(74q) The Van Der Waals Ten Commandments for Cubic Equations-of-State

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