Contents lists available atScienceDirect

## Fluid Phase Equilibria

j o u r n a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / f l u i d

## Can cubic equations of state be recast in the virial form?

### Romain Privat

^{a}

### , Yannick Privat

^{b}

### , Jean-Noël Jaubert

^{a,∗}

a*Laboratoire de Thermodynamique des Milieux Polyphasés, Nancy-Université, 1 rue Grandville, B.P. 20451, F-54001 Nancy Cedex, France*

b*MAPMO UMR 6628, Fédération Denis Poisson, CNRS, Université d’Orléans, UFR Sciences, Bâtiment de mathématiques - Route de Chartres. B.P. 6759 - 45067 Orléans Cedex 2, France*

a r t i c l e i n f o

*Article history:*

Received 27 March 2009

Received in revised form 15 April 2009 Accepted 16 April 2009

Available online 24 April 2009
*Keywords:*

Cubic equations of state Virial expansion Power series expansion Van der Waals Redlich–Kwong Redlich–Kwong–Soave Peng–Robinson Schmidt–Wenzel

a b s t r a c t

In this paper, we propose a mathematical study of the virial expansion of cubic equations of state. We attempt to provide an answer to the following questions:

- Is the virial equation only appropriate for the description of gases at low to moderate densities?

- What is the impact of the order of truncation on the representation ofP−*v*isotherms?

- What is the difference between a truncation at an even order and a truncation at an odd order?

- What is the theoretical volume range of validity of a virial expansion?

To illustrate and apply these concepts, we considered four classical cubic equations of state, namely:

Van der Waals, Redlich–Kwong–Soave, Peng–Robinson and Schmidt–Wenzel. For all of these equations, we detail the limitations and the capabilities of the virial expansions.

Finally, we propose a new general relation between the coefﬁcients of the virial equation in pressure and those of the virial equation in density.

© 2009 Elsevier B.V. All rights reserved.

**1. Introduction**

The virial equation of state gives the molar compressibility factor
*z*of a pure ﬂuid as an inﬁnite power series in the reciprocal molar
volume 1/*v*:

z(T,*v*)= P*v*

RT =1+B(T)
*v* ^{+}

C(T)
*v*^{2} ^{+}

D(T)

*v*^{3} ^{+ · · · =}^{1}^{+}

i=1

ci+1(T)
*v*^{i} ^{(1)}
Eq.(1), is frequently written in the equivalent form:

z(T, )= P

RT =1+B(T)·+C(T)·^{2}+D(T)·^{3}+ · · ·

=1+

+∞i=1

ci+1(T)·^{i} (2)

where, the molar density, is equal to 1/*v*.

By convention,*B*=c_{2}is called the second virial coefﬁcient,*C*=c_{3}
the third virial coefﬁcient,*D*=*c*4the fourth, and so on. By this con-
vention, the ﬁrst virial coefﬁcient is unity. All virial coefﬁcients are
independent of pressure or density; for pure components they are
function only of the temperature. The compressibility factor is also

∗ Corresponding author. Fax: +33 3 83 17 51 52.

*E-mail address:*jean-noel.jaubert@ensic.inpl-nancy.fr(J.-N. Jaubert).

sometimes written as an inﬁnite power series in the pressure:

z(T, P)= P*v*

RT =1+B^{}(T)·P+C^{}(T)·P^{2}+D^{}(T)·P^{3}+ · · ·

=1+

+∞i=1

c_{i+1}^{} (T)·P^{i} (3)

where coefﬁcients*B*^{},*C*^{},*D*^{},. . .depend on temperature but are inde-
pendent of pressure or density. We will follow the general practice
of reserving the name*virial coefﬁcients*for*B,C,D,*. . .of Eqs.(1) and
(2), and not for*B*^{},*C*^{},*D*^{},. . .of Eq.(3). Eqs.(2) and (3)provide two
equivalent expressions for*z*and the coefﬁcients in the two series
are related with the results:

B^{}= B

RT, C^{}= C−B^{2}

(RT)^{2}, D^{}= D−3BC+2B^{3}
(RT)^{3} ,
E^{}=E−4DB−2C^{2}+10CB^{2}−5B^{4}

(RT)^{4} (4)

The general relation between the two sets of coefﬁcients but also more details regarding the virial equation in pressure are given in Appendix A.

The actual representation of*z*by an inﬁnite series inor*P*is a
practical impossibility; moreover, values for the virial coefﬁcients
beyond the seventh are, to our knowledge, never available in data
compilation tables. Thus, one must in practice deal with truncations
of the virial equations, and for the same number of terms these are
0378-3812/$ – see front matter © 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.ﬂuid.2009.04.011

not equivalent for the two kinds of series. It is however obvious
that whatever the virial equation considered, its range of applica-
bility increases with the number of coefﬁcients used. Truncations to
two terms (z= 1 +*B*or*z*= 1 +BP/RT) and truncations to three terms
(z= 1 +*B*+*C*^{2} or*z*= 1 +*BP/RT+ (C*−*B*^{2})P^{2}/(RT)^{2}) are widely used.

The common observation shows that*z*for dense gases is better
represented by a polynomial in density[1,2]than by a polynomial
of the same degree in pressure. Thus*z= 1 +B*+*C*^{2}is the preferred
three-term virial equation. At low density,*z= 1 +BP/RT*is the pre-
ferred two-term virial equation[1]because it is easier to use and
probably more accurate than*z= 1 +B*.

