Thermal physics: Van der Waals

Answer

Van der Waals equation is

$[P+a*(n/V)^2]*(V/n -b) = RT$

with $n=N/N_a$ and $a$,$b$ constants depending on gas

By differentiating we get

$d P*(V/n -b) =R*d T$

$d P/d T = R/(V/n -b)$

$d T/d V = (-1/C_v)[RT / (V/n -b) -P]$

with $P =RT / (V/n -b) -a*(n/V)^2$

$d T/d V = (-1/C v)*[RT/(V/n -b) – RT/(V/n-b) +a*(n/V)^2]$

$d T = (-1/C_v)*a*(n/V)^2*d V$

By integrating we obtain

$Delta(T) = T2-T1 =(+1/C_v)*a*n^2 *(1/V2-1/V1)$

where constant $a$ depends on gas.