The CSvdW equation consists of two terms:

1. the Carnahan-Starling (CS) hard-sphere repulsion term

2. the van der Waals (vdW) attraction term

The CS term has a theoretical basis and is derived from the virial equation below. The first equation is the virial equation with the virial coefficients B_i which corresponds to the i-particle interaction. In the second equation the virial coefficients for hard spheres are inserted. Here b is the covolume of the spheres. With the definition of the packing fraction y=b/4V_m the third line is obtained.

The fourth equation is obtained after rounding the coefficients of the third equation. The integer coefficients can be written as function of number of the virial coefficient (n²+3n). The sixth equation is the closed form of the series above.

The vdW attraction can also be derived theoretically with some approximations (see ref. 2). It is the simplest possible expression for an attraction term:

The complete CSvdW equation of state is:

Ths equation is used in the calculations of the PT projections.

References:

1. J. D. van der Waals, "On the Continuity of the Gaseous and Liquid States" in Studies in Statistical Mechanics, Vol. XIV, ed. by J. S. Rowlinson, North-Holland, Amsterdam, 1988
2. G. H. Findenegg, Statistische Thermodynamik, Steinkopff Darmstadt, 1985
3. N. F. Carnahan, K. E. Starling, J. Chem. Phys. 51, 635, (1969)
4. N. F. Carnahan, K. E. Starling, AIChE Journal, 18, 1184, (1972)