The reason for the special importance of the virial equation of
state is that it has a thoroughly sound theoretical foundation. There
is a deﬁnite interpretation for each virial coefﬁcient in terms of
molecular properties. The second virial coefﬁcient represents the
deviations from perfection corresponding to interactions between
two molecules, the third represents the deviations corresponding
to interactions among three molecules, and so on. To summarize,
the importance of the virial equation of state lies in its theoret-
ical connection with the forces between molecules. This is the
reason why, in 1901 [3], Heike Kamerlingh Onnes, an eminent
professor of the Dutch school, Van der Waals’s fellow and future
holder of the Nobel prize (1913), decided to call for the ﬁrst time
the coefﬁcients of the polynomial expansion of *z* in the molar
density,*virial coefﬁcients. Indeed, the word virial (from the Latin*
*vis, genitive* *viris) means force. Onnes used this word by refer-*
ence to Clausius’s exact virial theorem, which relates the average
kinetic energy of a system of moving molecules to the average
of the inner product of intermolecular force and intermolecular
distance[4].

This paper is aimed at understanding why it is always claimed
that the virial equation of state (inﬁnite power series of*z*the in
molar density) is only appropriate for the description of gases at
low to moderate densities. Indeed, it is generally felt that the virial
equation of state actually diverges at high densities, although the
questions as the nature of the divergence and the region of con-
vergence have never been entirely settled, either theoretically or
experimentally. Some simple possibilities which have occasion-
ally been mentioned are that the series is only asymptotically
convergent in any case, or that terms have been omitted which
are negligible at low densities but important at high densities
[2]. Below the critical temperature it seems reasonably certain
from experiment that the series is convergent up to the den-
sity of the saturated vapour. But, is the series really divergent for
liquid densities? and why? What we can say is that the exact
region of convergence is still not well established. The question of
convergence is of both theoretical and practical importance and
will be addressed in this paper. Moreover, the problem of how
many virial coefﬁcients are sufﬁcient to give useful results has
been investigated very incompletely. The impact of the order of
truncation, but also the impact of truncation at an even or at
an odd order on the representation of P−*v* isotherms will be
discussed.

To address these questions, the popular two-parameter cubic
equations of state (Van der Waals[5], Soave–Redlich–Kwong[6],
Peng–Robinson[7]or Schmidt–Wenzel[8], etc) which are known
to be capable of representing both vapour and liquid behaviour
will be used. We will explain, in which conditions, such equa-
tions of state can be recast into the virial form. By knowing the
virial coefﬁcients to all orders for the four equations of state, the
models in question become natural choices to consider in the
inquiry of whether cubic equations of state, put in the form of
virial series expansion, contain information relevant to the con-
densed phase. We will debate whether the virial expansion contains
*all* the information about the complete equation of state for all
phases.

**Table 1**

Values of*r*1and*r*2for four cubic equations of state.

Equations *r*1 *r*2a

Van der Waals (VdW) 0 0

Soave–Redlich–Kwong (SRK) 0 –1

Peng–Robinson (PR) −1+√

2 −1−√

2

Schmidt–Wenzel^{b}(SW) =1+18ω+9ω^{2}

−^{1+3ω−}_{2}^{√}^{} −^{1+3ω+}_{2}^{√}^{}

a*r*1and*r*2are such as|r2| ≥ |r1|.

bFactorization only valid forω≥ −1+2√

2/3≈ −0.057.

To summarize, we shall ﬁrstly address some general mathe- matical aspects regarding the power series expansion of cubic EoS, then we shall study in-depth their range of validity. Through our study, we will attempt to ﬁnd out whether and if so, why these equations can effectively not represent liquid phase behaviours.

**2. Virial expansions of cubic EoS**

In this paper, we only address cubic equations of state deriv- ing from the Van der Waals equation. These ones take the general following form:

P(T,*v*)= RT
*v*−b− a

Q(*v*) (5)

*Q*is a quadratic polynomial of*v*that can generally be written as:

Q(*v*)=(*v*−r1b)(*v*−r2b). The parameters*r*_{1} and*r*_{2} are speciﬁc to
each equation.Table 1reminds their values in the case of the four
aforementioned models.

The molar compressibility factor of a pure ﬂuid, calculated from a cubic EoS is thus:

z(T,*v*)= *v*
*v*_{−}_{b}^{−}

a RT

*v*

(*v*_{−}_{r}_{1}_{b})(*v*_{−}_{r}_{2}_{b}) (6)
By introducing the positive dimensionless variable =b/*v*,
called*packing fraction, Eq.*(6)becomes:

z(T, )= 1 1−− a

RTb

(1−r1)(1−r2) (7)

•If*r*_{1} =*/* *r*_{2}, by using a partial fraction expansion, one has:

z(T, )= 1

1−− a

RTb(r2−r1)

_{r}

2

1−r2− r1

1−r1

(8)

It is moreover well known that, for any*x*in ]−1,1[, the function
*f*deﬁned by:f(x)=1/(1−x) can be expressed as an inﬁnite power
series by: 1/(1−x)=

_{+∞}

i=0x^{i}.
As a consequence: 1/(1−)=

_{+∞}

i=0^{i}and thus:

r_{j}
1−r_{j} =

i=0

r_{j}^{i+}^{1}^{i}withj∈ {1,2}

We thus obtain : 1 (r2−r1)

_{r}

2

1−r2− r1

1−r1

=

+∞i=0

r_{2}

^{i+1}−r

_{1}

^{i+1}r2−r1

^{i},and thus :

**Table 2**

Values of virial coefﬁcients*A,**B,**C,**D,**E*and*F*for four cubic equations of state.

