(352o) Mathematical Operator Simplified Mole Number to Mole Fraction in the Derivatives of Gibbs Partial Molar Properties for Mixture Fugacity Coefficient in Cubic Equation of State

National Academic of Engineers (NAE) define Engineering as application of Science, which in principle implies that the goal of Engineering Discipline is to explore the application of scientific principles in Engineering Analysis. Consequently, this Research Experience for Undergraduate (REU) explores the use of mathematical rules (such as Product and Quotient rules, Logarithmic rule) and mathematical operators (such as Jacobians, Forward and Backward operators, Differential, Laplace, Dirac delta and Kronecker delta operators, Gamma function, Del, Divergence and Curl operators, Cartesian product and Summation operators) developing a simplifying procedure for resolving the Gibbs partial molar properties (fugacity coefficient of a component in a mixture, partial Z-factor, partial enthalpy, partial excess properties, partial excess volumes) for multiplicity of combining rules used for the mixture parameters (a_m, b_m, c_m, d_m, α_m, β_m) in the Van der Waals theory of cubic equations of state.

The derivatives of molar property taken at fixed temperature, pressure and mole number are usually referred to as the Gibbs partial molar properties. Such derivative is required for the derivation of closed expressions for fugacity coefficient of a component in a mixture based on the various combining rules designed for the equations-of-state mixing rules, excess and residual mixture properties and pseudo-critical properties. Since the partial molar properties involve mole number rather than mole fraction, a differential operator is established solely from the definition of partial molar properties for simplifying the derivatives with respect to mole fraction for the various types of combining rules used for the mixture parameters. The differential operator facilitates analytic expressions of fugacity coefficient for combining rules (linear, geometric, rational and harmonic rules) of the thermodynamic and thermophysical properties. The differential operator also lends itself to symbolic computation or MATHEMATICA programming and thus provides a rapid means of establishing closed expressions for chemical potential or fugacity of a component in a mixture. The differential operator provides an alternative to using complex algebraic solutions for coefficients in the partial molar properties to achieve the closed-form expression of fugacity coefficients as reported in Chapter 6.28 of The Properties of Gases and Liquids (5^th edition, 2001) by Poling-Prausnitz-O’Connell.