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Homework Help: Equation of State of an Ideal Gas

  1. Nov 2, 2008 #1
    1. The problem statement, all variables and given/known data
    Show that:-
    a) the expansivity [tex]\beta[/tex] = [tex]\frac{1}{T}[/tex]
    b) the isothermal compressibilty [tex]\kappa[/tex] = [tex]\frac{1}{P}[/tex]

    2. Relevant equations
    P[tex]\upsilon[/tex] = RT where [tex]\upsilon[/tex] = molar volume

    3. The attempt at a solution
    A big mess!
    Last edited: Nov 3, 2008
  2. jcsd
  3. Nov 2, 2008 #2


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    Hi LeePhilip01, welcome to PF. Do you know how the expansivity and isothermal compressibility are defined in general? (Hint: it will involve derivatives.)
  4. Nov 3, 2008 #3
    Yes, however i wasn't sure whether they were important because they weren't given in th question.

    [tex]\beta[/tex] = [tex]\frac{1}{V}[/tex] . [tex]\frac{dV}{dT}[/tex]

    [tex]\kappa[/tex] = - [tex]\frac{1}{V}[/tex] . [tex]\frac{dV}{dP}[/tex]
  5. Nov 3, 2008 #4


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    To be precise, we should say

    [tex]\beta=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P[/itex]

    [tex]\kappa=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T[/itex]

    to acknowledge that V is a function of multiple variables and that we are taking the partial derivative with respect to one of the variables while holding the others constant.

    Now use


    [tex]\beta=\frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_P=\frac{1}{v}\,\frac{\partial }{\partial T}\left(\frac{RT}{P}\right)\right)_P[/itex]

    and so on.
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