Can cubic equations of state be recast in the virial form?irma.math.unistra.fr/~privat/documents/viriel.pdf · of these equations, we detail the limitations and the capabilities of

Can cubic equations of state be recast in the virial form? R. Privat, Y. Privat and J.-N. Jaubert

- 1 -

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

Romain PRIVAT(a), Yannick PRIVAT(b) and Jean-Noël JAUBERT(a,*).

(a) Laboratoire de Thermodynamique des Milieux Polyphasés, Nancy-Université.

1 rue Grandville, B.P. 20451, F-54001 Nancy Cedex.

(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.

E-mail: [email protected] - Fax number: +33 3 83 17 51 52

(*) author to whom the correspondence should be addressed.

Abstract 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 of P-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. Keywords: Cubic equations of state, virial expansion, power series expansion, Van der Waals, Redlich-Kwong, Redlich-Kwong-Soave, Peng-Robinson, Schmidt-Wenzel.


- 2 -

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 associated to a virial series expansion in the variable P b , covolume. ci, i

th virial coefficient i.e. coefficient associated to a virial series expansion in the variable 1 / v

ic′′ , ith coefficient associated to a virial series expansion in the variable P

ic′ , ith coefficient associated to a virial series expansion in the variable η

EoS, equation of state f(x), mathematical function of the variable x R, gas constant R, R0, R1, radius of convergence P, pressure Pc, critical pressure PR, Peng-Robinson Q(v), quadratic polynomial of v r1, r2, specific parameters of a given equation of state SRK, Soave-Redlich-Kwong SW, Schmidt-Wenzel T, absolute temperature Tc, critical temperature v , molar volume VdW, Van der Waals x, real number

( )z Pv / RT= , molar compressibility factor

αi, ith coefficient of a series b / vη = , packing fraction 1/ vρ = , reciprocal molar volume (molar density)

ω, acentric factor


- 3 -

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 :

i 12 3 i

i 1

Pv B(T) C(T) D(T) c (T)z(T,v) 1 ... 1

RT v v v v

+∞+

=

= = + + + + = +∑ (1)

Eq. (1), is frequently written in the equivalent form:

2 3 ii 1

i 1

Pvz(T, ) 1 B(T) C(T) D(T) ... 1 c (T)

RT

+∞

+=

ρ = = + ⋅ρ + ⋅ρ + ⋅ρ + = + ⋅ρ∑ (2)

where ρ, the molar density, is equal to 1 v .

By convention, 2B c= is called the second virial coefficient, 3C c= the third virial

coefficient, 4D c= the fourth, and so on. By this convention, 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 sometimes written as an infinite power series in the pressure:

2 3 ii 1

i 1

Pvz(T,P) 1 B (T) P C (T) P D (T) P ... 1 c (T) P

RT

+∞

+=

′ ′ ′ ′′= = + ⋅ + ⋅ + ⋅ + = + ⋅∑ (3)

where coefficients B′ , C′ , D′ , … depend on temperature but are independent 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

= 2

2

C BC'

(RT)

−= 3

3

D 3BC 2BD'

(RT)

− += 2 2 4

4

E 4DB 2C 10CB 5BE'

(RT)

− − + −= (4)

The general relation between the two sets of coefficients but also more details regarding the virial equation in pressure are given in Appendix 1. The actual representation of z by an infinite series in ρ or 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 not equivalent for the two kinds of series. It is however obvious that whatever the virial equation considered, its range of applicability increases with the number of coefficients used. Truncations to two terms

(z 1 B= + ρ or z 1 BP RT= + ) and truncations to three terms ( 2z 1 B C= + ρ + ρ or

( )2 2 2z 1 BP RT C B P (RT)= + + − ) 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 2z 1 B C= + ρ + ρ is the preferred three-term virial equation.

At low density, z 1 BP RT= + is the preferred two-term virial equation [1] because it is easier to use and probably more accurate than z 1 B= + ρ . 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


- 4 -

deviations corresponding to interactions among three molecules, and so on. To summarize, the importance of the virial equation of state lies in its theoretical 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. H. K. Onnes used this word by reference 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 convergence have never been entirely settled, either theoretically or experimentally. Some simple possibilities which have occasionally 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 density 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], …) which are known to be capable of representing both vapour and liquid behaviour will be used. We will explain, in which conditions, such equations 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 condensed phase. We will debate whether the virial expansion contains all the information about the complete equation of state for all phases. To summarize, we shall firstly address some general mathematical 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.


- 5 -

1. Virial expansions of cubic EoS In this paper, we only address cubic equations of state deriving from the Van der Waals equation. These ones take the general following form:

( ) ( )RT a

P T, vv b Q v

= −−

(5)

Q is a quadratic polynomial of v that can generally be written as: ( ) ( ) ( )1 2Q v v r b v r b= − − .

The parameters 1r and 2r are specific to each equation. Table 1 reminds 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:

( ) ( ) ( )1 2

v a vz T,v

v b RT v r b v r b= −

− − − (6)

By introducing the positive dimensionless variable b

vη = , called packing fraction, Eq. (6)

becomes:

( ) ( ) ( )1 2

1 az T,

1 RTb 1 r 1 r

ηη = −− η − η − η

(7)

• If 1 2r r≠ , by using a partial fraction expansion, one has:

( ) ( )2 1

2 1 2 1

r r1 az T,

1 RTb r r 1 r 1 r

ηη = − − − η − − η − η (8)

It is moreover well known that, for any x in ] [1,1− , the function f defined by: 1

f (x)1 x

=−

can be expressed as an infinite power series by: i

i 0

1x

1 x

+∞

=

=− ∑ .

