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

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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,∗

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

bMAPMO 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−visotherms?

- 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 zof a pure fluid as an infinite power series in the reciprocal molar volume 1/v:

z(T,v)= Pv

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=c₂is called the second virial coefficient,C=c₃ the third virial coefficient,D=c4the 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)= Pv

RT =1+B(T)·P+C(T)·P²+D(T)·P³+ · · ·

=1+

+∞

i=1

c_i+1 (T)·Pⁱ (3)

where coefficientsB,C,D,. . .depend on temperature but are inde- pendent of pressure or density. We will follow the general practice of reserving the namevirial coefficientsforB,C,D,. . .of Eqs.(1) and (2), and not forB,C,D,. . .of Eq.(3). Eqs.(2) and (3)provide two equivalent expressions forzand the coefficients in the two series are related with the results:

B= B

RT, C= C−B²

(RT)², D= D−3BC+2B³ (RT)³ , E=E−4DB−2C²+10CB²−5B⁴

(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 ofzby an infinite series inorPis 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 +Borz= 1 +BP/RT) and truncations to three terms (z= 1 +B+C² orz= 1 +BP/RT+ (C−B²)P²/(RT)²) are widely used.

The common observation shows thatzfor dense gases is better represented by a polynomial in density[1,2]than by a polynomial of the same degree in pressure. Thusz= 1 +B+C²is the preferred three-term virial equation. At low density,z= 1 +BP/RTis the pre- ferred two-term virial equation[1]because it is easier to use and probably more accurate thanz= 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 ofzthe 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 ofr1andr2for four cubic equations of state.

Equations r1 r2a

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+9ω²

−^1+3ω−₂^√ −^1+3ω+₂^√

ar1andr2are 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)

Qis a quadratic polynomial ofvthat can generally be written as:

Q(v)=(v−r1b)(v−r2b). The parametersr₁ andr₂ 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_−r1b)(v_−r2b) (6) By introducing the positive dimensionless variable =b/v, calledpacking fraction, Eq.(6)becomes:

z(T, )= 1 1−− a

RTb

(1−r1)(1−r2) (7)

•Ifr₁ =/ r₂, 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 anyxin ]−1,1[, the function fdefined by:f(x)=1/(1−x) can be expressed as an infinite power series by: 1/(1−x)=

_+∞

i=0xⁱ. As a consequence: 1/(1−)=

_+∞

i=0ⁱand thus:

r_j 1−r_j =

+∞

i=0

r_jⁱ⁺¹ⁱwithj∈ {1,2}

We thus obtain : 1 (r2−r1)

_r

2

1−r2− r1

1−r1

=

+∞

i=0

r₂ⁱ⁺¹−r₁ⁱ⁺¹ r2−r1

ⁱ,and thus :

Table 2

Values of virial coefficientsA,B,C,D,EandFfor four cubic equations of state.

Equation A=c1 B=c2 C=c3 D=c4 E=c5 F=c6

VdW 1 b−_RT^a b² b³ b⁴ b⁵

SRK 1 b−_RT^a b²+^ab_RT b³−^ab_RT² b⁴+^ab_RT³ b⁵−^ab_RT⁴

PR 1 b−_RT^a b²+^2ab_RT b³−^5ab_RT² b⁴+^12ab_RT³ b⁵−^29ab_RT⁴

SW 1 b−_RT^a b²+^abς_RT¹ b³

⎧

−^ab_RT^2ς2 b⁴+^ab_RT^3ς3 b⁵−^ab_RT^4ς4

⎨

⎩

ς1=1+3ω ς2=1+9ω+9ω² ς3=1+15ω+45ω²+27ω³ ς4=1+21ω+117ω²+189ω³+81ω⁴

− a

RTb(r2−r1)

r2

1−r2− r1

1−r1

= − a RTb

+∞

i=0

rⁱ⁺¹₂ −r₁ⁱ⁺¹ r2−r1

ⁱ⁺¹

= − a RTb

+∞

i=1

_ri 2−r₁ⁱ r2−r1

ⁱ

=− a RTb

+∞

i=0

_ri 2−rⁱ₁ r2−r1

ⁱsincer₂⁰−r₁⁰ r2−r1⁰=0

Eq.(8)can be rewritten:

z(T, )=

+∞

i=0

1− a RTb

r₂ⁱ−r₁ⁱ r2−r1

ⁱ

=1+

+∞

i=1

1− a RTb

r₂ⁱ−rⁱ₁ r2−r1

ⁱ (9)

Remembering that=b, one has:

z(T, )=

+∞

i=0

bⁱ−abⁱ⁻¹ RT

r₂ⁱ−r₁ⁱ r2−r1

ⁱ

=1+

+∞

i=1

bⁱ−abⁱ⁻¹ RT

r₂ⁱ−r₁ⁱ r2−r1

ⁱ (10)

