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I read with great interest Rainder Abrol's piece entitled "Computation of Vapor Pressure" (J. Chem. Educ. 1995, 72, 1077) which can predict vapor pressure based on the van der Waals equation. Such computations may also interest people without FORTRAN and whose computer languages cannot solve cubic equations. It is the solving of cubic equations that permits the user to find the local maximum and local minimum in graph of pressure versus volume for an isotherm, as well as the three volumes, V1, V2, and V3 in order of magnitude, for which the pressure is the same. Abrol's algorithm compares the ratio of the areas between the trial vapor pressure line and the isotherm between V1 and V2 and between V2 and V3.
For those without cubic equation-solving capability, an alternative method would be faster. After selecting a V1 for a trial vapor pressure, V3 (in Abrol's
terminology) may be determined by programmed trial and error. When V3 is found, one need only see how well this trial pressure fulfills the condition
RTlog[(V3 - b)/(V1 - b)] + a(1/V3 - 1/V1) - P(V3-V1) = 0
finding the V1, P, and V3 that best fulfill the condition by programmed trial and error.
The volumes representing the local minimum, Vmin, and local maximum, Vmax, can be approximated by estimation based on the absolute temperature and the
van der Waals constants of the gas. The estimates can be refined by programmed trial and error.
Vmax ~ 8a/9RT + 1.08(a2 - 27RTab/8)1/2/RT
Vmin can be approximated as follows
F = 27RTb/8a
x ~ 0.2887641(1 - F0.03096279295)-0.50495929259
(Where the exact value of x is one that fulfills the condition F1-2x + 6F1-x + 12F + 8Fx+1 - 27 = 0)
Vmin ~ b(1 + 2Fx)
I readily acknowledge that there may be better methods of estimation or refinements on these approximations, but they should suffice for student use where solving the cubic equation is not an option. A "derivation" of these approximations is available on request.
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