As a consequence:

i

i 0

1

1

+∞

=

= η− η ∑ and thus: j i 1 i

jj i 0

rr

1 r

+∞+

=

= η− η ∑ with { }j 1,2∈

We thus obtain: ( )

i 1 i 1i2 1 2 1

2 1 2 1 2 1i 0

r r r r1

r r 1 r 1 r r r

+∞ + +

=

−− = η − − η − η − ∑


- 6 -

And thus:

( )i 1 i 1

i 12 1 2 1

2 1 2 1 2 1i 0

i ii2 1

2 1i 1

i i 0 0i 02 1 2 1

2 1 2 1i 0

r r r ra a

RTb r r 1 r 1 r RTb r r

r ra

RTb r r

r r r ra since 0

RTb r r r r

+∞ + ++

=+∞

=+∞

=

−η− − = − η − − η − η −

−= − η −

− −= − η η = − −

∑

∑

∑

Eq. (8) can be rewritten:

( )i i i i

i i2 1 2 1

2 1 2 1i 0 i 1

r r r ra az T, 1 1 1

RTb r r RTb r r

+∞ +∞

= =

− −η = − η = + − η − − ∑ ∑ (9)

Remembering that bη = ρ , one has:

( )i 1 i i i 1 i i

i i i i2 1 2 1

2 1 2 1i 0 i 1

ab r r ab r rz T, b 1 b

RT r r RT r r

+∞ +∞− −

= =

− −ρ = − ρ = + − ρ − − ∑ ∑ (10)

By comparison with Eq. (2), the virial coefficients for cubic equations of state, when 1 2r r≠ ,

are:

1c 1= and for all i ii 1

i 2 1i 1

2 1

r rabi 1, c b

RT r r

−

+ −≥ = − −

(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 η :

( ) ( ) ii 1

i 1

Pvz T, 1 c T

RT

+∞

+=

′η = = + ⋅η∑ (12)

The coefficients i 1c +′ are simply related to the virial coefficients ci+1 by:

i i

i 2 1i 1 i 1

2 1

a r rc c b 1

RTb r r−

+ + −′ = = − −

(13)

• If 1 2r r r= = , Eq. (7) writes:

( )( )2

1 az T,

1 RTb 1 r

ηη = −− η − η

(14)

Moreover, for any x in ] [1,1− , the function f defined by: ( )2

xf (x)

1 x=

− can be expressed as

an infinite power series by: ( )

i2

i 1

xi x

1 x

+∞

=

= ⋅−

∑ .

As a consequence : ( ) ( )

i i2 2

i 1

a a r air

RTb RTbr RTbr1 r 1 r

+∞

=

η η− = − = − η− η − η ∑


- 7 -

Since, i

i 1

11

1

+∞

=

= + η− η ∑ , Eq. (14) becomes:

( )i 1

i

i 1

airz T, 1 1

RTb

+∞ −

=

η = + − η

∑ (15)

meaning that: i 1

i 1air

c 1RTb

−

+′ = − (16)

By the end:

( )i 1 i 1

i i

i 1

air bz T, 1 b

RT

+∞ − −

=

ρ = + − ρ

∑ (17)

By comparison with Eq. (2), the virial coefficients for cubic equations of state, when 1 2r r= ,

are:

1c 1= and for all i 1 i 1

ii 1

air bi 1, c b

RT

− −

+≥ = − (18)

In Table 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 second 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.


- 8 -

2. 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 equations of state are defined for any b vη = 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 any η in [ [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:

( ) ii

i 0

f x x+∞

=

= α∑ (19)

where x is the variable and αi, the ith coefficient of the series. The radius of convergence, denoted R, is a nonnegative real number 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:

i 1

i i

1lim

R+

→+∞

α=α

(if the limit exists) (20)

Once the general mathematical context reminded, the calculation 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 1 2r r≤ (according to

the sets of parameters given in Table 1). In order to properly address the calculation of R, four different cases have to be distinguished: •••• CASE 1: 2r 1> and 21 rr < (this is the case for the Peng-Robinson and the Schmidt-

Wenzel equations of state). According to Eq. (13), the coefficient ic′ at infinite order takes the following form:

( ) ( )[ ] ( )12

1i2

i

1i21

12

1i2

i rrRTb

arr/r1

rrRTb

ar1c

−−∼−

−−=′

−

+∞→−

− (21)

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

i 12

i i

clim r

c+

→+∞

′=

′

And thus: 2

1R

r= (22)


- 9 -

•••• CASE 2: 2r 1≤ and 21 rr < (this is the case for the SRK equation of state)

- If 1r2 < , the coefficient ic′ at infinite order is equal to one:

( ) ( )[ ] 1r/r1rrRTb

ar1c

i1i

2112

1i2

i →−−

−=′+∞→

−−

(23)

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

i

i i 1

cR lim 1

c→+∞ +

′= =

′ (24)

- If 1r2 = , the ratio 1ii c/c +′′ has a finite limit when i tends to infinity:

1

r1r1

RTba

1

r1r1

RTba

1

c

ci

1

i1

1

1i1

1i

i →

−−−

−−−

=′′

+∞→

−

+ (since 0rlim i

1i

=+∞→

) (25)

and one finds back: R = 1. - If 2r 1= − , the radius of convergence may be found by splitting the expression of ( )z T,η 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: 1 2r r r= − =

This configuration requires a special treatment because in this case, Eqs. (21) and (23) cannot

be used since the quantity ( ) ( )i i1 2r / r 1= − has no limit when i tends to infinity.

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

( ) ( )( ) ( )2

1 a 1 a rz T,

1 RTb 1 r 1 r 1 RTbr1 r

η ηη = − = −− η − η + η − η − η

(26)

We know that the function f defined by 2

xf (x)

1 x=

− can be expressed as an infinite power

series by: 2i 12

i 0

xx

1 x

+∞+

=

=− ∑ .

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

( ) i 2i 2i 1

i 0 i 0

az T, r

RTb

+∞ +∞+

= =

η = η − η∑ ∑ (27)

The radius of convergence of the series i

i 0

+∞

=

η∑ is 1R 1= , whereas the one of the series

2i 2i 1

i 0

ar

RTb

+∞+

=

− η∑ is equal to 2 21

Rr

= , according to d’Alembert's ratio test. The radius of

convergence of the sum of these two series is the smallest out of the two: { }1 2R min R ,R= .

