(352p) A Recursive Formula Simplified n-Order Derivatives of Virial Coefficients from the Van Der Waals Theory of Cubic Equations of State

This Research Experience for Undergraduates (REU) project is design to gain experience in using mathematical operation of partial differentiation to establish higher order virial coefficients from the Van der Waals theory of cubic equations of state. By reforming empirical parameters in the Van der Waals generic cubic equation of state to reflect the pertinent molecular parameters of the Lennard-Jones (1924) potential function and Pople potential function (1954), a robust closed-form recursive expression is established for virial coefficients of polar and non-polar pure substances. The generic cubic equation of the type formulated by the Lawal-Lake-Silberberg (LLS) cubic equation of state is applied to derive the recursive expression. It is shown that the second virial coefficient is the building block for the third and higher virial coefficients and also that all even-terms of virial coefficients are negative at sub-critical temperature but positive at sub-critical temperatures. While all odd-terms of virial coefficients are positive at all temperatures in opposition to some reported results in the literature that show negative values for third-virial coefficient. It is also shown that each succeeding even and odd terms of virial coefficients are smaller than the preceding terms. It is confirmed that the Boyle temperature depend on the shape of pure substances as reported sixty-years ago by Chueh-Prausnitz (AIChE J., 1963). The derivation of the recursive formula is briefly explained in the following paragraph.

We can state the n^th virial coefficient B_n for hard-sphere equation of state as

Eq. 1 can be derived from the thermodynamic relations (see Hill, T. L., An Introduction to Statistical Thermodynamics, Dover Publications, Inc., New York, 1960; pp. 261) of universal law of equation of state for dilute gas and B_n(T) are virial coefficients which depend on intermolecular forces.

By analogy to Eqs.1-2, we can write the general n^th virial coefficient B_n for cubic equations of state (see O’Connell-Haile: Thermodynamics (Fundamentals for Applications), pp. 154-155, Eq. 4.5.8) as

Or simply written as (where C_n is the nth virial coefficients):

We can then apply Eq. 4 to the Z-factor derived from the Lawal-Lake-Silberberg cubic equation of state as detailed below. Let us represent the Lawal-Lake-Silberberg (LLS) equation as

By transform to the Z-form

Or in ρ-form

Using partial fraction for further simplification and setting α = d + g and β = - dg, we have

or

By applying the thermodynamic criterion of Eq. 4 to Eq. 9, we arrives at the Recursive Formula for the virial coefficient C_n of n^th order virial coefficients is derived from

This can be simplified into

We use binomial formula to expand any power of the summation (d + g) into a sum of the form

The binomial coefficients, which is denoted by is used to derive the coefficient of by the following formula

Then, we translate the result of the expansion into the individual pure substances molecular structure parameter, α and the molecular shape parameter, β; thus, α = (d + g) and β = - dg

For validation of the Recursive Formula, the solution of C_n in terms of parameters a = [a(T)], b = [b(T)], α and β

For n = 2, C₂ = b (if n = 1, C₁ = 1)

n = 3, C₃ = b² +

n = 4, C₄ = b³

n = 5, C₅ = b⁴ +

n = 6, C₆ = b⁵ (