The Van Der Waals Ten Commandments for Cubic Equations-of-State (Electronic Poster Presentation)

The reading of the Van der Waals (VDW) 1910 Nobel Prize Lecture^1,2 reveals the hindsight and foresights of the VDW theory of cubic equations of state for the individual pure substances and mixtures. A correct reading of the VDW Nobel Prize Lecture reveals some Do's and Don'ts and their consequences led the author to the following itemized Van der Waals Ten Commandments for Cubic Equations-of-State, which begins with Oh the Faithful:

o   Be Loyal to the VDW Cubic Form of Attractive and Repulsive Expressions

o   Be Loyal to the VDW Ultimate Objective: Construct A Substance-Based Cubic Equation 

o   Be Loyal to the VDW Asymptotic Critical Volume-Limit: Weak-Point of VDW Theory

o   Be Loyal to the Four Properties of VDW Theory of Cubic Equations of State

o   Be Loyal to the VDW Empirically-Based Molecular Parameters for Reformed Equations

o   Be Loyal to the VDW Critical Point as Limit of Phenomenological Gas-Liquid Transition

o   Be Loyal to the VDW Defined Gas-Constant for Unifying Caloric and Transport EOS

o   Be Loyal to the VDW Continuity of Gas and Liquid States for Temperature Functions

o   Be Loyal to the VDW Constraining Temperature Parameters to a(T) and b(T)

o   Be Loyal to the VDW Corresponding States Principles for Generalized Property Charts

1. INTRODUCTION AND OBJECTIVE

We recently celebrated the hundred year anniversary^1,2 of the Nobel Prize Lecture delivered by J. D. Van der Waals on December 10, 1910 entitled “The Equation of State for Gases and Liquids.”

This poster highlights the salient points, including the obvious and non-obvious facts about the author's reading of the 1910 Nobel Prize Lecture. In particular, areas of the Nobel Lecture where noticeable quotes have emerged over the years into textbooks, seminars, conferences and symposium are addressed from the viewpoint of the author's experience. The objective of the poster is to highlight fact that can be applied in the future development of the Van der Waals cubic equations of state.

 

2. DISCUSSION OF THE TEN COMMANDMENTS

It is revealed in this poster that the difficulties encountered with the modified VDW type of cubic equations can be traced to the violation of one or more of those VDW Commandments: as for instance the lack of continuously differentiable temperature functions a(T) and b(T) leads to difficulties in the supercritical regions or to the infinite isochoric heat capacity at the critical point or to the impossibility of simultaneously predicting accurate vapor pressure and virial coefficients with the same functional form of temperature-dependent parameters.

Applying the VDW gas constant, R^VDW provides a way to unite thermodynamic and transport equations of state and also it is a way for the direct use of the VDW cubic equations of state for predicting thermodynamic properties as opposed to the widely accepted route of the PVT derivatives: after all the PVT derivatives are based upon the temperature functions and not the VDW parameters a (T_c) and b (T_c).

Also, it is by using empirically-based physical parameters in the reformed VDW 1873 equation that would stop the proliferation of the multiplicity of 2-P and 3-P cubic equations of state.  VDW theory is a modified form of the ideal gas-law with molecular-based empirical parameters for correcting the incomplete VDW theory.

Furthermore, those investigators that are passionately advocating the replacement of VDW repulsive term forget that the theory of liquids of the VDW theory is based solely on the VDW repulsion term as shown by the following limiting behavior:

But, in accordance with the VDW theory, the Z_c of cubic equation solely depends on the ratio of b/v_c, as shown by

 

Perhaps knowing that fact, VDW stated in the 1910 Nobel Prize Lecture²: “the weak point of my theory is b,” so whenever b/v_c = 1/3 (or v_c = 3b), Z_c = 3/8 of the VDW 1873 cubic equation of state.            