Finally, one has: - if r 1≤ , then the radius of convergence is equal to one: 1R R 1= = (this is the case for the

VdW equation of state).


- 10 -

- if r 1> , then the radius of convergence is: 2 21

R Rr

= = .

•••• CASE 4: 1 2r r r= =

For similar reasons to case 3, this case has to be treated specifically. The compressibility factor is given by Eq. (14) and the ic′ 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 is: 1

Rr

= .

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;Rη∈ (η being a

positive variable), or identically: R

0;b

ρ∈ or

bv ;

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;1η∈ ) only if

R is equal to one.


- 11 -

3. Discussion 3.a) 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), (14) and Table 1, VdW EoS takes the following form:

( )

( )

2

RT aP T, v

v b v1 a

z T,1 RTb

= − − η η = −

− η

(28)

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

( ) i

i 2

az T, 1 1

RTb

+∞

=

η = + − η + η

∑ (29)

The radius of convergence of this series is according to the previous study: R 1= . The virial expansion of z in the molar density is according to Eq. (17):

( ) i i

i 2

az T, 1 b b

RT

+∞

=

ρ = + − ρ + ρ

∑ (30)

The radius of convergence of this series is according to the previous study: R 1 b= . Consequently, the virial equation can be used in the same domain: [ [0;1η∈ , [ [0;1 bρ ∈ 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. 3.b) Redlich-Kwong type EoS Redlich-Kwong (RK) type EoS can be written under the general form:

( ) ( )( )

( ) ( )

a TRTP T, v

v b v v b

a T1z T,

1 RTb 1

= − − +

η η = −

− η + η

(31)

The original Redlich-Kwong EoS version is not still very used today. The SRK EoS (RK EoS modified by G. 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):

( ) ( )i i

i 1

az T, 1 1 1

RTb

+∞

=

η = + + − η ∑ (32)

According to our previous study (case 2, 2r 1= − ), the radius of convergence of this series is:

R 1= . The virial expansion of z in the molar density is according to Eq. (10):

( ) ( )i 1

ii i

i 1

abz T, 1 b 1

RT

+∞ −

=

ρ = + + − ρ

∑ (33)

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


- 12 -

Consequently, the virial equation can be used in the same domain [ [0;1η∈ , [ [0;1 bρ ∈ or

] [v b;∈ +∞ as the cubic EoS. It is 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 ρ (Eq. 33) is a practical impossibility. This is why, we are going to study the influence of the order of truncation on the prediction of P-v isotherms. We will distinguish truncations to odd terms and truncations to even terms. In Figs. 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 truncated virial expansions. From Fig. 1, it is clear that by considering a

truncation to the fifth term: 2 3 4z(T,v) 1 B v C v D v E v= + + + + , 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 (see Figs. 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 coefficients is considered. To better illustrate this point, we have plotted in Fig. 3, on a very large domain of pressure (a logarithmic scale is used) the isotherm of pure ethane calculated with the SRK EoS and 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 covolume b, which represents the minimum volume that a pure fluid can take up. Truncated virial expansions are polynomial functions in the variable 1/ vρ = (e.g.

2 3z 1 B C D= + ρ + ρ + ρ ). As a consequence, z and P tend to infinity (+∞ or −∞ ) when ρ tends to +∞ i.e. when v tends to zero. The P-v isotherm, calculated with a truncated virial expansion, will always admit v 0= as a vertical asymptote. The isotherm calculated with a cubic EoS will admit v b= as a vertical asymptote. 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 considered), which is predominant when v is null, is positive. In Fig. 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 ci virial coefficients 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 (see Table 2) are temperature dependent but they all can be expressed as a simple function of B. Indeed, one has (see Table

2): 24D c b B= = , 4

6F c b B= = , and more generally (2k 2)2kc b 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 ci 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


- 13 -

expansion to odd terms than to even terms. This is exactly what is observed in Fig. 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 volumes. When the virial series is truncated to an even term, the last considered virial coefficient has to be positive to make the expansion able to reproduce uncompressible liquid phase behaviour. 3.c) Peng-Robinson EoS The Peng-Robinson EoS [7], still very popular [9-20] writes:

( ) ( )( ) ( )

( ) ( )2

a TRTP T, v

v b v v b b v b

a T1z T,

1 RTb1 2

= − − + + −

η η = −

− η + η − η

(34)

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

( ) [ ] [ ]( )i i i

i 1

a1z T, 1 1 2 1 22 2RTb

+∞

=

+η = + η−− − − +

∑ (35)

According to Eq. (22), the radius of convergence of this series is:

1

R 2 1 0.41421 2

= = − ≈+

(36)

The virial expansion of z in the molar density is according to Eq. (10):

( ) [ ] [ ]( )i 1

ii i i

i 1

abz T, 1 b 1 2 1 22 2RT

+∞ −

=

ρ = + ρ+ − − − − +

∑ (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 η∈ − ,

0;( 2 1) b ρ∈ − or ( )v b 2 1 ; ∈ + +∞

smaller than that of the cubic EoS. For this

reason, we will say that the PR EoS can not 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 calculated 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. At this stage, it seems interesting to have a look at supercritical isotherms. Dealing with the SRK EoS and supercritical temperatures, one observes (see Fig. 6a) that a very low order of truncation of the virial series expansion enables to model the P-v isotherm with quite a good accuracy from low to very high pressures (e.g. n 3= in Fig. 6a). In addition, when the


- 14 -

temperature is above the Boyle temperature [ )bR/(aTB = ], all the virial coefficients are positive 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 the P-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 limitation 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= in Fig. 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 density is much smaller than the one corresponding to the radius of convergence of the series. All these observations can be found back in Fig. 6b. 3.d) Schmidt-Wenzel EoS This cubic EoS [8] is a little bit more elaborate than the SRK and the PR EoS because the parameters r1 and r2 of Eq. (6) are not anymore two universal constants characterising the EoS but depend on the nature of the component via the acentric factor ω. The SW equation writes:

( ) ( )( )

( ) ( )( )

2 2

2

a TRTP T, v

v b v 1 3 bv 3 b

a T1z T,

1 RTb1 1 3 3

= − − + + ω − ω

η η = −

− η + + ω η − ωη

(38)

As a specificity, this EoS tends to behave as the SRK EoS for values of ω close to zero and looks rather like the PR EoS when ω increases.

