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[PChem] Van de Waals Partial Derivative

  1. Aug 29, 2014 #1
    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!
     
  2. jcsd
  3. Aug 29, 2014 #2

    Ray Vickson

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    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
    [tex] \frac{\partial V}{\partial p} = \frac{(V - nb)V}{Vp+Va\,n^2-2nbp-2n^3\,b a} [/tex]
    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?
     
  4. Aug 29, 2014 #3
    Ray,

    I do! Thanks :) Is Maple a free program I can use to check my answers?
     
  5. Aug 29, 2014 #4

    vela

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    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.
     
  6. Aug 29, 2014 #5

    Ray Vickson

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    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.
     
  7. Aug 30, 2014 #6
    Thanks for both your help, guys.

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