Equation *A*=*c*1 *B=**c*2 *C=**c*3 *D*=*c*4 *E*=*c*5 *F=**c*6

VdW 1 b−_{RT}^{a} *b*^{2} *b*^{3} *b*^{4} *b*^{5}

SRK 1 b−_{RT}^{a} b^{2}+^{ab}_{RT} b^{3}−^{ab}_{RT}^{2} b^{4}+^{ab}_{RT}^{3} b^{5}−^{ab}_{RT}^{4}

PR 1 b−_{RT}^{a} b^{2}+^{2ab}_{RT} b^{3}−^{5ab}_{RT}^{2} b^{4}+^{12ab}_{RT}^{3} b^{5}−^{29ab}_{RT}^{4}

SW 1 b−_{RT}^{a} b^{2}+^{abς}_{RT}^{1} b^{3}

### ⎧

−^{ab}

_{RT}

^{2}

^{ς}

^{2}b

^{4}+

^{ab}

_{RT}

^{3}

^{ς}

^{3}b

^{5}−

^{ab}

_{RT}

^{4}

^{ς}

^{4}

### ⎨

### ⎩

ς1=1+3ω
ς2=1+9ω+9ω^{2}
ς3=1+15ω+45ω^{2}+27ω^{3}
ς4=1+21ω+117ω^{2}+189ω^{3}+81ω^{4}

− a

RTb(r2−r1)

r21−r2− r1

1−r1

= − a RTb

+∞i=0

r^{i+1}

_{2}−r

_{1}

^{i+1}r2−r1

^{i+1}

= − a RTb

+∞i=1

_{r}i 2−r

_{1}

^{i}r2−r1

^{i}

=− a RTb

+∞i=0

_{r}i 2−r

^{i}

_{1}r2−r1

^{i}sincer_{2}^{0}−r_{1}^{0}
r2−r1^{0}=0

Eq.(8)can be rewritten:

z(T, )=

+∞i=0

1− a RTb

r_{2}

^{i}−r

_{1}

^{i}r2−r1

^{i}

=1+

+∞i=1

1− a RTb

r_{2}

^{i}−r

^{i}

_{1}r2−r1

^{i} (9)

Remembering that=*b*, one has:

z(T, )=

+∞i=0

b^{i}−ab^{i−1}
RT

_{2}

^{i}−r

_{1}

^{i}r2−r1

^{i}

=1+

+∞i=1

b^{i}−ab^{i−}^{1}
RT

_{2}

^{i}−r

_{1}

^{i}r2−r1

^{i} (10)

By comparison with Eq.(2), the virial coefﬁcients for cubic equa-
tions of state, when*r*_{1} =*/* *r*_{2}, are:

c1=1 and for alli≥1, ci+1=b^{i}−ab^{i−}^{1}
RT

_{2}

^{i}−r

_{1}

^{i}r2−r1

(11)
Eq.(9)shows that when working with cubic EoS, it is convenient
to write the molar compressibility factor*z*of a pure ﬂuid as an
inﬁnite power series in the packing fraction:

z(T, )= Pb RT =1+

+∞i=1

c^{}_{i+1}(T)·^{i} (12)

The coefﬁcientsc_{i+}^{} _{1}are simply related to the virial coefﬁcients
*c** _{i+1}*by:

c_{i+1}^{} =ci+1b^{−i}=1− a
RTb

^{i}

_{2}−r

_{1}

^{i}r2−r1

(13)

•If*r*1=*r*2=*r, Eq.*(7)writes:

z(T, )= 1 1−− a

RTb

(1−r)^{2} (14)

Moreover, for any*x* in ]−1,1[, the function *f*deﬁned by: f(x)=
x/(1−x)^{2} can be expressed as an inﬁnite power series by:

x/(1−x)^{2}=

_{+∞}

i=1i·x^{i}.

As a consequence:−_{RTb}^{a} _{(1−r)}^{} 2 = −_{RTbr}^{a} _{(1−r)}^{r} 2 = −_{RTbr}^{a}

i=1

ir^{i}^{i}
Since, 1/(1−)=1+

_{+∞}

i=1^{i}, Eq.(14)becomes:

z(T, )=1+

+∞i=1

1−air^{i−1}
RTb

^{i} (15)

and thus:

c^{}_{i+}_{1}=1−air^{i−}^{1}

RTb (16)

By the end:

z(T, )=1+

+∞i=1

b^{i}−air^{i−1}b^{i−1}
RT

^{i} (17)

By comparison with Eq.(2), the virial coefﬁcients for cubic equa-
tions of state, when*r*_{1}=*r*_{2}, are

c1=1 and for alli≥1, c_{i+}1=b^{i}−air^{i−1}b^{i−1}

RT (18)

InTable 2, the expressions of the six ﬁrst virial coefﬁcients for
the four aforementioned cubic EoS are provided. We can notice that
all cubic EoS have the same mathematical formulation for the sec-
ond virial coefﬁcient:*B*=b−*a/(RT). For the SRK, PR and SW EoS,*
all the virial coefﬁcients (except the ﬁrst which is obviously 1), are
temperature-dependant. This feature contrasts markedly with the
virial coefﬁcients due to the Van der Waals EoS, whereof only the
second virial coefﬁcient depends on the temperature.

**3. Existence domain of virial expansion series**

As a limitation, power series expansions are not necessarily
deﬁned on the same range as the expanded function. Cubic equa-
tions of state are deﬁned for any=*b/v*ranging from zero (the ﬂuid
is thus an ideal gas) to one (the ﬂuid is thus a compressed liquid
under inﬁnite pressure). The question we thus need to answer is: are
the virial expansions previously deﬁned (Eqs.(9) and (15)) valid for
anyin [0;1[. In order to properly address this question, let us start
with some general mathematical reminders about that subject.

Let us consider a function*f, expressed as an inﬁnite power series*
expansion by:

f(x)=

+∞i=0

˛_{i}x^{i} (19)

where*x*is the variable and˛*i*, the*ith coefﬁcient of the series.*

The*radius of convergence, denotedR, is a nonnegative real num-*
ber such that the series converges if|*x*|<*R*and diverges if|*x*|>R. At

|*x*|=*R, the series may converge or diverge but this latter case which*
requires a speciﬁc study is out of interest for this work and will not
be considered through this article.