In the case where 2

1 2 0.0573

ω ≥ − + ≈ − , the polynomial function ( ) 2Q 1 1 3 3= + + ω η − ωη

can be factorised under the form ( ) ( )1 2Q 1 r 1 r= − η − η and the values of parameters r1 and r2

are provided in Table 1. This is the unique case discussed hereafter but more information

about the expansion of the SW EoS in virial series when 2

1 23

ω < − + may be found in

Appendix 2. The infinite power series of z in the packing fraction is according to Eq. (9):

( )i i

i2 1

2 1i 1

a r rz T, 1 1

RTb r r

+∞

=

−η = + − η − ∑ with

2

1

2

2

1 3 1 18 9r

2

1 3 1 18 9r

2

− − ω + + ω + ω = − − ω − + ω + ω=

(39)

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

2

1R min 1;

r

=

with 22

1 2

r 1 3 1 18 9=

+ ω + + ω + ω, when

21 2

3ω ≥ − + (40)


- 15 -

For ω in [ ]0;1 , the radius of convergence is plotted as a function of ω in Fig. 7.

The virial expansion of z in the molar density is according to Eq. (10):

( )i 1 i i

i i2 1

2 1i 1

ab r rz T, 1 b

RT r r

+∞ −

=

−ρ = + − ρ − ∑ with

ω+ω+−ω−−=

ω+ω++ω−−=

2

918131r

2

918131r

2

2

2

1 (41)

As a consequence, the virial equations can only be used in the domain [ [0;Rη∈ , [ [0;R bρ∈

or ] [v b / R;∈ +∞ ) smaller than that of the cubic EoS. It is interesting to notice that for values

of ω close to zero, the radius of convergence is close to one. This reminds of the behaviour of the SRK EoS. The more ω increases, 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 M1 and M2 having the following features:

1

1

1

c,M c,ethane

1 c,M c,ethane

M

T T 305.3 K

M P P 48.7 bar

0

= =

= =

ω =

and 2

2

2

c,M c,ethane

2 c,M c,ethane

M

T T 305.3 K

M P P 48.7 bar

0.5

= =

= =

ω =

In Fig. 8, a subcritical isotherm and a supercritical isotherm calculated with the SW EoS are represented for each of the two molecules M1 and M2. 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 M1, for which the radius of convergence is exactly equal to one ( )0=ω , the isotherms calculated from the EoS or its truncated virial expansion are completely merged on the whole range of molar volume. Regarding molecule M2, results are strongly different. In this case, the radius of convergence is smaller than one (R 1 3 0.33 for 0.5= ≈ ω = ): the EoS

and its truncated virial expansion only match on the range ] [v 3b;∈ +∞ . Outside this domain,

strong divergences are observed.


- 16 -

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 appropriate for the description of gases at low to moderate densities. These two statements are obviously contradictory since the popular two-parameter cubic equations of state (Van der Waals, Soave-Redlich-Kwong, Peng-Robinson or Schmidt-Wenzel, …) 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 cannot be modelled by a first order virial series. Let us notice that nothing prevents from using a truncated virial series to represent 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 component P-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 coefficients, the liquid state calculation was considerably improved. (ii) Secondly, when using analytical expressions of virial coefficients, 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 variable η is 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 any η in [ [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.


- 17 -

References [1] H.C. Van Ness, M.M. Abbott, Classical thermodynamics of non electrolyte solutions with

applications to phase equilibria, McGraw-Hill Chemical Engineering Series, New York, 1982.

[2] E.A. Mason, T.H. Spurling, The virial equation of state in "The international encyclopedia of physical chemistry and chemical physics", Pergamon Press, 1969.

[3] H. Kamerlingh Onnes, Royal Netherlands Academy of Arts and Sciences (KNAW). 4 (1902) 125-147.

[4] J. Levelt Sengers, How fluids unmix, Edited by the Royal Netherlands Academy of Arts and Sciences, Amsterdam, 2002.

[5] J.D. Van der Waals, On the continuity of the gaseous and liquid states (Over de continuiteit van den gas- en vloeistoftoestand), Ph.D. thesis, Leiden, 1873.

[6] G. Soave, Chem. Eng. Sci. 27(6) (1972) 1197-1203. [7] D.-Y. Peng, D. B. Robinson, Ind. Eng. Chem. Fundam. 15(1) (1976) 59-64. [8] G. Schmidt, H. Wenzel, Chem. Eng. Sci. 35 (1980) 1503-1512. [9] J.-N. Jaubert, L. Coniglio, F. Denet, Ind. Eng. Chem. Res. 38 (1999) 3162-3171. [10] J.-N. Jaubert, L. Coniglio, Ind. Eng. Chem. Res. 38 (1999) 5011-5018. [11] J.-N. Jaubert, P. Borg, L. Coniglio, D. Barth, J. Supercrit. Fluids 20 (2001) 145-155. [12] J.-N. Jaubert, F. Mutelet, Fluid Phase Equilib. 224 (2004) 285-304. [13] J.-N. Jaubert, S. Vitu, F. Mutelet, J.P. Corriou, Fluid Phase Equilib. 237 (2005) 193-

211. [14] F. Mutelet, S. Vitu, R. Privat, J.-N. Jaubert, Fluid Phase Equilib. 238 (2005) 157-168. [15] S. Vitu, J.-N. Jaubert, F. Mutelet, Fluid Phase Equilib. 243 (2006) 9-28. [16] S. Vitu, R. Privat, J.-N. Jaubert, F. J. Supercrit. Fluids 45(1) (2008) 1-26. [17] R. Privat, J.-N. Jaubert, F. Mutelet, Ind. Eng. Chem. Res. 47(6) (2008) 2033-2048. [18] R. Privat, J.-N. Jaubert, F. Mutelet, J. Chem. Thermodynamics 40 (2008) 1331-1341. [19] R. Privat, J.-N. Jaubert, F. Mutelet, Ind. Eng. Chem. Res. 47 (2008) 7483-7489. [20] R. Privat, F. Mutelet, J.-N. Jaubert, Ind. Eng. Chem. Res. 47 (2008) 10041-10052. [21] W.E. Putnam, J.E. Kilpatrick, J. Chem. Phys. 21 (1953) 951. [22] A. Bulinski, A. Shashkin, Limit Theorems for Associated Random Fields and Related

Systems (Advanced Series on Statistical Science and Applied Probability). World Scientific Publishing, Singapore, 2007.