By comparison with Eq.(2), the virial coefficients for cubic equa- tions of state, whenr₁ =/ r₂, are:

c1=1 and for alli≥1, ci+1=bⁱ−abⁱ⁻¹ RT

r₂ⁱ−r₁ⁱ r2−r1

(11) Eq.(9)shows that when working with cubic EoS, it is convenient to write the molar compressibility factorzof 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+1by:

c_i+1 =ci+1b⁻ⁱ=1− a RTb

rⁱ₂−r₁ⁱ r2−r1

(13)

•Ifr1=r2=r, Eq.(7)writes:

z(T, )= 1 1−− a

RTb

(1−r)² (14)

Moreover, for anyx in ]−1,1[, the function fdefined by: f(x)= x/(1−x)² can be expressed as an infinite power series by:

x/(1−x)²=

_+∞

i=1i·xⁱ.

As a consequence:−_RTb^a _(1−r) 2 = −_RTbr^a _(1−r)^r 2 = −_RTbr^a

+∞

i=1

irⁱⁱ Since, 1/(1−)=1+

_+∞

i=1ⁱ, Eq.(14)becomes:

z(T, )=1+

+∞

i=1

1−airⁱ⁻¹ RTb

ⁱ (15)

and thus:

c_i+1=1−airⁱ⁻¹

RTb (16)

By the end:

z(T, )=1+

+∞

i=1

bⁱ−airⁱ⁻¹bⁱ⁻¹ RT

ⁱ (17)

By comparison with Eq.(2), the virial coefficients for cubic equa- tions of state, whenr₁=r₂, are

c1=1 and for alli≥1, c_i+1=bⁱ−airⁱ⁻¹bⁱ⁻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/vranging 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 functionf, expressed as an infinite power series expansion by:

f(x)=

+∞

i=0

˛_ixⁱ (19)

wherexis the variable and˛i, theith coefficient of the series.

Theradius of convergence, denotedR, is a nonnegative real num- ber such that the series converges if|x|<Rand 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 variablex, 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 ofRis 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|r1| ≤ |r2| (according to the sets of parameters given inTable 1). In order to properly address the calculation ofR, four different cases have to be distinguished:

•Case 1:|r2|> 1 and|r1|<|r2|(this is the case for the Peng–Robinson and the Schmidt–Wenzel equations of state).

According to Eq.(13), the coefficientc_iat infinite order takes the following form:

c_i=1− arⁱ⁻₂¹ RTb(r2−r1)

1−

_r

1

r2

_i−1

i→+∞∼ − ar₂ⁱ⁻¹ RTb(r2−r1)

(21)

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

i→+∞lim

^ci+1_c

i

^{= |r2|}

And thus:

R= 1

|r2| (22)

•Case 2:|r₂| ≤1 and|r₁|<|r₂|(this is the case for the SRK equation of state).

- If|r2|< 1, the coefficientc_iat infinite order is equal to one:

c_i=1− arⁱ⁻¹₂ 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^ci

i+1

^{=1 (24)}

- Ifr₂= 1, the ratioc_i/c_i+1 has a finite limit whenitends to infinity:

c_i

c_i+1 = 1−_RTb^{a 1−r}_1−r¹ⁱ⁻₁¹ 1−_RTb^a 1−r1i

1−r1

i→+∞−→1 (since lim

i→+∞r1ⁱ=0) (25)

and one finds back:R= 1.

- Ifr₂=−1, the radius of convergence may be found by splitting the expression ofz(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:r₁=−r₂=r

This configuration requires a special treatment because in this case, Eqs.(21) and (23)cannot be used since the quantity (r₁/r₂)ⁱ= (−1)ⁱhas no limit whenitends 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 functionfdefined byf(x)=x/(1−x²)can be expressed as an infinite power series by:

x 1−x² =

+∞

i=0

x²ⁱ⁺¹.

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

z(T, )=

+∞

i=0

ⁱ− a RTb

+∞

i=0

r²ⁱ²ⁱ⁺¹ (27)

The radius of convergence of the series

_+∞

i=0ⁱ is R₁= 1, whereas the one of the series −_RTb^a

+∞

i=0r²ⁱ²ⁱ⁺¹ is equal to R2=1/r², 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{R₁,R₂}. Finally, one has:

- if|r| ≤1, then the radius of convergence is equal to one:R=R₁= 1 (this is the case for the VdW equation of state);

- if|r|> 1, then the radius of convergence isR=R2=1/r².

•Case 4:r₁=r₂=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

orv_∈

_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 ifRis 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 v² z(T, )= 1

1−− a RTb

(28)

The infinite power series ofzin 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 ofzin the molar density is according to Eq.