In other words, for a speciﬁed value of the variable*x, so that x*
is in ]−*R;R[,f(x) is rigorously equal to its power series expansion.*

To ﬁnd out the value of the radius of convergence, one may, in
many cases, refer to d’Alembert’s ratio test. This one states that the
reciprocal of*R*is given by the inﬁnite limit of the absolute ratio of
two consecutive terms of the series:

1 R= lim

i→+∞

^{˛}

_{˛}

^{i+}

^{1}

i

### (if the limit exists) (20)

Once the general mathematical context reminded, the calcula-
tion of the domain of validity (that is, the calculation of the radius
of convergence) for cubic EoS expanded in the virial form (Eq.(12))
can now be performed. Thereafter, we shall assume that|*r*1| ≤ |*r*2|
(according to the sets of parameters given inTable 1). In order to
properly address the calculation of*R, four different cases have to*
be distinguished:

•*Case 1:*|*r*2|> 1 and|*r*1|<|*r*2|(this is the case for the Peng–Robinson
and the Schmidt–Wenzel equations of state).

According to Eq.(13), the coefﬁcientc^{}_{i}at inﬁnite order takes
the following form:

c^{}_{i}=1− ar^{i−}_{2}^{1}
RTb(r2−r1)

1−

_{r}

1

r2

_{i−}1

i→+∞∼ − ar_{2}^{i−}^{1}
RTb(r2−r1)

(21)

As a consequence, by applying d’Alembert’s ratio test, we get:

i→+∞lim

^{c}

^{i+1}

^{}

_{c}

^{}

i

^{= |r}

^{2}

^{|}

And thus:

R= 1

|r2| (22)

•*Case 2:*|*r*_{2}| ≤1 and|*r*_{1}|<|*r*_{2}|(this is the case for the SRK equation
of state).

- If|*r*2|< 1, the coefﬁcientc_{i}^{}at inﬁnite order is equal to one:

c^{}_{i}=1− ar^{i−1}_{2}
RTb(r2−r1)

1−(r1/r2)^{i−1}

i→+∞1 (23)

and therefore, the radius of convergence is equal to one:

R= lim

i→+∞

_{c}

^{c}

^{}

^{}

^{i}

i+1

^{=}

^{1}

^{(24)}

- If*r*_{2}= 1, the ratioc_{i}^{}/c_{i+1}^{} has a ﬁnite limit when*itends to inﬁnity:*

c^{}_{i}

c_{i+1}^{} = 1−_{RTb}^{a} ^{1}^{−r}_{1−r}^{1}^{i−}_{1}^{1}
1−_{RTb}^{a} 1−r1i

1−r1

i→+∞−→1 (since lim

i→+∞r1^{i}=0) (25)

and one ﬁnds back:*R*= 1.

- If*r*_{2}=−1, the radius of convergence may be found by splitting
the expression of*z(T,*) as an inﬁnite power series in the packing
fraction (cf. Eqs.(12) and (13)) in an odd-term series plus an
even-term series. By doing so, one may observe that for each of
the two series the radius of convergence is equal to one, thus:

*R*= 1.

•*Case 3:r*_{1}=−*r*_{2}=r

This conﬁguration requires a special treatment because in
this case, Eqs.(21) and (23)cannot be used since the quantity
(r_{1}/r_{2})* ^{i}*= (−1)

*has no limit when*

^{i}*i*tends to inﬁnity.

For sake of convenience, the cubic EoS can be rewritten:

z(T, )= 1 1−− a

RTb (1−r)(1+r)

= 1 1−− a

RTbr r

1−(r)^{2} (26)

We know that the function*f*deﬁned byf(x)=x/(1−x^{2})can be
expressed as an inﬁnite power series by:

x
1−x^{2} =

i=0

x^{2i+1}.

The expansion in power series of Eq.(26)is thus:

z(T, )=

+∞i=0

^{i}− a
RTb

i=0

r^{2i}^{2i+1} (27)

The radius of convergence of the series

_{+∞}

i=0^{i} is *R*_{1}= 1,
whereas the one of the series −_{RTb}^{a}

i=0r^{2i}^{2i+1} is equal to
R2=1/r^{2}, according to d’Alembert’s ratio test. The radius of con-
vergence of the sum of these two series is the smallest out of the
two:*R*= min*{R*_{1},R_{2}*}*. Finally, one has:

- if|*r*| ≤1, then the radius of convergence is equal to one:*R*=*R*_{1}= 1
(this is the case for the VdW equation of state);

- if|*r*|> 1, then the radius of convergence isR=R2=1/r^{2}.

•*Case 4:r*_{1}=r_{2}=r

For similar reasons to Case 3, this case has to be treated specif-
ically. The compressibility factor is given by Eq.(14)and thec^{}_{i}
coefﬁcients are given by Eq.(16). D’Alembert’s ratio test leads to
the following results:

- if|*r*| ≤1, then the radius of convergence is equal to one:*R*= 1
(this is the case for the VdW equation of state).

- if|*r*|> 1, then the radius of convergence isR=1/r.

*To sum up: all the EoS which can be written under the form of*
Eq.(6)can be expanded in virial series according to Eq.(12). The
series range of validity is∈[0;R[ (being a positive variable), or
identically:∈

0;^{R}_{b}

or*v*_{∈}

_{b}

R;+∞

. The virial expansion can by
no means be used outside this domain. As a consequence, before
using a virial equation stemming from a cubic EoS, one needs to
carefully check its range of applicability. The range of validity of the
virial expansion will be the same as that of the cubic EoS (∈[0;1[)
only if*R*is equal to one.