- 18 -

Appendix 1 – 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 choosing pressure as the independent variable and by expanding z in an infinite power series, one has:

( ) ( )∑+∞

=+ ⋅′′=

0i

i1i PTcP,Tz (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 3n = , 4n = and 12n = terms are represented. One may observe that even with a few coefficients ( 3n = , 4n = ), 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 conclusion, one can state that the virial equation

in pressure is in the best case, only valid for pressures belonging to the range ( )[ ]TP;0 si .

In order to calculate the coefficients ( )Tc 1i +′′ of the virial equation in pressure, one may

express them with respect to the virial coefficients jc ( 1ij +≤ ). Eq. (4) gives the equations

enabling the calculation of the four coefficients ( )TcB 2′′=′ , ( )TcC 3′′=′ , ( )TcD 4′′=′ and

( )TcE 5′′=′ from the knowledge of ( )2B c T= , ( )3C c T= , ( )4D c T= and ( )5E c 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 consuming 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 the ( )Tc 1i +′′ coefficients of the virial equation in pressure from the jc virial

coefficients. Our method is much simpler and easier to use than the one developed in 1953. Explicit calculation of the coefficients of the virial series expansion in pressure On one hand, according to the virial series expansion in density, the pressure of a pure component is:

( ) ( ) ( )i ii 1 i 1

i 0 i 0

Pz T, c T P RT c T

RT

+∞ +∞

+ += =

ρ = = ⋅ρ ⇒ = ρ ⋅ρρ ∑ ∑ (43)

As a consequence, Eqs. (42) and (43), lead to:

( ) ( ) ( ) ( )i

i i ji 1 j 1

i 0 j 0

z T, c T RT c T+∞ +∞

+ += =

′′ρ = ⋅ ⋅ρ ⋅ρ

∑ ∑ (44)

In order to express the quantity ( )i

jj 1

j 0

c T+∞

+=

⋅ρ ∑ as a power series expansion, the Cauchy’s

product is used. Let us denote ( ) jj j 1c T+α = ⋅ρ .


- 19 -

Hence:

( )2

jj 1 j j k,1

j 0 j 0 j 0 k 0

c T m+∞ +∞ +∞ +∞

+= = = =

⋅ρ = α × α = ∑ ∑ ∑ ∑ with ∑

=−α⋅α=

k

0kkkk1,k

1

11m

Similarly, according to the Cauchy’s product:

( )3

jj 1 j k,1 k,2

j 0 j 0 k 0 k 0

c T m m+∞ +∞ +∞ +∞

+= = = =

⋅ρ = α × = ∑ ∑ ∑ ∑

with 1

1 1 2 1 2 1

1 1 2

kk k

k,2 k ,1 k k k k k k kk 0 k 0 k 0

m m − − −= = =

= ⋅ α = α ⋅ α ⋅α∑ ∑ ∑

By using a recurrence relation, the ith term of the sequence

α∑

∞+

=

i

0jj is:

( )

( ) ( ) ( ) ( )

( ) ( )

1 i 2

1 1 2 i 2 i 1 i 1

1 2 i 1

1 i 2

1 1 2 i 2 i 1 i 1

1 2 i 1

1 1 2 i

ik kk

jj 1 k k k k k k k

j 0 k 0 k 0 k 0 k 0

k kkk

k k 1 k k 1 k k 1 k 1k 0 k 0 k 0 k 0

k k 1 k k 1 k

c T ... ...

... c T c T ... c T c T

c T c T ... c

−

− − −

−

−

− − −

−

−

+∞ +∞

+ − − −= = = = =

+∞

− + − + − + += = = =

− + − +

⋅ρ = α ⋅ α ⋅ ⋅ α ⋅ α

= ⋅ ⋅ ⋅ ⋅ ⋅ρ

= ⋅ ⋅ ⋅

∑ ∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

( ) ( )2 i 1 i 1

i 1 1

kk 1 k 1

0 k ... k k

T c T− −

−

− + +≤ ≤ ≤ ≤ ≤+∞

⋅ ⋅ρ∑ This quantity writes thus in the equivalent form:

( )i

j i kj 1 k

j 0 k 0

c T+∞ +∞

+= =

⋅ρ = δ ⋅ρ ∑ ∑ (45)

with ( ) ( ) ( ) ( )1 1 2 i 2 i 1 i 1

i 1 1

ik k k 1 k k 1 k k 1 k 1

0 k ... k k

c T c T ... c T c T− − −

−

− + − + − + +≤ ≤ ≤ ≤

δ = ⋅ ⋅ ⋅ ⋅∑

The sequence ( )ikδ has got the following features:

( )

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1

1

00

ii0 1

0k

1k k 1

k2k k 1 k k 1 1 k 1 2 k k 1 k k 1

k 0

1

c 1

0, for all k 0

c T

c T c T c T c T c T c T ... c T c T

+

+ − + + + − +=

δ =δ = =δ = >δ =

δ = ⋅ = ⋅ + ⋅ + + ⋅

∑

⋮

(46)


- 20 -

Combining Eqs. (44) and (45), we obtain:

( ) ( ) ( )∑∑+∞

=

+∞

=

++ ρ⋅δ⋅⋅′′=ρ

0k 0i

ikik

i1i RTTc,Tz (47)

