[PChem] Van de Waals Partial Derivative

1. The problem statement, all variables and given/known data



Find [tex](\frac{dV}{dp})_{n,T}[/tex] for the Van de Waals gas law



2. Relevant equations



Van de Waals gas law: [tex](\frac{p+an^2}{V^2})(V-nb)=nRT[/tex]



3. The attempt at a solution



I just started doing problems like these so I would like to know if I am doing them right...



What I did was I took the implicit derivative of dV WRT dp for both sides...



[tex](\frac{V^2-(p+an^2)2V(\frac{dV}{dp})}{V^4}(V-nb)+(\frac{p+an^2}{V^2})(\frac{dV}{dp})=0[/tex]



...Solve for dV/dp and I ended up getting...



[tex]\frac{dV}{dp}=\frac{V-nb}{\frac{2(p+an^2)(V-nb)}{V}-p+an^2}[/tex]



...Can anyone verify this is correct? Or if it is not correct, can you verify I took the right approach? I haven't done calculus in a while before this course, but is this the correct approach to implicit differentiation? Differentiate p as normal but tack on a "dV/dp" after differentiating a V?



Thanks!