7264
Ind. Eng. Chem. Res. 2006, 45, 7264-7274
An Alternative Approach to Nonrandomness in Solution Thermodynamics Ioannis Tsivintzelis, Giorgos S. Dritsas, and Costas Panayiotou* Department of Chemical Engineering, UniVersity of Thessaloniki, 54124 Thessaloniki, Greece
An alternative formulation of the configurational partition function of mixtures is proposed, which leads to analytical expressions for the basic thermodynamic quantities, including the local compositions or the nonrandomness factors in multicomponent mixtures. By implementing the new formalism into an equationof-state framework, a simple and versatile nonrandom model of fluids and their mixtures is developed. The new model is tested against experimental data for both pure fluids and mixtures. Among others, the model calculates a significant degree of nonrandomness in the distribution of free volume, even in nonpolar systems. The new expressions for local compositions are compared with corresponding expressions from the literature, including Guggenheim’s quasi-chemical expressions and Monte Carlo simulations. These expressions are similar to the well-known Wilson-type expressions. Thus, our approach provides, to some degree, a semitheoretical basis for the Wilson expressions for local compositions, and their consistency problem is discussed. Introduction The molecular species are, as a rule, distributed nonrandomly in their mixtures. In other words, the local composition in the immediate neighborhood of a molecule is generally different from the overall (or bulk) composition of the mixture. Even in pure fluids, there is a degree of nonrandomness in the distribution of their functional groups. In fact, modern experimental techniques, such as positron annihilation spectroscopy, reveal significant nonrandomness in the distribution of free volume throughout the volume of the pure fluid, even in nonpolar systems.1,2 One of the earliest and still most important attempts to describe nonrandomness in mixtures is Guggenheim’s “QuasiChemical” approximation.3 This approximation leads to expressions for the local compositions, which are analytical for binary mixtures only, whereas iterative procedures must be used to solve the equations for multicomponent mixtures.4,5 Analytical expressions for local compositions in binary mixtures are rather easy to find in the literature. As an example, such expressions have been proposed recently by Yan et al.,6 based on Monte Carlo simulation results; however, their extension to multicomponent mixtures is not yet available or straightforward. This is why the most widely used equations for local compositions in the literature, especially for engineering calculations, are probably the Wilson-type analytical equations,7,8 despite the fact that they lead to inconsistencies or violate simple material balance or intermolecular contact conservation requirements.4,8 In a recent work,9 a new approach to nonrandomness in systems of r-mer fluids has been proposed. This method permitted the development of a new nonrandom thermodynamic model for multicomponent mixtures of fluids in which the calculations for all thermodynamic quantities can be performed without any recourse to intermediate calculation of nonrandomness factors or local compositions. Yet, new analytical expressions for the nonrandomness factors were obtained. In the same work,9 and in its appendix, an alternative approach has been indicated, which leads to an expression for the temperaturedependent factor of the configurational partition function, essentially identical in form to the normalization factor proposed by Wang and Vera10 and Vera11 that makes the partition function * To whom all correspondence should be addressed. Tel./Fax: +30 2310 996223. E-mail:
[email protected].
