express p:

(10)

$p=\frac{kT}{v-b}-\frac{a}{v^2}$

At constant T equation (8) becomes:

(11)

$(\frac{\partial{u}}{\partial{v}})_T=T(\frac{\partial{P}}{\partial{T}})_v-p$

Differentiate (10) at constant v and put into (11):

(12)

$T(\frac{\partial{P}}{\partial{T}})_v-p=\frac{kT}{v-b}-p=\frac{a}{v^2}$

QED

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