(352r) Derivative-Free Computation of High-Pressure Density, Isothermal Compressibility, Volume Expansivity of Pure Substances and Binary Systems By Van Der Waals Cubic Equation

Accurate prediction of thermodynamic properties for pure substances requires accurate measurement of vapor pressures and coexistence phase volumes as well as residual volumes, enthalpies and entropies. It is also desirable to acquire density extrema in isothermal variations of the isochoric heat capacity, extrema of the isothermal compressibility, speed of sound, and densities where the reduced bulk modulus (the reciprocal isothermal compressibility is called the bulk modulus) and isobaric expansivity are essentially independent of temperature (or can be described to have very weak maxima). Even though tailored multiparameter equation of state models show all of those desirable qualities, VdW cubic equations and other state equations based on two- and three-parameter corresponding states principles (CSP) do not usually meet the desirable qualities. The routinely modifications of the attractive temperature-dependent parameter and co-volume dependence in the generalized van der Waals cubic models for improving vapor pressures and saturated phase volumes do not provide improved descriptions of those extrema for the computation of derivative properties (or thermodynamic properties). Since both compressibilities (isothermal compressibility and isobaric expansivity) are intensive measurable state functions, they can be obtained from PvT Experiments or from VdW cubic models that satisfactorily conform to the specified boundary conditions at low- and high-pressure limits (as P → ∞, v → b(T) at high-pressures). Consequently, rather than considering PvT derivatives for the computation of the compressibilities, this Poster explores the possibility of designing VdW gas-constant (R_vdw) into the Lawal-Lake-Silberberg cubic equation of state for the calculation of the isothermal compressibility and isobaric expansivity properties (without taking any derivatives).

By definition, the computation of isothermal compressibility [-1/v (∂v/∂P)_T] (because volume decreases with increasing pressure, the definition contain negative sign to make isothermal compressibilities positive) and volume expansivity [1/v (∂v/∂T)_P] involves respective computation of first-order derivative of Volume with respect to Pressure (at fixed Temperature) and first-order derivative of Volume with respect to Temperature (at fixed Pressure) from the Van der Waals (VdW) theory of cubic equations of state. But, the direct computation of volume expansivity from the VdW cubic equations is very challenging because T (and P) are in the expansion of cubic equation in ν and consequently, a remedial action using cyclic triple product rule of partial differentiations is usually applied [1-10]. Even though the simplicity of the cyclic product rule relation can be observed as geometrical representation in the P−V−T diagram [4-5], the effects of the errors in the PvT relation are carried through to all thermodynamic property variations because they involve derivatives. Hence, major errors for the heat capacities (isochoric and isothermal specific heat), isothermal compressibility, and sound speed have been shown in 1996 by Gregorowycz, et al. [11]; and even if analytic VdW cubic equations are accurate for saturation volumes, they do not give reliable changes of volume upon compression and thus making the errors in the predicted isothermal compressibilities extremely large.

Consequently, this poster describes characteristics of the derivative properties for pure substances and binary systems by reforming the VdW 1873 equation as universal cubic equation of state typified by the Lawal-Lake-Silberberg (LLS) cubic equation. The results of the LLS cubic equation are compared with the results from several well-known cubic equation models. The extrema is achieved with the LLS cubic equation by the designed co-volume temperature-dependent parameter without requiring any density dependence in the attractive parameter of the LLS equation. The analysis of results show accurate description of the extrema is possible with such cubic model (the LLS cubic equation) without the co-volume parameter being any complex dependence on both temperature and density, as commonly done with some cubic models of the VdW theory of cubic equations of state.

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