applicable to systems that exhibit any degree of order. This factor also leads to analytical expressions for local compositions in multicomponent systems; however, despite their simplicity, they were not given any consideration so far.9 Thus, in the present work, we will focus on this alternative approach and we will detail the rationale that leads to a new simple and versatile nonrandom equation-of-state model of fluids. In the limit of random distribution of molecular species and with many simplifications, the new model can be reduced to the lattice-fluid type models, such as the widely used model of Sanchez and Lacombe.12 Again, although not needed in the calculation of any basic thermodynamic quantity, we will derive new analytical expressions for the nonrandomness factors and will compare them with corresponding expressions from the literature. In this way, we will be able to critically compare the present approach with the previous approaches, in regard to the derived expressions for local compositions. The calculations of the model for pure fluids and mixtures will be compared with experimental data from the literature. Theory 1. The Rationale of the Approach. Before presenting the equation-of-state formalism for systems of real fluids differing in size and shape, we will detail the rationale of our approach for nonrandomness, taking a very simple case of spherical molecules of equal size. Let us consider a binary system of such molecules N1 and N2 of components 1 and 2, respectively, at a temperature T, which are arranged on a quasi-lattice of coordination number z, occupying a total volume V. The configurational partition function, Q, of the system can be written in the classical manner (as a sum over states) as follows:13
Q(N1,N2,T,V) )
∑i Ωie-βE
i
(1)
The summation in the aforementioned equation spans all microstates i in the phase space of our system characterized by a corresponding energy level Ei. Ωi is the degeneration or multiplicity factor for all microstates corresponding to the energy level Ei. β is the thermal energy factor, equal to 1/(kT), where k is the Boltzmann’s constant. Many assumptions are usually made to perform the summation in eq 1.
10.1021/ie060490p CCC: $33.50 © 2006 American Chemical Society Published on Web 08/25/2006
{[ ( ) [()
]
Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 7265
The first key assumption that we will adopt here is that we may factorize the partition function into a random or athermal part and an energetic or thermal part, as follows:3,6-11,13-21
Q = QAthermal QThermal ) QR
∑i Wie-βE ) QRQE i
Ei ) {N1111 + N1212 + N2222}i
(3)
On the other hand, the number of intermolecular contacts should meet the “mass balance” or site conservation criteria:
z N11 + N12 + N22 ) N 2
(4a)
2N11 + N12 ) zN1
(4b)
2N22 + N12 ) zN2
(4c)
N is the total number of molecules in the system, N ) N1 + N2, whereas zNi/2 is the total number of intermolecular contacts in which the molecule i participates. To proceed, we will need an expression for Wi in eq 2. One reasonable approximation for Wi can be obtained by finding, first, the number of ways we may select out the 1-1 contacts from the zN1/2 contacts and the 2-2 contacts from the zN2/2 contacts. From Combinatorics, this is given by3,9,13,20
ω{Nkl}i )
)
[
(2z N )! 1
][
(2z N )! 2
z z N11! N1 - N11 ! N22! N2 - N22 ! 2 2
(
)
(
[{ ( ) ( ) } ] (2z N )! (2z N )! 1
)
]
2
N12 N12 N11! ! N22! ! 2 2
(5)
i
By picking up a contact at random, let the probability for the contact of type 1-1, 2-2, and 1-2 be w11, w22, and w12, respectively. Inserting these probabilities into eq 5, we obtain, for Wi, the following expression:
{()
}
z z N ! N ! 2 1 2 2 w N11w12N12/2w22N22w12N12/2 W{Nkl}i ) N12 N12 11 N11! ! N22! ! 2 2 i (6)
( ) ( )
( )
z
QE )
(2)
The random part, QR, represents the limiting value of the partition function when all intermolecular interaction energies vanish and the molecules are distributed randomly throughout the volume of the system. The energetic part, QE, is then a correction term that retains the original form of the partition function and should become equal to 1 in the limiting case of zero interaction energies. The summation in eq 2 is very much facilitated by our lattice picture and by further adopting the assumption that only nearest-neighbor interactions are important. This, essentially, implies that a microstate i of our system is fully determined by the set {Nkl}i of the numbers Nkl of contacts between molecules of types k and l. If each contact k-l is characterized by an energy kl, the corresponding energy level is given by
Combining eqs 3 and 6, we may express QE in eq 2 as follows:
( )
∑
{Nkl}i
2
N1 !
(w11e-b11)N11(w12e-β12)N12/2 ×
N12
N11!
2 z
!
( ) 2
N2 !
N22!
2
]}
(w22e-β22)N22(w12e-β12)N12/2
N12
!
