[PChem] Van de Waals Partial Derivative

[Coop]

Homework Statement

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

Homework Equations

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

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...

$$(\frac{V^2-(p+an^2)2V(\frac{dV}{dp})}{V^4}(V-nb)+(\frac{p+an^2}{V^2})(\frac{dV}{dp})=0$$

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

$$\frac{dV}{dp}=\frac{V-nb}{\frac{2(p+an^2)(V-nb)}{V}-p+an^2}$$

...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!

 

[Ray Vickson]

Homework Statement

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

Homework Equations

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

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...

$$(\frac{V^2-(p+an^2)2V(\frac{dV}{dp})}{V^4}(V-nb)+(\frac{p+an^2}{V^2})(\frac{dV}{dp})=0$$

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

$$\frac{dV}{dp}=\frac{V-nb}{\frac{2(p+an^2)(V-nb)}{V}-p+an^2}$$

...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!

Here is what I get using Maple:

> f:=n*R*T=(V-n*b)*(p+a*n^2)/V^2; <--- input
f:=n*R*T = (V-n*b)*(p+a*n^2)/V^2 <--- echo of input

> implicitdiff(f,V,p); <---command
(V-n*b)*V/(V*p+V*a*n^2-2*n*b*p-2*n^3*b*a) <---- the implicit derivative

In LaTeX this is
$$\frac{\partial V}{\partial p} = \frac{(V - nb)V}{Vp+Va\,n^2-2nbp-2n^3\,b a}$$
If you multiply both the numerator and denominator of your expression by $V$ you get Maple's numerator. Do you also get Maple's denominator after expanding out yours?

 

[Coop]

Ray,

I do! Thanks :) Is Maple a free program I can use to check my answers?

 

[vela]

You can make the calculation a bit easier if you rearrange the initial expression as
$$\frac{1}{nRT}(p+an^2)=\frac{V^2}{V-nb}$$ and then differentiate implicitly.

 

[Ray Vickson]

Ray,

I do! Thanks :) Is Maple a free program I can use to check my answers?

No, not free, but some departments/universities have site licences. Alternatively, you can use Mathematica (perhaps through an institutional site licence). However, I do not have access to that, so I don't know what would be the appropriate commands.

Wolfram Alpha is a free, on-line computer algebra/calculus package, but has limitations---it is Mathematica Lite. I don't know the "implicit differentiation" commands for it, but I am sure some on-line help is available.

There are also a number of free computer algebra/calculus packages available for downloading to your own computer. Just do a Google search.

 

[Coop]

Thanks for both your help, guys.

P.S. I found a free download for Maple from my university.