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Virial coeffficients

  1. Apr 12, 2007 #1
    1. The problem statement, all variables and given/known data
    A gas obeys the equation of state

    [tex](P + \frac{a}{kTv^2})(v-v_{0})=kT[/tex].

    Where a and v0 are constants and v=V/N is the volume per particle.

    Find the second and third virial coefficients for this equation of state.


    2. Relevant equations

    [tex]B_{2}=V( 1/2 - Q_{2}/Q_{1}^2 )[/tex]

    [tex]B_{3}=V^2[ 2Q_{2}/Q_{1}^2 (2Q_{2}/Q_{1}^2 - 1) - 1/3(6Q_{3}/Q_{1}^3 - 1)[/tex]

    [tex]Q_{n}[/tex]=canonical partition function of a subsystem of n particles.


    3. The attempt at a solution
    I was looking to the virial expansion:

    [tex]PV/nRT = 1 + B(T)n/V + C(T)n^2/V^2 +...[/tex]

    In this expansion B(T) is the 2nd virial coeff., and C(T) is the 3rd virial coeff.
    I was trying to find some relationship between this equation and the equation of state that was given in the problem.
    My question is: how can I start this problem? What is the first thing that I have to do to find the virial coefficients?
    Any hint will be apreciated.
     
    Last edited: Apr 12, 2007
  2. jcsd
  3. Apr 13, 2007 #2
    I solve the problem this way...

    Solving to P:
    [tex]P=NkT/(V-Nv_{0}) - aN^2/(kTV^2)[/tex]

    The compressibility is:
    Z=PV/NkT

    Multilplying both sides by V and divide by NkT:

    [tex]Z=PV/NkT=1/(1-Nv_{0}/V) - aN/(k^2T^2V)[/tex]

    For very low density
    [tex]Nv_{0}/V << 1[/tex]
    Using approximation: 1/(1-x) ~ 1+x

    [tex]Z= 1 + Nv_{0}/V - aN/(k^2T^2V) = 1 + (N/V)(v_{0} - a/k^2T^2)[/tex]

    So, the second virial coefficient is:

    [tex]B_{2}(T)= v_{0} - a/k^2T^2[/tex]

    Is it right? And, how can I find the third virial coefficient?
     
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