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Expansion/Compressibility Coefficients: Carnahan & Starling Model

  1. Oct 31, 2007 #1
    1. The problem statement, all variables and given/known data
    2. Relevant equations

    Using the Carnahan and Starling equation, estimate the coefficient of volumetric expansion, α, and the coefficient of compressibility, β, defined as

    α ≡ 1/V * (δV/δT) [holding P constant] and
    β ≡ -1/V * (δV/δP) [holding T constant]

    (I've used δ here for partial differentials)

    for the hard sphere fluid at a packing fraction η = 0.40.

    3. The attempt at a solution

    I've tried using Z = PV/nkT and then the CS expression for Z: Z = (1+η+η²-η³) / (1-η)³ however we find α = 1/T and β = 1/P which seems far too easy.

    Could someone please offer a suggestion of which direction I should be looking in? Any help would be greatly appreciated!!
  2. jcsd
  3. Nov 1, 2007 #2


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    I get something different.

    But, the packing fraction [itex]\eta[/itex] is a function of V.

    [tex]\eta = \frac{ \pi n d^3}{6V} [/tex]

    So, when you use the equation of state to find the partial derivatives holding P or T constant, remember to account for this.
    Last edited: Nov 1, 2007
  4. Nov 1, 2007 #3
    hmm, OK, let me see if I'm on the right track here,

    setting Z = PV/NkT = (1+η+η²-η³) / (1-η)³
    where, as you described, η = pi*N*d³ / 6V

    Doing some quick algebra, I get:

    V^4 (x) - V³ (1 + 3xy) + V² (3y²x - y²) - V (y³x) + y³ = 0

    Where I've said,
    x = P / NkT (volume "coefficient" in ideal gas law)
    y = N*pi*d³ / 6 (volume "coefficient" in η)

    Am I on the right track there, or have I gone completely wrong?! Thanks for your help, siddharth :)
  5. Nov 1, 2007 #4


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    I think so. However, it's easier to work with [itex]\eta[/itex] rather than V. Just express T as a function of p and [itex]\eta[/itex], calculate the derivative wrt to V (ie, eta) at constant p and then invert it to get your required partial derivative.
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