By the way, before we agree with the notion that the VDW theory shows anomaly behavior at the fluid critical point, we need to examine the adherence to the following limiting critical behavior that should be imposed on all the VDW cubic equations of state:

That stipulated physical boundary condition states that the VDW cubic equations of state should predict at fluid critical point: accurate critical volume and critical compressibility factor of the individual pure substances or mixture of fixed overall composition.  In that case, the parameters of cubic equations of state should be constrained by four critical constraint criteria because there are four properties of the VDW 1873 equation: Z_c =3/8; Ω_a = 27/64; Ω_b = 1/8; Ω_w (or b/v_c) = 1/3 and that is in accord with the theory of cubic polynomial equations. Thus, four unrelated parameters are required in the VDW 1873 cubic equation; simply, parameters a and b are necessary but insufficient to resolve the VDW 1873 cubic equation at fluid critical point); the required four critical constraint criteria expressions are stated as:

Besides the combining rules for parameters a_m and b_m that VDW introduced in 1888, the justification for more composition-dependent parameters in the VDW 1873 cubic equation is based on the insufficiency of the expressions for the critical properties:

While P_c, v_c and T_c vary with composition of mixture in the expression for the critical properties, the Z_c does not vary with composition, being identically the same as for the pure substance; that is the major source of errors in predicting critical properties from the VDW 1873 equation. Consequently, additional parameters are required to be embedded in the reformed VDW 1873 equation to express Z_c = f (mixture composition). That is another justification for the design of the four-parameter Lawal-Lake-Silberberg equation of state.       

The statistical mechanical derivation of the virial coefficients from the VDW 1873 equation is reported by Hill (1947, 1948)³ as inaccurate after the second virial coefficient; as seen from the virial expansion of the VDW 1873 cubic equation,

The virial expression shows that the attractive parameter a (T_c) disappears from the third and higher virial coefficients: that is not VDW theory because the VDW concept is Z_rep + Z_att. Therefore, the remedy is for more parameters in denominator of a/v² term (as the theory of cubic polynomial equations stipulates four unrelated parameters for cubic equations), which was understood by Clausius (1880) and Berthelot (1900) in their reformed VDW cubic equations of state.  However, it is not always obvious from the coefficients of the virial expansion of the VDW 1873 equation that the second virial is the building blocks for higher virial coefficients; the fact is shown by the following limiting values of PVT data for the second (B), third (C) and fourth (D) virial coefficients:

Those limiting values of PVT data uniquely justified more parameters in the denominator of the a/v² term of the VDW 1873 equation of state because the expression of the second virial coefficient (B) requires parameters a(T_c) and b(T_c) as the building blocks for the third and higher virial coefficients: that is another justification for the construction of the four-parameter Lawal-Lake-Silberberg equation of state.

 

By predicting physical properties using the VDW theory of cubic equations of state, the author agrees with the statement by Guggenheim (1945)⁴ that the corresponding states principle “may safely be regarded as the most useful by-product of the Van der Waals 1873 equation of state.”  Therefore, besides the temperature-dependent parameters a (T) and b (T), no other parameters embedded in the reformed VDW cubic equations of state should depend on temperature, otherwise the meaning of the Law of Corresponding States (LCS) is ambiguous. Hence, temperature-dependent binary interaction parameter is meaningless in the context of the LCS. The design of cubic equations of state reported by Himpan (1951) and Heyen (1980, 1981), having more temperature-dependent parameters than stipulated by the LCS, violates the corresponding states principle.

 

3. CONCLUSION

By adherence to the unspoken words of Van der Waals, which are casted as Ten Commandments in this poster, we can resolve the major disagreement between the Van der Waals theory and fluid properties, including the fluid critical point. Evidently, there should be a particular reason to construct another VDW cubic equation of state, otherwise we would reached the point of diminishing returns and joint others in the high degree of trivialization of the VDW theory of cubic equations of state.

REFERENCES:

1.      Journal of Supercritical Fluids,  Volume 55, Issue 2, Pages 401-860 (December 2010)

100th year Anniversary of van der Waals' Nobel Lecture (Edited by Sona Raeissi, Maaike C. Kroon and Cor J. Peters)

2.      Van der Waals, J. D., 1910 Nobel Prize Lecture available from "J. D. van der Waals - Nobel Lecture: The Equation of State for Gases and Liquids". http://www.nobelprize.org/nobel_prizes/physics/laureates/1910/waals-lecture.html

3.      Hill, T. L., "Free-Volume Models for Liquids," J. Phys. and Colloid Chem., 51, 1219 1947; "Derivation of the Complete Van der Waals' Equation from Statistical Mechanics," J. Chem. Educ., 25 (6), 347, 1948

4.      Guggenheim, E. A., The Principle of Corresponding States. J. Chem. Phys. 13, 253, 1945