(352d) Phenomenological Theory of Computation By Vdw Cubic Equation of State Free of Thermodynamic Derivatives in the Estimation of First and Second Order Derived Property Functions

As yet, thermodynamic properties [1-33] are traditionally calculated by performing partial derivatives of PVT equations or by integrating volume deviation, Z – 1. Consequently, the prediction of derived properties (such as the isochoric heat capacity) remains a challenge for cubic equations of state. Rather than taking PVT derivatives, a new approach is undertaken based on a defined Van der Waals (VdW) gas-constant (R_vdw) for the calculation of the individual derived properties (or thermodynamic properties: enthalpy, entropy, heat capacity, sound speed, Joule-Thomson). Since all pure substances that have similar value of acentric factor have about the same compressibility factor (Z-factor) at identical T_R and P_R and deviate from ideal-gas behavior to about the same degree [4-11, 13-31]. That observation is the basis for the hypothetical state of matter designated by the ideal-gas state: which is the state of a gas comprised of real molecules that have negligible molecular volume and no intermolecular forces at all temperatures and pressures. Consequently, it is not the gas that is ideal [3], but the state; that has foundation in the formulation of the dimensionless thermodynamic function as enunciated in 1963 by McKetta [1-2] Thermodynamic Group in the Department of Chemical Engineering at the University of Texas in Austin, Texas.

Computation of first- and second-order PVT derivatives of thermodynamic properties from VdW cubic equations is very challenging because PVT derivatives exposed the imperfections of the cubic equations of states. The first-order PVT derivatives are performed for the estimation of enthalpy (partial derivatives (∂H/∂T)_P and (∂H/∂P)_T), enthalpy of vaporization, entropy, entropy of vaporization, internal energy, adiabatic reversible process (∂T/∂V)_S and Joule effect (∂T/∂V)_U while the second-order PVT derivatives are performed for estimation of isochoric and isothermal heat capacities, sound speed (or sonic velocity), Joule-Thomson coefficient, Joule-Thomson inversion point, Gibbs energy and Helmholtz energy functions. Interrelations of those derived properties usually involve derivatives, of which those involving P, v, and T are the most easily obtained while other required derivatives are based on T, P, S, and the three derivatives most susceptible of measurement are: (∂v/∂T)_P, (∂v/∂p)_T, and C_P = (∂H/∂T)_P.

But, the foundation for the new approach is based on the forgotten theory of dimensionless thermodynamic function (see Table I) first introduced in 1963 and 1967 by John McKetta and his Students [1-2]. The derived properties computed from PVT data are obtained in the form of deviations of the properties from ideal behavior. While the thermodynamic properties of a substance in the ideal gas state can be calculated by using the methods of statistical mechanics [28-29], here the ideal gas state is computed by Lemmon fundamental equation and equated to critical point in Eq. 3. The derivation of the expressions relating the functions of Table I to the compressibility factor, pressure, and temperature has been presented in detail by Heichelheim and McKetta [1-2] while the ideal gas-state computed by McKetta group is used in PVT relation of Table I to estimate the individual thermodynamic properties.

As opposed to the McKetta method, the approach here is described by Eqs.1-4.

Thus, in the PVT equation of state of the functional form

P = f (R, V, T, x, a, b)....................................................................................................(1)

where a and b are Van der Waals parameters, R denotes gas constant, V denotes molar volume, T denotes temperature and x denotes composition

While in the thermodynamic property equation of the functional form

P = f (R_vdw, φ, T, x, a, b), where φ = H, S, U, C_p, C_v, w, µ_JT, G, A.................................................................(2)

Where x denotes substance composition, w is sound speed and µ_JT is the Joule-Thomson coefficient while C_p and C_v respectively denotes isothermal and isochoric heat capacities; H denotes enthalpy, S denotes entropy, and U denotes internal energy. The thermodynamic relation use in the functional form of Eq. 2 to establish the van der Waals gas-constant R_vdw is defined as (superscript c denotes critical point):

R_vdw = (P_c φ^c / Z_c T_c)

The gas-liquid critical properties for the respective thermodynamic property is obtained from the following PVT relations (subscript std. denotes standard condition while crtpt. denotes critical point and superscript o denotes ideal gas state)

(P φ^o / Z T)_std. = (P_c φ^c / Z_c T_c)_crtpt ................................................................................(3)

The thermodynamic property (φ) equation of state is an adaptation of the LLS cubic equation (α and β are substance-dependent),

P = R_vdw T/ (φ – b) – a(T)/ [φ² + αbφ – βb²] ...................................................................(4)

The Van der Waals cubic equations of state used to accomplish the task of this poster consists of two-parameter version (Van der Waals, Soave-Redlich-Kwong, and Peng-Robinson), three-parameter version (Schmidt-Wenzel, Harmens-Knapp, and Patel-Teja) and four-parameter version (Lawal-Lake-Silberberg (LLS), Trebble-Bishnoi). The analysis of results shows the four-parameter cubic equations of state produce accurate results than other versions of the Van der Waals theory of cubic equations of state. The advantage of using the functional model of Eq. 2 in the way of none derivatives supports the analysis reported by Gregorowicz, et al. [33] who stated that taking derivatives exacerbate cubic equation model weaknesses and departure (or derived) functions established with the derivatives of EoS models lead to considerable errors.

Table I. Dimensionless Functions for Deviation of Thermodynamic Properties from Ideal Gas Behavior

Function Dimensionless Form

Volume P(V – V^o)/ RT

Internal energy (U – U^o)/ RT

Enthalpy (H – H^o)/ RT

Entropy (S – S^o)/ R

Gibbs free energy (G – G^o)/ RT

Helmholtz free energy (A – A^o)/ RT

^o indicates property of an ideal gas

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