Question

for constants a ,b, n, R, Van der waals equation relates the pressure P to the volume V of a fixed quantity of a gas at constant temperature T:

Answer 1

\[(P + (n^2a)/V^2) (V -nb) = nRT\]

Find the rate of change of volume with pressure dV/dP

Answer 2

So you just need to take a derivative with respect to P right? The only variables that depend on P are P and V right? So differentiate:
\[\frac{d}{dV}(P+\frac{n^2a}{V^2})(V-nb)=\frac{d}{dV}nRT\]
First look at the right side, since n,R, and T are constants then the derivative with respect to V of that product is zero.
For the left side, we have a term of n^2a/V^2 and V-nb, so we need a PRODUCT RULE:
\[\frac{d}{dx}f(x)g(x)=f'(x)g(x)+f(x)g'(x) \]
So:
\[\implies 0 = \frac{d}{dV}(P+\frac{n^2 a}{V^2})(V-nb)+(P+\frac{n^2 a}{V^2}) \frac{d}{dV}(V-nb)=\]
\[(\frac{dP}{dV}-\frac{2 n^2 a}{V^3})(V-nb)+(P+\frac{n^2 a}{V^2})(1)=0 \implies \frac{dP}{dV}=\frac{P+\frac{n^2 a}{V^2}}{nb-V}+\frac{2n^2 a}{V^3}\]
The way I see it.

Answer 3

thank you that makes so much more sense then when my teachers was on this topic.

Answer 4

You're welcome :D