Expansion/Compressibility Coefficients: Carnahan & Starling Model

[raintrek]

Homework Statement

Homework 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.

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!

 

[siddharth]

however we find α = 1/T and β = 1/P which seems far too easy.

I get something different.

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

But, the packing fraction \eta is a function of V.

\eta = \frac{ \pi n d^3}{6V}

So, when you use the equation of state to find the partial derivatives holding P or T constant, remember to account for this.

 

[raintrek]

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 :)

 

[siddharth]

Am I on the right track there, or have I gone completely wrong?!

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