By introducing variable ikn += , Eq. (47) becomes:

( ) ( ) ( )∑∑+∞

=

+∞

=

−−+− ρ⋅δ⋅⋅′′=ρ

0k kn

nknk

kn1kn RTTc,Tz (48)

By applying Fubini’s theorem [22], Eq. (48) writes:

( ) ( ) ( )n

n k n k nn k 1 k

n 0 k 0

z T, c T RT+∞

− −− +

= =

′′ρ = ⋅ ⋅ δ ⋅ρ∑∑ (49)

By identifying Eqs. (43) and (49), one has:

( ) ( ) ( )n

n k n kn 1 n k 1 k

k 0

c T c T RT− −

+ − +=

′′= ⋅ ⋅ δ∑

Or similarly: ( ) ( ) ( )n

k kn 1 k 1 n k

k 0

c T c T RT+ + −=

′′= ⋅ ⋅ δ∑ (50)

Since 1n0 =δ , one gets a general expression of coefficient ( )Tc 1n+′′ :

( )( )

( ) ( ) ( )n 1

k kn 1 n 1 k 1 n kn

k 0

1c T c T c T RT

RT

−

+ + + −=

′′ ′′= − ⋅ ⋅ δ

∑ (51)

The use of Eq. (51) to calculate the four first coefficients of the virial equation in pressure, namely: ( )Tc1′′ to ( )Tc4′′ , is illustrated here after. • 0n = :

According to Eq. (51): ( ) ( )( )

( )11 10

c TA c T c T A 1

RT′ ′′= = = = = (52)

• 1n = : According to Eq. (51):

( ) ( ) ( ) ( )( )

( )0 02 1 1 2

2 1

c T c T RT c T BB c T

RT RTRT

′′− ⋅ ⋅ δ′ ′′= = = = since 01 0δ = (53)


( ) ( ) ( ) ( ) ( ) ( )( )

0 10 13 1 2 2 1

3 2

c T c T RT c T RTc T

RT

′′ ′′− ⋅ ⋅ δ − ⋅ ⋅ δ′′ =

Since: 02

11 2

0

c (T)

δ =

δ = and by expressing 2c (T)′′ according to Eq. (53), one finds:

( ) ( ) ( )( ) ( )

2 23 2

3 2 2

c T c T C BC c T

RT RT

− − ′ ′′= = = (54)


- 21 -


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

0 1 20 1 24 1 3 2 2 3 1

4 3

c T c T RT c T RT c T RTc T

RT

′′ ′′ ′′− ⋅ ⋅ δ − ⋅ ⋅ δ − ⋅ ⋅δ′′ =

Since:

1 1

1

03

12 3

121 k 1 2 k 1 2 2 1

k 0

0

c (T)

c (T) c (T) 2c (T) c (T) 2c (T) (since c (T) 1)+ −=

δ =δ =δ = ⋅ = ⋅ = =

∑

and by expressing 2c (T)′′ and 3c (T)′′ according to Eqs. (53) and (54), one finds:

( )( ) ( ) ( ) ( ) ( ) ( )

( )

24 2 3 2 3 2

4 3

c T c T c T 2c T c T c Tc T

RT

− ⋅ − − ′′ =

Thus: ( ) ( ) ( ) ( ) ( )( ) ( )

3 34 2 3 2

4 3 3

c T 3c T c T 2 c T D 3BC 2BD c T

RT RT

− ⋅ + − + ′ ′′= = = (55)


- 22 -

Appendix 2 – Virial series expansion of the SW EoS and calculation of its radius of convergence when ωωωω < 22/31+− The SW EoS writes:

( ) ( )( )

( ) ( )( )

2 2

2

a TRTP T, v

v b v 1 3 bv 3 b

a T1z T,

1 RTb1 1 3 3

= − − + + ω − ω

η η = −

− η + + ω η − ωη

The acentric factor of a pure component i is defined by: ( )

1P

T7.0TPlog

c

csi −

=−=ω . Since

( ) ccsi PT7.0P < , ω belongs to the range ] [1;− +∞ . Let us notice that in practice, this range

may often be reduced to [ ]1;0 for most of the usual components.

When 057.023

21 −≈+−<ω , the denominator of the attractive term of the SW EoS cannot

be factorised anymore and as a consequence, the general Eqs. (12) and (13) cannot be applied to find out the virial series expansion. We thus propose an alternative method concerning this specific case, that is when

+−−∈ω 23

21;1 .

We denote:

ω−=ω+=

3w

31u (56)

As a first step, we look for a power series expansion of the attractive term of z, as follows:

∑+∞

=

η=η+η+

η

0n

nn2

awu1

(57)

Eq. (57) may be rewritten as:

∑∑∑+∞

=

++∞

=

++∞

=

η+η+η=η0n

2nn

0n

1nn

0n

nn awaua (58)

When identifying the right hand side with the left hand side of Eq. (58), one gets the expression of the sequence ( )na :

0

1

n 2 n 1 n

a 0

a 1

a u a w a 0 for all n 0+ +

= = + ⋅ + ⋅ = ≥

(59)

To characterize explicitly this linear recurrent sequence, we form the characteristic equation:

0wrur2 =+⋅+


- 23 -

The discriminant of this equation is then: 1189w4u 22 +ω+ω=−=∆ and is negative in the

present case (i.e. when 23

21+−<ω ). The solutions of the characteristic equation are

complex and write:

ir e± θ= µ with:

( )

( ) ( ) ( )

2

2 2

11 3

21

9 18 12

r 3

arg r arctan / modulo

α = − + ωβ = ω + ω +µ = = α + β = − ω

θ = = β α π

(60)

The nth term of the sequence ( )na takes then the following expression:

( ) ( )[ ]θ⋅+θ⋅µ= nsinBncosAa nn where A and B are two real constants.