(7)
i
Assuming that the wkl values are independent of the Nkl values, and applying the multinomial theorem of Combinatorics in each of the brackets in eq 7 separately, we obtain
QE ) (w11e-b11 + w12e-β12)zN1/2(w22e-β22 + w12e-β12)zN2/2 ) (w11τ11 + w12τ12)zN1/2(w22τ22 + w12τ12)zN2/2 (8) where
τkl ) e-βkl
(for k,l ) 1 or 2)
(9)
We will come back to the summation in eq 7 in a later section. The problem now is to determine expressions for the probabilities wkl and replace them in eq 8. One reasonable approximation is the following:9
wkl ) xkxlΓkl
(for k,l ) 1 or 2)
(10)
In this equation, Γkl is either equal to zero for the physically meaningless terms or equal to the nonrandomness factor4,5,8,9 for the contact k-l, which is defined by
Nkl ) ΓklNkl0
(for k,l ) 1 or 2)
(11)
where the superscript 0 indicates the corresponding number of intermolecular contacts in the limit of perfectly random mixture. In this perfectly random mixture, Γ ) 1. In the random case, these numbers of contacts are given by the following set of equations:
( 2z)N x z ) ( )N x 2
0 ) N11
1 1
0 N22
2 2
0 N12 ) zN1x2 ) zN2x1
}
(random case)
(12)
However, an expression similar to that of eq 10, if replaced in eq 7, would make the summation extremely difficult to be performed. In a zeroth approximation,9 we might set Γkl ) 1 in eq 10 and obtain from eq 8 the following expression:
QE ) (x12τ11 + x1x2τ12)zN1/2(x22τ22 + x1x2τ12)zN2/2 (13) This expression does not become equal to one in the limit of zero interaction energies as it should. Instead, it becomes
QE ) (x12 + x1x2)zN1/2(x22 + x1x2)zN2/2 ) (x1)zN1/2(x2)zN2/2 (randomness limit) (14) Obviously, we have overcounted QE, and several terms have contributed to it, although they should not have made any
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contribution. Following Guggenheim,3 we normalize eq 13 by dividing it with the randomness limit from eq 14 and obtain
∂ ln QE
E ) kT 2
)
QE )
l
zqiNi
)
∑j qjNj
z
zqiNi zqN
(17)
The probability to have one specific type of intermolecular contact i-j between one segment of type i and a central segment of type j is now given by the “local” surface fraction,4 θij, rather than the local mole fraction, xij. As a consequence, eq 16, in this case, becomes t
QE )
( ∑ θmτml) ∏ l)1 m ) 1
zqlNl/2
∑i θi ) ∑j θji ) 1
z
(19)
(20)
t
∑Niqim)1 ∑ θmΓmimi
2 i)1
(nonrandom)
∑l θlτli
mi
(22)
τmi (23)
∑l θlτli
( )∏
Nr! Nq!
z/2 t
N0! Nr!
i)1
ωiNrliNi Ni!
