# Calculate the molar volume using the van der Waals equation.

## 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.

$$V_m=b+\frac{RT}{P-\frac{a}{V_m^2}}$$
$$V_m^{n+1}=b+\frac{RT}{P-\frac{a}{(V_m^n)^2}}$$
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