Taking into account the expressions of a0 and a1, one deduces:

( )

θ⋅µ==

sin/1B

0A

And finally:

( )

θθµ= −

sin

nsina 1n

n (61)

According to Eqs. (38), (57) and (61), the power series expansion of the SW EoS when

23

21+−<ω is:

( )

( ) ( )

nn 1

n 0

n 1n 1

z T, c

a T sin nc 1

bRT sin

+∞

+=

−+

′η = η

θ′ = − µ θ

∑ (62)

And the virial series expansion is then:

( ) n

n 1n 0

nn 1 n 1

z T, c

c c b

+∞

+=

+ +

ρ = ⋅ρ

′= ⋅

∑ (63)

Calculation of the radius of convergence R of the virial series expansion of the SW EoS when ωωωω < 22/31+− : To do so, three steps are considered. In steps 1 and 2, we shall study the convergence of the

series ( ) ∑+∞

=

η=η0n

nn1 aS for some specific values of η . We recall that the radius of

convergence R1 of the series if such that:

>η<η

convergenot does series the,R if

converges series the,R if

1

1

In step 3, the radius of convergence of the SW EoS will be deduced from steps 1 and 2.


- 24 -

Step 1. Let us consider the series:

( ) ∑+∞

=

η=η0n

nn1 aS with

( )θθ⋅µ= −

sin

nsina 1n

n (64)

According to the triangle inequality:

∑∑∑+∞

=

+∞

=

+∞

=

η⋅µ⋅µ

≤η⋅≤η0n

n

0n

nn

0n

nn

1aa

If 1<η⋅µ or similarly: µ

<η 1, then the series ∑

+∞

=

η⋅µ0n

n converges and thus the series ( )η1S

also converges. As a consequence, [ [1R;01

;0 ⊆

µ and the radius of convergence verifies:

µ

≥ 1R1 (65)

Step 2. Let us consider the particular packing fraction: µ

=η 1. In that case, Eq. (64) writes:

( ) ( )∑+∞

=

θθµ

=µ=η0n

1 nsinsin

1/1S

This series has no limit (the terms of the series alternate between a lower and an upper bound

when n tends to infinity). As a consequence, the series ( ) ∑+∞

=

η=η0n

nn1 aS does not converge

when µ

=η 1 thus, the radius of convergence is necessary lower than or equal to this quantity:

µ

≤ 1R1 (66)

Finally, from Eqs. (65) and (66), it seems obvious that:

µ

= 1R1 (67)

Step 3. The radius of convergence R of the virial series expansion can now be calculated. According to Eq. (62), the power series expansion of the SW EoS for acentric factors

23

21+−<ω is:

( ) ( )

( )

0 1

n0 0

n 0

n1 n 1

n 0

a Tz T, S ( ) S ( )

bRT

S ( ) , with radius of convergence : R 1

1S a , with radius of convergence : R

+∞

=+∞

=

η = η − η η = η = η = η = µ

∑

∑


- 25 -

The radius of convergence R is then:

{ } { } { }0 1R min R ;R min 1;1 min 1;1 3= = µ = − ω (68)

As an illustration, a molecule M3 having the following features is considered:

9.0

bar 7.48PP

K3.305TT

M

3

3

3

M

ethane,cM,c

ethane,cM,c

3

−=ω

==

==

In Fig. 10, which shows the P-v isotherms of this molecule at 287 K calculated with the SW EoS and with the virial series expansion truncated to 100 terms, the impossibility to use the

virial expansion of the SW EoS to represent liquid densities when µ

=>η 1R (or for molar

volumes µ< bv ) clearly appears. Conclusion:

By noticing that if 1

3ω = − (i.e. 1µ = ), one has: 1RR 10 == and according to Eq. (40), one

can state that:

2

22

0

1

1 1 3 1 18 9if 0, then R with: r

r 2

1if 0, then R R 1

31 1

if 1 , then R R with: 33

− − ω − + ω + ω ω > = = − ≤ ω ≤ = = − < ω < − = = µ = − ω µ

(69)

These results are summed up in Fig. 11.


- 26 -

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

Equations 1r 2r*

Van der Waals (VdW) 0 0 Soave-Redlich-Kwong

(SRK) 0 –1

Peng-Robinson (PR) 1 2− + 1 2− − 21 18 9∆ = + ω + ω

Schmidt-Wenzel** (SW) 1 3

2

+ ω − ∆− 1 3

2

+ ω + ∆−

* 1r and 2r are such as: 2 1r r≥ .

** Factorization only valid for 1 2 2 3 0.057ω ≥ − + ≈ − . Table 2. Values of virial coefficients A, B, C, D, E, F for four cubic equations of state.

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

VdW 1 a

bRT

− 2b 3b 4b 5b

SRK 1 a

bRT

− 2 abb

RT+

23 ab

bRT

− 3

4 abb

RT+

45 ab

bRT

−

PR 1 a

bRT

− 2 2abb

RT+

23 5ab

bRT

− 3

4 12abb

RT+

45 29ab

bRT

−

2 1abb

RT

ς+ 2

3 2abb

RT

ς− 3

4 3abb

RT

ς+ 4

5 4abb

RT

ς−

SW 1 a

bRT

− 1

22

2 33

2 3 44

1 3

1 9 9

1 15 45 27

1 21 117 189 81

ς = + ω

ς = + ω + ω

ς = + ω + ω + ως = + ω + ω + ω + ω


- 27 -

Figure Captions Figure 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= ). Figure 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= , n 22= ). Figure 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= ). Figure 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. Figure 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. Figure 6. P-v isotherms of pure ethane at supercritical temperature cT 2T 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. Figure 7. Radius of convergence of the infinite power series of z in the packing fraction for the SW EoS as a function of ω. Figure 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 M1 ( 0=ω ) at subcritical temperature. (b) molecule M1 at supercritical temperature. (c) molecule M2 ( 5.0=ω ) at subcritical temperature. (d) molecule M2 at supercritical temperature. Figure 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= , 4n = , 12n = ). Figure 10. lnP-v isotherms of a fictive molecule M3 ( 9.0−=ω ) at 287 K calculated from the SW EoS (straight line) and from a truncated virial expansion to 100 terms (dashed line). Figure 11. Radius of convergence of the infinite power series of z in the packing fraction for the SW EoS as a function of ω ranging from ] ]1;2− .


