Van der Waals Roots
Van der Waals Roots 
Back
to previous section: Van der Waals Equation of State
Continue
to next section: Other Equations of State
From the previous page, we saw that the van der Waals eos can be written
as a cubic polynomial; i.e.,
f = Pv3 - (Pb + RT)v2 + av - ab = 0To find the molar volume for a given temperature and pressure, we must find the roots of the equation. We have learned that in the two-phase region, we have three real roots. A portion of an isotherm is shown below. The liquid root is the smallest of the three; the vapor root is the largest. A non-physical root exists between the other two. This non-physical root is often quite close in numerical value to the liquid root.
The key to getting the right root is the initial guess. Mathcad or Excel can be used to solve for the desired root, but we must supply an initial guess. Root-finding algorithms often use a Newton-Raphson technique which finds the derivative of the equation at the initial guess and uses that slope to linearly extrapolate to where the value of the function f(v) would be equal to zero. This is graphically illustrated in the figure above. The important point to understand from this is that if the guessed value for the liquid root is above the first minimum shown in the figure, then the slope will be positive and the algorithm will converge on the middle root rather than on the desired liquid root.
So, how do we make good initial guesses? If we desire a vapor
root, we should start with a value larger than the vapor root. Actually,
the vapor guess does not need to be very accurate because the second hump
on the curve above is so broad. We will almost always want to choose
the ideal gas molar volume as the initial guess for the vapor root.
Thus, vguess = RT/P for the gas root.
For the liquid root, we generally choose a compressibility factor of about
0.001 or smaller. We shall see that the compressibility factor is
defined as z = Pv/(RT), so a convenient initial guess would be v
= 0.001RT/P.