Calculate the molar volume using the van der Waals equation.

[EmmanuelD]

Homework Statement

Calculate the molar volume using the van der Waals equation of a gas at P=3000psia, T=60F. The critical pressure and temperature, Pc and Tc, are Pc=408psia, Tc=504F.

Homework Equations

The given van der Waals equations(s):

(P+a/Vm^2)(Vm-b)=RT ----(1)

Vm^3-Vm^2(b+RT/P)+Vm(a/P)-(ab/P) ----(2)

a=(27/64)R^2Tc^2/Pc^2 ----(3)

b=(1/8)RTc^2/8Pc^2 ----(4)

The Attempt at a Solution

I first determined the constants a and b to be:

a=110600psia(ft^3/lb-mol)^2

b=3.168ft^3/lb-mol

And since the critical volume Vc=3b:

Vc=3(3.168)=9.504ft^3/lb-mol

I then basically simplified the cubic equation of state as much as I could and came up with this (leaving the units out for the moment):

Vm^3-Vm^2(9.504)+Vm(270.978)-(858.458)=0

Now, I don't know how to solve for the roots of this particular cubic equation. I tried factoring out the term Vm, leaving me with a root equal to zero and a quadratic polynomial. I tried solving for the roots of the quadratic polynomial using the quadratic formula but calculated complex roots which are incorrect, I believe.

Does anyone know of a practical method for calculating the molar volume of a real gas using the van der Waals equation of state?

Thanks for taking the time to read this.

 

[Chestermiller]

Probably, iterative successive substitution would work. Make an initial estimate of the molar volume using the ideal gas law. Then re-express the Van der Waals equation as:
$$V_m=b+\frac{RT}{P-\frac{a}{V_m^2}}$$
Than solve it iteratively according to:
$$V_m^{n+1}=b+\frac{RT}{P-\frac{a}{(V_m^n)^2}}$$
where n is the number of the iteration. If the method converges, you will have your answer. Just keep iterating until the estimate of the specific volume stops changing from one iteration to the next.