**4. Discussion**

*4.1. The Van der Waals EoS*

The VdW EoS[5], although not very used nowadays, is at the root of many recent models. According to Eqs.(6) and (14)andTable 1, VdW EoS takes the following form:

### ⎧ ⎪

### ⎨

### ⎪ ⎩

P(T,*v*)= RT
*v*_{−}_{b}^{−}

a
*v*^{2}
z(T, )= 1

1−− a RTb

(28)

The inﬁnite power series of*z*in the packing fraction is according
to Eq.(15):

z(T, )=1+

1− a RTb

+

+∞i=2

^{i} (29)

The radius of convergence of this series is according to the pre-
vious study:*R*= 1.

The virial expansion of*z*in the molar density is according to Eq.

(17):

z(T, )=1+

b− a RT

+

+∞i=2

b^{i}^{i} (30)

The radius of convergence of this series is according to the pre-
vious study:*R*= 1/b.

Consequently, the virial equation can be used in the same
domain:∈[0;1[,∈[0;1/b[ or*v*_{∈}]b;+∞[ as the cubic EoS. The
VdW EoS can thus be recast in the virial form but except the second,
all the virial coefﬁcients are temperature independent.

*4.2. Redlich–Kwong type EoS*

Redlich–Kwong (RK) type EoS can be written under the general form:

### ⎧ ⎪

### ⎨

### ⎪ ⎩

P(T,*v*)= RT
*v*_{−}_{b}^{−}

a(T)
*v*(*v*_{+}_{b})
z(T, )= 1

1−−a(T) RTb

1+

(31)

The original Redlich–Kwong EoS version is not still very used today. The SRK EoS (RK EoS modiﬁed by Soave[6]) is currently much more spread through industrialist and academic worlds. Theoretical results that we shall mention apply both to RK and SRK EoS. Figures shown in this paper are calculated with the SRK equation.

The inﬁnite power series of*z*in the packing fraction is according
to Eq.(9):

z(T, )=1+

+∞i=1

1+(−1)^{i} a
RTb

^{i} (32)

According to our previous study (Case 2,*r*_{2}=−1), the radius of con-
vergence of this series is*R*= 1.

The virial expansion of*z*in the molar density is according to Eq.

(10):

z(T, )=1+

+∞i=1

b^{i}+(−1)^{i}ab^{i−}^{1}
RT

^{i} (33)

The radius of convergence of this series is thus:*R*= 1/b.

Consequently, the virial equation can be used in the same
domain∈[0;1[,∈[0;1/b[ or*v*∈]b;+∞[ as the cubic EoS. It is

**Fig. 1.**P−*v*isotherms of pure ethane at 287 K calculated from the SRK EoS (straight
line) and from four truncated virial expansions (dashed and dotted lines) to odd
terms (n= 1,*n= 3,**n= 5,**n= 21).*

**Fig. 2.**P−*v*isotherms of pure ethane at 287 K calculated from the SRK EoS (straight
line) and from four truncated virial expansions (dashed and dotted lines) to even
terms (n= 2,*n= 4,**n= 6 and**n= 22).*

thus possible to claim that the RKS EoS can be recast in the virial
form with temperature dependent virial coefﬁcients. The RKS EoS is
well-known to be able to represent the gas and the liquid state. We
thus see no reason to claim that the virial expansion is only valid for
gases. As stated in the introduction, the representation of*z*by an
inﬁnite series in(Eq.(33)) is a practical impossibility. This is why,
we are going to study the inﬂuence of the order of truncation on
the prediction ofP−*v*isotherms. We will distinguish truncations
to odd terms and truncations to even terms. InFigs. 1 and 2, the
P−*v*isotherm of pure ethane at 287 K calculated with the SRK EoS
(Eq.(31)) is compared with some isotherms calculated from trun-
cated virial expansions. FromFig. 1, it is clear that by considering a
truncation to the ﬁfth term:z(T,*v*)=1+B/*v*_{+}_{C/}*v*^{2}_{+}_{D/}*v*^{3}_{+}_{E/}*v*^{4},
satisfying results can be obtained for molar volumes in the gas but
also in the liquid region. Nevertheless, the vapour pressure tends
to be overestimated. By truncating the virial expansion to the 21st
or 22nd term (seeFigs. 1 and 2), it becomes possible to perfectly
reproduce the isotherm calculated with the SRK EoS.Figs. 1 and 2
clearly highlight that the behaviour of the isotherm is not the same
whether an odd (Fig. 1) or an even (Fig. 2) number of virial coefﬁ-
cients is considered. To better illustrate this point, we have plotted
inFig. 3, on a very large domain of pressure (a logarithmic scale is
used) the isotherm of pure ethane calculated with the SRK EoS and

**Fig. 3.**P−*v*isotherm of pure ethane at 287 K calculated with the SRK EoS and two
of its virial expansions. Comparison between expansions to an odd term (n= 5) and
to an even term (n= 6).

with the virial equation truncated to the ﬁfth and to the sixth term.

First let us recall that cubic EoS predict that the pressure of a liquid
phase becomes inﬁnite when the molar volume tends to the covol-
ume*b, which represents the minimum volume that a pure ﬂuid can*
take up. Truncated virial expansions are polynomial functions in the
variable=1/*v*(e.g.*z= 1 +B*+*C*^{2}+D^{3}). As a consequence,*z*and
*P*tend to inﬁnity (+∞or−∞) whentends to +∞i.e. when*v*tends
to zero. TheP−*v*isotherm, calculated with a truncated virial expan-
sion, will always admit*v*_{=}0 as a vertical asymptote. The isotherm
calculated with a cubic EoS will admit*v*=bas a vertical asymp-
tote. This means that at very high pressure, all the truncated virial
expansions will always underestimate the molar liquid volume. In
order to make the truncated virial equation able to represent liquid
phase behaviours, the pressure and the compressibility factor must
tend to +∞for a null molar volume. This requires that the highest
degree term of the virial expansion (the last virial coefﬁcient con-
sidered), which is predominant when*v*is null, is positive. InFig. 3,
the ﬁfth virial coefﬁcient (E) is positive and the pressure*P*tends to
+∞when*v*tends to zero. However, the sixth (F) virial coefﬁcient
is negative and in this case the pressure tends to−∞when*v*tends
to zero. As a consequence, a much less accurate prediction of the
liquid volume is observed.

