Page 93 - Handbook of Natural Gas Transmission and Processing Principles and Practices

(2.22)

where P and v are the externally measured pressure and molar volume, is the
gas constant, and α is a measure of the kinetic energy of the molecule. This equation was rewritten
later as follows:

(2.23)

It appears that the pressure results from the addition of a repulsive term and an
attractive term . Writing the critical constraints (i.e., that the critical isotherm has a
horizontal inflection point at the critical point in the P-v plane), it becomes possible to express the a
and b parameters with respect to the experimental values of T and P , respectively, the critical
c
c
temperature and pressure:

(2.24)

The Van der Waals equation is an example of cubic equation. It can be written as a third-degree
polynomial in the volume, with coefficients depending on temperature and pressure:

(2.25)

The cubic form of the Van der Waals equation has the advantage that three real roots are found at
the most for the volume at a given temperature and pressure. EoS including higher powers of the
volume comes at the expense of the appearance of multiple roots, thus complicating numerical
calculations or leading to nonphysical phenomena. In the numerical calculation of phase
equilibrium with cubic equations, one simply discards the middle root, for which the
compressibility is negative (this root is associated to an unstable state). Van der Waals' EoS and his
ideas on intermolecular forces have been the subject of many studies and development all through
the years. Clausius (1880), proposed an EoS closely similar to the Van der Waals equation in which
an additional constant parameter (noted c) was added to the volume in the attractive term, and the
attractive term is made temperature dependent. Containing one more adjustable parameter than
Van der Waals' equation, Clausius' equation showed a possible way for increasing the model
accuracy.

(2.26)

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