(17):

z(T, )=1+

b− a RT

+

+∞

i=2

bⁱⁱ (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[ orv_∈]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 ofzin 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,r₂=−1), the radius of con- vergence of this series isR= 1.

The virial expansion ofzin the molar density is according to Eq.

(10):

z(T, )=1+

+∞

i=1

bⁱ+(−1)ⁱabⁱ⁻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[ orv∈]b;+∞[ as the cubic EoS. It is

Fig. 1.P−visotherms 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−visotherms 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 andn= 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 ofzby 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−visotherms. We will distinguish truncations to odd terms and truncations to even terms. InFigs. 1 and 2, the P−visotherm 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/v²_+D/v³_+E/v⁴, 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−visotherm 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- umeb, 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+C²+D³). As a consequence,zand Ptend to infinity (+∞or−∞) whentends to +∞i.e. whenvtends to zero. TheP−visotherm, calculated with a truncated virial expan- sion, will always admitv₌0 as a vertical asymptote. The isotherm calculated with a cubic EoS will admitv=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 whenvis null, is positive. InFig. 3, the fifth virial coefficient (E) is positive and the pressurePtends to +∞whenvtends to zero. However, the sixth (F) virial coefficient is negative and in this case the pressure tends to−∞whenvtends 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 thec_ivirial coeffi- cients are always positive wheniis odd. Wheniis 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 (exceptB) 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 ofB. Indeed, one has (seeTable 2):D=c₄=b²B,F=c₆=b⁴B, and more generally c_2k=b^(2k−2)Bfork≥1. This means that all the even virial coefficients have the same sign asB. 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 evenc_iwill 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 withn= 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 ofzin 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−visotherms 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 ofzin the molar density is according to Eq.

(10):

z(T, )=1+

+∞

i=1

bⁱ+ abⁱ⁻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−visotherms 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−visotherms of pure ethane at supercritical temperatureT= 2Tc= 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−visotherm 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 [TB=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−visotherm.

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 orn= 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 parametersr1andr2of 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)

v²₊(1+3ω)bv₋3ωb² z(T, )= 1

1−−a(T) RTb

1+(1+3ω)−3ω²

(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+²₃√

2≈ −0.057, the polynomial function Q() = 1 + (1 + 3ω)−3ω² can be factorised under the formQ() = (1−r₁)(1−r₂) and the values of parametersr₁andr₂ 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+²₃√

2 may be found inAppendix B.

The infinite power series ofzin the packing fraction is according to Eq.(9):

z(T, )=1+

+∞

i=1

1− a RTb

r₂ⁱ−r₁ⁱ r2−r1

ⁱwith

⎧ ⎪

⎨

⎪ ⎩

r1= −1−3ω+

1+18ω+9ω² 2

r2= −1−3ω−

1+18ω+9ω² 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+9ω²,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 ofzin the packing fraction for the SW EoS as a function ofω.

Fig. 8.P−visotherms of pure ethane calculated with the SW EoS (straight line) and from a truncated virial expansion to 101 terms (dashed line). (a) MoleculeM1(ω= 0) at subcritical temperature. (b) MoleculeM1at supercritical temperature. (c) MoleculeM2(ω= 0.5) at subcritical temperature. (d) MoleculeM2at supercritical temperature. The grey vertical line materializes the radius of convergence of the series.

The virial expansion ofzin the molar density is according to Eq.

(10):

z(T, )=1+

+∞

i=1

bⁱ−abⁱ⁻¹ RT

r₂ⁱ−r₁ⁱ r2−r1

ⁱwith

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

r1=−1−3ω+

1+18ω+9ω² 2

r2=−1−3ω−

1+18ω+9ω² 2

(41)

As a consequence, the virial equations can only be used in the domain∈[0;R[,∈[0;R/b[ orv∈]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 moreRdecreases. 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 M1andM2having 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 moleculesM₁ andM₂. 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 ton= 101 terms. Concerning molecule M₁, 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 moleculeM₂, 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 rangev∈]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 variableP

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 variableP

c_i ith coefficient associated to a virial series expansion in the variable

EoS equation of state

f(x) mathematical function of the variablex R gas constant

R,R₀,R₁ radius of convergence

P pressure

Pc critical pressure PR Peng–Robinson

Q(v) quadratic polynomial ofv

r1,r2 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 expandingzin an infinite power series, one has:

z(T, P)=

+∞

i=0

c_i+1(T)·Pⁱ (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 ofT 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

0;P^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 coefficientsc_j(j≤i+ 1). Eq.(4)gives the equations enabling the cal- culation of the four coefficientsB=c₂(T),C=c₃(T),D=c₄(T) and E=c₅(T) from the knowledge ofB=c2(T), C=c3(T), D=c4(T) and E=c₅(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 thecjvirial coef- ficients. Our method is much simpler and easier to use than the one developed in 1953.

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