By looking at Table 2, we can notice that the*c** _{i}*virial coefﬁ-
cients are always positive when

*i*is odd. When

*i*is even, the virial coefﬁcients may be positive or not, according to the EoS and the temperature considered. For instance, it is obvious that the VdW EoS only leads to positive even order coefﬁcients (except

*B) that are*not temperature dependent. On the other hand, regarding the RKS EoS, even virial coefﬁcients (seeTable 2) are temperature depen- dent but they all can be expressed as a simple function of

*B. Indeed,*one has (seeTable 2):

*D*=

*c*

_{4}=

*b*

^{2}

*B,F*=

*c*

_{6}=b

^{4}

*B, and more generally*

*c*

_{2k}=b

^{(2k−2)}

*B*for

*k*≥1. This means that all the even virial coefﬁcients have the same sign as

*B. It is however well-known that the second*virial coefﬁcient,

*B(T), is an increasing function of temperature. It is*negative at low temperatures, passes through zero at the so-called

“Boyle temperature” and then becomes positive. As a conclusion,
for any temperature below the Boyle temperature, the even*c** _{i}*will
be negative. Because the Boyle temperature is much higher than the
critical temperature, subcritical isotherms will be better predicted
in the liquid region using a truncated virial expansion to odd terms
than to even terms. This is exactly what is observed inFig. 3(with

*n*= 5, the liquid phase is better predicted than with

*n*= 6).

To sum up, when the virial series is truncated to an odd term, feasible results are always obtained in the vicinity of null molar vol- umes. When the virial series is truncated to an even term, the last considered virial coefﬁcient has to be positive to make the expan- sion able to reproduce uncompressible liquid phase behaviour.

*4.3. Peng–Robinson EoS*

The Peng–Robinson EoS[7], still very popular[9–20]writes:

### ⎧ ⎪

### ⎨

### ⎪ ⎩

P(T,*v*)= RT

*v*−b− a(T)
*v*(*v*+b)+b(*v*−b)
z(T, )= 1

1−−a(T) RTb

1+2−^{2}

(34)

The inﬁnite power series of*z*in the packing fraction is according
to Eq.(9):

z(T, )=1+

+∞i=1

1+ a 2√

2RTb

(−1−√

2)^{i}−(−1+√
2)^{i}

^{i} (35)

According to Eq.(22), the radius of convergence of this series is R= 1

1+√ 2=√

2−1≈0.4142 (36)

**Fig. 4.**P−*v*isotherms of pure ethane at 287 K calculated from the PR EoS (bold
straight line) and from ﬁve truncated virial expansions (dashed and dotted lines)
to odd terms (n= 3,*n*= 9,*n*= 11,*n*= 27,*n= 51). The grey vertical line materializes the*
radius of convergence of the series.

The virial expansion of*z*in the molar density is according to Eq.

(10):

z(T, )=1+

+∞i=1

b^{i}+ ab^{i−1}
2√

2RT

(−1−√

2)^{i}−(−1+√
2)^{i}

^{i}(37)

The radius of convergence of this series is thus:R=(√ 2−1)/b. Consequently, the virial equations can be only used in the domain ∈[0;√

2−1[, ∈[0; (√

2−1)/b[ or *v*_{∈}]b(√

2+1);+∞[ smaller than that of the cubic EoS. For this reason, we will say that the PR EoS cannot be recast in the virial form. As a consequence, regardless of the order of truncation, the PR EoS can by no means be written as a virial series to represent liquid phase behaviour.

In Figs. 4 and 5, the isotherm of pure ethane at 287 K calcu- lated with the PR EoS is compared with isotherms calculated from truncated virial expansions. It clearly appears that in the vicinity of the radius of convergence, the virial series truncated to odd terms (Fig. 4) is unable to follow the shape of the isotherm calculated from the PR EoS. Using virial series truncated to even terms (Fig. 5), the liquid branch of the isotherm is never reached.

**Fig. 5.**P−*v*isotherms of pure ethane at 287 K calculated from the PR EoS (bold
straight line) and from ﬁve truncated virial expansions (dashed and dotted lines) to
even terms (n= 4,*n= 10,**n= 12,**n= 28,**n*= 52). The grey vertical line materializes the
radius of convergence of the series.

**Fig. 6.**P−*v*isotherms of pure ethane at supercritical temperatureT= 2T*c*= 610.6 K calculated from the SRK and PR EoS (straight line) and from some truncated virial expansions.

(a) SRK EoS compared with truncated virial expansion to 3 terms. (b) PR EoS compared with truncated virial expansions to 3, 8 and 71 terms. The grey vertical line materializes the radius of convergence of the series.

At this stage, it seems interesting to have a look at supercritical
isotherms. Dealing with the SRK EoS and supercritical tempera-
tures, one observes (seeFig. 6a) that a very low order of truncation
of the virial series expansion enables to model theP−*v*isotherm
with quite a good accuracy from low to very high pressures (e.g.

*n*= 3 inFig. 6a). In addition, when the temperature is above the
Boyle temperature [T*B*=*a/(bR)], all the virial coefﬁcients are pos-*
itive and regardless of the parity of the order of truncation, the
pressure calculated from the virial expansion always tends to +∞
when the molar volume tends to zero. It is thus possible to expect
an accurate prediction of theP−*v*isotherm.

Using the PR EoS, the conclusions in the supercritical area are
quite similar to those drawn in the subcritical area: there is a lim-
itation due to the radius of convergence of the series, and one still
has to take care to truncate the series to an order such that the last
considered virial coefﬁcient is positive (e.g.*n*= 3 or*n*= 71 inFig. 6b).

