Expansion/Compressibility Coefficients: Carnahan & Starling Model

[mathtrek]

Hoping for a bit of insight with a problem I've been tasked with...

 

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.

 

 

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!!

[insane_alien]

don't see what is wrong with 1/T and 1/P

 

seems quite reasonable. it is usually something like that in nearly all cases(if you have multicomponent mixtures variable pressures AND temperatures then it gets a bit(HA!) more complicated.

 

i take it you have just been introduced to these equations?

basing that on the fact you were surprised by the results.

[mathtrek]

^ Yeah, just been introduced. My concern is that the calculation is far too simple - I was lead to believe it would be far more involved... I'm just wondering if I've dreadfully oversimplified it and in doing so have missed a crucial step

[insane_alien]

no it's not oversimplified at all. if it is then part of my project in uni last year should have failed miserably.

 

just give it a few classes and you'll be wishing it were back to being that simple.

[mathtrek]

Hmm, well after further research it does seem that for an ideal gas that, indeed, α = 1/T and β = 1/P.

 

However, reconsidering the question - it's directing the use of the Carnahan and Starling equation to estimate these coefficients. Would that imply that one doesn't use the Ideal equation..?

[insane_alien]

the ideal gas laws are incredibly useful and are used wuite a bit in the real world.

 

especially for high temperature/low pressure processes.

[Klaynos]

Hmm, well after further research it does seem that for an ideal gas that, indeed, α = 1/T and β = 1/P.

 

However, reconsidering the question - it's directing the use of the Carnahan and Starling equation to estimate these coefficients. Would that imply that one doesn't use the Ideal equation..?

 

As it says estimate I would say that implies that you should use the ideal gas law, because it is only an approximation of reality...

[swansont]

IIRC hard-sphere scattering follows the ideal gas law as long as you can ignore the size of the spheres. You are doing something similar by holding the packing fraction constant at 0.4. Doesn't a constant packing fraction imply a constant volume? Things will get a lot more interesting when you vary that parameter, i.e. have Z = Z(V), which may be the point of the problem. Look at the variation on the vicinity of [math]\eta[/math] = 0.4 but have [math]\eta[/math] still be a variable.