- 28 -

COLOR FIGURES

Figure 1.

0.0 400.0 800.0

20.0

30.0

40.0

50.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar

n = 1 (ideal gas) n = 3

n = 5

n = 21

SRK (almost completely merged with virial expansion truncated to 21 terms)

T = 287 K

Figure 2.


- 29 -

0.0 400.0 800.0

10.0

20.0

30.0

40.0

50.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar

n = 22


n = 4

n = 2

n = 6

T = 287 K


- 30 -

Figure 3.

0.0 400.0 800.0 2.50

3.50

4.50

5.50

v/(cm 3⋅⋅⋅⋅mol –1)

ln(P/bar) SRK n = 5 n = 6

T = 287 K

Figure 4.

0.0 400.0 800.0

10.0

20.0

30.0

40.0

50.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar

n = 3 n = 9 n = 11 n = 27 n = 51 PR EOS

( )b 2 + 1

T = 287 K


- 31 -

Figure 5.

0.0 400.0 800.0

10.0

20.0

30.0

40.0

50.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar n = 4 n = 10 n = 12 n = 28 n = 52 PR EOS

( )b 2 +1

T = 287 K

Figure 6.

100.0 300.0 0.0

5.0

10.0

15.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/PcSRK EOS

T = 2Tc

n = 3

(a)

200.0 300.0 0.0

5.0

10.0

15.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/Pc

( )b 2 +1

T = 2Tc

n = 3 n = 8 n = 71 PR EOS

(b)


- 32 -

Figure 7.

0.00 1.00

0.20

0.40

0.60

0.80

1.00

ωωωω

R


- 33 -

Figure 8.

0.0 400.0 800.0 20.0

30.0

40.0

50.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar(a)T = 287 K

b/R = b

0.0 400.0 800.0 5.0

15.0

25.0

35.0

45.0

55.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar

b/R = 3b

(c)T = 287 K

0.0 400.0 800.0 0.0

400.0

800.0

P/bar

v/(cm 3⋅⋅⋅⋅mol –1)

b/R = b

(b)T = 350 K

0.0 400.0 800.0 0.0

400.0

800.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar

b/R = 3b

(d)T = 350 K


- 34 -

Figure 9.

0.0 500.0 1000.015.0

25.0

35.0

45.0

55.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/barn = 3

PR

T = 287 K n = 4

n = 12

Figure 10.

0.0 400.0 800.03.00

5.00

v/(cm 3⋅⋅⋅⋅mol –1)

ln(P/bar)

b/R = bµµµµ

T = 287 K Molecule M 3

(ωωωω = -0.9)

SW n = 100


- 35 -

Figure 11.

-1.00 0.00 1.00 2.00

0.00

0.50

1.00

ωωωω

R

-1/3

1 / |r2|

1 / µµµµ


- 36 -

BLACK AND WHITE FIGURES

Figure 1.

0.0 400.0 800.0

20.0

30.0

40.0

50.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar

n = 1 (ideal gas) n = 3

n = 5

n = 21


T = 287 K

Figure 2.


- 37 -

0.0 400.0 800.0

10.0

20.0

30.0

40.0

50.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar

n = 22


n = 4

n = 2

n = 6

T = 287 K


- 38 -

Figure 3.

0.0 400.0 800.0 2.50

3.50

4.50

5.50

v/(cm 3⋅⋅⋅⋅mol –1)

ln(P/bar) SRK n = 5 n = 6

T = 287 K

Figure 4.

0.0 400.0 800.0

10.0

20.0

30.0

40.0

50.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar

n = 3 n = 9 n = 11 n = 27 n = 51 PR EOS

( )b 2 + 1

T = 287 K


- 39 -

Figure 5.

0.0 400.0 800.0

10.0

20.0

30.0

40.0

50.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar n = 4 n = 10 n = 12 n = 28 n = 52 PR EOS

( )b 2 +1

T = 287 K

Figure 6.

100.0 300.0 0.0

5.0

10.0

15.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/PcSRK EOS

T = 2Tc

n = 3

(a)

200.0 300.0 0.0

5.0

10.0

15.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/Pc

( )b 2 +1

T = 2Tc

n = 3 n = 8 n = 71 PR EOS

(b)


- 40 -

Figure 7.

0.00 1.00

0.20

0.40

0.60

0.80

1.00

ωωωω

R


- 41 -

Figure 8.

0.0 400.0 800.0 20.0

30.0

40.0

50.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar(a)T = 287 K

b/R = b

0.0 400.0 800.0 5.0

15.0

25.0

35.0

45.0

55.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar

b/R = 3b

(c)T = 287 K

0.0 400.0 800.0 0.0

400.0

800.0

P/bar

v/(cm 3⋅⋅⋅⋅mol –1)

b/R = b

(b)T = 350 K

0.0 400.0 800.0 0.0

400.0

800.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/bar

b/R = 3b

(d)T = 350 K


- 42 -

Figure 9.

0.0 500.0 1000.015.0

25.0

35.0

45.0

55.0

v/(cm 3⋅⋅⋅⋅mol –1)

P/barn = 3

PR

T = 287 K n = 4

n = 12

Figure 10.

0.0 400.0 800.03.00

5.00

v/(cm 3⋅⋅⋅⋅mol –1)

ln(P/bar)

b/R = bµµµµ

T = 287 K Molecule M 3

(ωωωω = -0.9)

SW n = 100


- 43 -

Figure 11.

-1.00 0.00 1.00 2.00

0.00

0.50

1.00

ωωωω

R

-1/3

1 / |r2|

1 / µµµµ

Documents

Can cubic equations of state be recast in the virial form?irma.math.unistra.fr/~privat/documents/viriel.pdf · of these equations, we detail the limitations and the capabilities of