Furthermore, contrary to the SRK EoS, a truncation at a low order
of the virial expansion (e.g.*n*= 3) of the PR EoS does not enable to
get a proper representation of the isotherm at high pressure (i.e. at
high density). An important deviation appears even when the den-
sity is much smaller than the one corresponding to the radius of
convergence of the series. All these observations can be found back
inFig. 6b.

*4.4. Schmidt–Wenzel EoS*

This cubic EoS[8] is a little bit more elaborate than the SRK
and the PR EoS because the parameters*r*1and*r*2of Eq.(6)are not
anymore two universal constants characterising the EoS but depend
on the nature of the component via the acentric factorω.

The SW equation writes:

### ⎧ ⎪

### ⎨

### ⎪ ⎩

P(T,*v*)= RT
*v*_{−}_{b}^{−}

a(T)

*v*^{2}_{+}(1+3ω)b*v*_{−}3ωb^{2}
z(T, )= 1

1−−a(T) RTb

1+(1+3ω)−3ω^{2}

(38)

As a speciﬁcity, this EoS tends to behave as the SRK EoS for values ofωclose to zero and looks rather like the PR EoS whenωincreases.

In the case where ω≥ −1+^{2}_{3}√

2≈ −0.057, the polynomial
function *Q(*) = 1 + (1 + 3ω)−3ω^{2} can be factorised under the
form*Q(*) = (1−*r*_{1})(1−*r*_{2}) and the values of parameters*r*_{1}and*r*_{2}
are provided inTable 1. This is the unique case discussed hereafter
but more information about the expansion of the SW EoS in virial
series whenω <−1+^{2}_{3}√

2 may be found inAppendix B.

The inﬁnite power series of*z*in the packing fraction is according
to Eq.(9):

z(T, )=1+

+∞i=1

1− a RTb

r_{2}

^{i}−r

_{1}

^{i}r2−r1

^{i}with

### ⎧ ⎪

### ⎨

### ⎪ ⎩

r1= −1−3ω+

1+18ω+9ω^{2}
2

r2= −1−3ω−

1+18ω+9ω^{2}
2

(39)

Keeping in mind that the radius of convergence of a virial series expansion inis at the most equal to one, by using Eq.(22), one ﬁnds:

R=min

1; 1

|r2|

with : 1

|r2|= 2 1+3ω+

1+18ω+9ω^{2},for anyω≥ −1+2
3

√2 (40)

Forωin [0;1], the radius of convergence is plotted as a function ofωinFig. 7.

**Fig. 7.** Radius of convergence of the inﬁnite power series of*z*in the packing fraction
for the SW EoS as a function ofω.

**Fig. 8.**P−*v*isotherms of pure ethane calculated with the SW EoS (straight line) and from a truncated virial expansion to 101 terms (dashed line). (a) Molecule*M*1(ω= 0) at
subcritical temperature. (b) Molecule*M*1at supercritical temperature. (c) Molecule*M*2(ω= 0.5) at subcritical temperature. (d) Molecule*M*2at supercritical temperature. The
grey vertical line materializes the radius of convergence of the series.

The virial expansion of*z*in the molar density is according to Eq.

(10):

z(T, )=1+

+∞i=1

b^{i}−ab^{i−1}
RT

_{2}

^{i}−r

_{1}

^{i}r2−r1

^{i}with

### ⎧ ⎪

### ⎪ ⎨

### ⎪ ⎪

### ⎩

r1=−1−3ω+

1+18ω+9ω^{2}
2

r2=−1−3ω−

1+18ω+9ω^{2}
2

(41)

As a consequence, the virial equations can only be used in the
domain∈[0;R[,∈[0;R/b[ or*v*∈]b/R;+∞[) smaller than that of
the cubic EoS. It is interesting to notice that for values ofωclose to
zero, the radius of convergence is close to one. This reminds of the
behaviour of the SRK EoS.

The moreωincreases, the more*R*decreases. For large molecules,
the domain of validity of the virial series is very limited, as observed
with the PR EoS.

As an illustration, we propose to consider two ﬁctive molecules
*M*1and*M*2having the following features:

M1

Tc,M1=Tc,ethane=305.3 K
Pc,M1=P_{c,ethane}=48.7 bar
ωM1=0

andM2

Tc,M2=Tc,ethane=305.3 K
Pc,M2=P_{c,ethane}=48.7 bar
ωM2=0.5

In Fig. 8, a subcritical isotherm and a supercritical isotherm
calculated with the SW EoS are represented for each of the two
molecules*M*_{1} and*M*_{2}. These isotherms are compared with those
generated with the truncated virial expansion of the SW EoS. In

order not to see the limitations due to the order of truncation, the
virial equation is expanded to*n*= 101 terms. Concerning molecule
*M*_{1}, for which the radius of convergence is exactly equal to one
(ω= 0), the isotherms calculated from the EoS or its truncated virial
expansion are completely merged on the whole range of molar vol-
ume. Regarding molecule*M*_{2}, results are strongly different. In this
case, the radius of convergence is smaller than one (R= 1/3≈0.33
forω= 0.5): the EoS and its truncated virial expansion only match
on the range*v*∈]3b;+∞[. Outside this domain, strong divergences
are observed.

**5. Conclusion**

Many textbooks state that cubic EoS as any pressure-explicit
EoS which yields *z*= 1 in the limit as the molar volume *v*_{→}
+∞can be recast into the virial form. On the other hand, it is
always written that the virial equation in density is only appro-
priate for the description of gases at low to moderate densities.

These two statements are obviously contradictory since the pop- ular two-parameter cubic equations of state (Van der Waals, Soave–Redlich–Kwong, Peng–Robinson or Schmidt–Wenzel, etc.) are known to be capable of representing both vapour and liquid behaviour. The aim of this paper was to clarify this situation.

