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

#### AIChE Spring Meeting and Global Congress on Process Safety

#### 2015

#### 2015 AIChE Spring Meeting and 11th Global Congress on Process Safety

#### Spring Meeting Poster Session and Networking Reception

#### Poster Session - Physical Properties and Complex Fluids

#### Monday, April 27, 2015 - 5:00pm to 7:00pm

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 Lecture** ^{2}**:

?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

**to resolve the VDW 1873**

*insufficient*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/*v*^{2}

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/*v*^{2} 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.,

Chem. Educ.

*25*(6), 347,

**1948**

4. Guggenheim,

E. A., The

Principle of Corresponding States. *J. Chem. Phys*. 13, 253, **1945**