Our study has shown that when one wants to use a truncated virial expansion to calculate pure ﬂuids properties, at least two precautions have to be taken:

(i) Firstly, the conditions of temperature and pressure of the ﬂuid of interest have to be compatible with the order of truncation of the virial series. As an example, it is obvious that a liquid can-

not be modelled by a ﬁrst order virial series. Let us notice that
nothing prevents from using a truncated virial series to rep-
resent liquid phase behaviours. As shown previously, a mere
truncation to ﬁve terms of the SRK EoS allows to reproduce
rather accurately the liquid branch of a pure componentP−*v*
isotherm. Unfortunately these coefﬁcients are scarcely known
but they could be ﬁtted on experimental data. Our paper has
also shown that by considering an odd number of virial coefﬁ-
cients, the liquid state calculation was considerably improved.

(ii) Secondly, when using analytical expressions of virial coef- ﬁcients, the radius of convergence of the series has to be calculated in order to deduce the volume range on which the virial equation is applicable. As underlined in this article, trying to use a virial expansion issued from the PR EoS to represent an incompressible liquid would be a pure waste of time. We have indeed shown that the VdW and RKS EoS could be recast in the virial form whereas the PR and the SW EoS could not. To avoid confusion, we here mean that, according to Eq.(9), proof was given that the PR and the SW EoS could be put in the form of virial series expansions. However, the radius of convergence of these two series in the variableis smaller than one. This means that the series expansions (virial form of the EoS) have a smaller range of validity than the cubic EoS and can only be used in a shrunken domain.

Dealing with the VdW or the RKS EoS, the virial expansion is valid for anyin [0;1[ and thus contains all the information about the complete EoS for all phases. This is however not the case for the PR and SW EoS explaining why we wrote that such EoS could not be recast in the virial form.

*Notations*

*a* attractive parameter of a cubic EoS

*A,B,C,D,E* ﬁrst, second, third, fourth and ﬁfth virial coefﬁcient
*A*^{},*B*^{},*C*^{},*D*^{},*E*^{} ﬁrst, second, third, fourth and ﬁfth coefﬁcient asso-

ciated to a virial series expansion in the variable*P*

*b* covolume

*c*_{i}*ith virial coefﬁcient, i.e. coefﬁcient associated to a virial*
series expansion in the variable 1/*v*

c_{i}^{} *ith coefﬁcient associated to a virial series expansion in the*
variable*P*

c_{i}^{} *ith coefﬁcient associated to a virial series expansion in the*
variable

EoS equation of state

*f(x)* mathematical function of the variable*x*
*R* gas constant

*R,R*_{0},*R*_{1} radius of convergence

*P* pressure

*P**c* critical pressure
PR Peng–Robinson

Q(*v*) quadratic polynomial of*v*

*r*1,*r*2 speciﬁc parameters of a given equation of state
SRK Soave–Redlich–Kwong

SW Schmidt–Wenzel
*T* absolute temperature
*T** _{c}* critical temperature

*v* molar volume

VdW Van der Waals

*x* real number

*z=Pv/(RT) molar compressibility factor*

˛*i* *ith coefﬁcient of a series*
=b/*v* packing fraction

=1/*v* reciprocal molar volume (molar density)
ω acentric factor

**Appendix A. General relation between the coefﬁcients of**
**the virial equation in pressure and those of the virial**
**equation in density**

As mentioned in the introduction, the virial equation in pressure
is sometimes preferred to the virial equation in density. By choos-
ing pressure as the independent variable and by expanding*z*in an
inﬁnite power series, one has:

z(T, P)=

+∞i=0

c^{}_{i+}1(T)·P^{i} (42)

In Fig. 9, the subcritical isotherm of pure ethane at 287 K
calculated with the PR EoS and with three truncated virial
equations in pressure to *n*= 3, *n*= 4 and *n*= 12 terms are rep-
resented. One may observe that even with a few coefﬁcients
(n= 3,*n*= 4), quite accurate representations of the isotherm gas
branch are obtained. It is however obvious that contrary to
virial equation in density, virial equation in pressure can by no
means simultaneously represent both liquid-like and vapour-like
molar volumes. Indeed for speciﬁed values of*T* and *P, Eq.*(42)
always yields one volume root whereas three roots would be
necessary for a vapour–liquid equilibrium calculation. As a con-
clusion, one can state that the virial equation in pressure is in
the best case, only valid for pressures belonging to the range

^{s}

_{i}(T)

.

In order to calculate the coefﬁcientsc^{}_{i+}_{1}(T) of the virial equa-
tion in pressure, one may express them with respect to the virial
coefﬁcients*c** _{j}*(j≤

*i*+ 1). Eq.(4)gives the equations enabling the cal- culation of the four coefﬁcientsB

^{}=c

^{}

_{2}(T),C

^{}=c

_{3}

^{}(T),D

^{}=c

^{}

_{4}(T) and E

^{}=c

^{}

_{5}(T) from the knowledge of

*B*=

*c*2(T),

*C*=

*c*3(T),

*D=c*4(T) and

*E*=

*c*

_{5}(T). A general relation between the two sets of coefﬁcients has been worked out in 1953 by Putnam and Kilpatrick[21]. Their method is in practice quite tricky to apply and highly time consum- ing when one wants to calculate high order coefﬁcients. Although to our mind, simple and rapid methods do not exist, we propose here after a new rigorous relation allowing to calculate thec

^{}

_{i+}

_{1}(T) coefﬁcients of the virial equation in pressure from the

*c*

*j*virial coef- ﬁcients. Our method is much simpler and easier to use than the one developed in 1953.

**Fig. 9.**P−*v*isotherms of pure ethane at 287 K calculated from the PR EoS (bold
straight line) and from three truncated to*n*terms virial equations in pressure (n= 3,
*n= 4 and**n*= 12).