(24)
where t
N r ) N0 +
riNi ) N0 + rN ∑ i)1
(25)
and t
where Γji is again the nonrandomness factor for the distribution of segments of type j around a central segment of type i. In terms of nonrandomness factors, the potential energy of our system may be written as follows:4
E)
ΩR )
(18)
Generally, we may write4
θji ) θjΓji
τmi
These are simple analytical expressions that are valid for any number of components in the mixture. These Γ parameters should not be taken literally as consistent expressions for nonrandomness factors, neither in the present work nor in ref 9. Regarding the present work, by examining carefully eq 23, we observe that Γmi is different from Γim, in contrast with the basic contact-conservation requirement (cf. eq 4). Again, however, we will not need to calculate explicitly these factors for the calculation of other thermodynamic quantities. 3. Implementation to an Equation-of-State Model. In this section, we will use the central eq 18 to develop an equationof-state model of the lattice-fluid type.12,14-16 Thus, we will assume that the molecules in our system are distributed on a quasi-lattice, leaving a number, N0, of empty sites in it. This means that our pure fluid is a pseudo-binary system with molecular segments of type 1 and empty sites of type 0, while our binary system is a pseudo-ternary mixture of components 1, 2, and 0. For the random combinatorial term, QR, in eq 2 we will adopt the generalized Staverman expression,17 as previously described,15 which, for a system of t components, takes the form
t
This is the central equation that will form the basis for the developments in the remainder of this work. It is worth noting that eq 18 is similar in form to the correction term that has been used in Vera’s research10,11 for the normalization of their partition function. 2. Expressions for Nonrandomness Factors. By definition, the following must hold true:
t
Γmi )
(16)
Let us now consider a mixture of t types of molecules with Ni molecules of type i that differ in size and shape. Each molecule of type i is considered to be divided into ri identical segments and that have zqi external intermolecular contacts. Thus, the product zqiNi, which represents the total number of contact points of molecules i, may be viewed as the interacting “area” or contacting “surface” of molecules i. When this surface is divided by the sum of surfaces of all molecules, we obtain the surface fraction, θi, of component i in the system, or
θi )
∑
m)1
θm
Although not needed in any calculations with the present model, we may obtain expressions for the nonrandomness factors by comparing eqs 21 and 22. From this direct comparison, we obtain
t
( ∑ xmτml)zN /2 ∏ l)1 m)1
∑
t
Niqi
(15)
In fact, we could obtain eq 15 directly and without normalization by using in eq 5 ther term9,20 (zN/2)!, instead of the product (zN1/2)!(zN2/2)!, as before.9 This is a very important expression, which becomes indeed equal to 1 in the randomness limit. It can be generalized rather easily for a mixture of t types of molecules as follows: t
t
2 i)1
∂T QE ) (x1τ11 + x2τ12)zN1/2(x2τ22 + x1τ12)zN2/2
z
(21)
On the other hand, an expression for E may be obtained from the energetic or thermal term QE in the usual manner of statistical thermodynamics:13
N q ) N0 +
qiNi ) N0 + qN ∑ i)1
(26)
ωi is a characteristic parameter of the fluid i, which is dependent on the geometrical characteristics and the flexibility of the molecule but will cancel out in all applications of our interest. The Staverman parameter, li, is defined by17
z li ) (ri - qi) - (ri - 1) 2
(27)
This parameter is equal to zero for the usual lattice definition of q for linear chains (zq ) (z - 2)r + 2). If V* is the hardcore or van der Waals volume per segment in the latticefluid12,14-16 sense, the total volume of the system is given by
V ) NrV* ) N0V* + rNV* ) N0V* + V* The reduced volume is defined as
(28)
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V˜ )
1 V ) V* F˜
(29)
where F˜ is the reduced density. By assigning zero energy to any contact with empty sites, or, 10 ) 20 ) 00 ) 0, the corresponding Boltzmann factors are assigned a value of unity, or, τ10 ) τ20 ) τ00 ) 1. As a consequence, in the case of a binary (pseudo-ternary) mixture, eq 18 reduces to the following equation:
QE ) (θ0 + θ1 + θ2)zN0/2(θ0 + θ1e-β11 + θ2e-β12)zq1N1/2(θ0 + θ1e-β12 + θ2e-β22)zq2N2/2 (30)
The binary parameter ξ12 in eq 39 is expected to have values close to unity. For the applications in this work, we will also consider, as done previously,14 that V1* ) V2* ) V*. In addition, the reduced quantities are defined as follows:
T˜ ij )
(31)
where
θr ) 1 - θ0 )
zq1N1 + zq2N2 zqN q/r ) ) zq1N1 + zq2N2 + zN0 zNq (q/r) + V˜ - 1 (32)
Thus, eq 30 simplifies to the equation -β11
QE ) (θ0 + θ1e
)
(θ0 +
θ1e-β12 + θ2e-β22)zq2N2/2 ) (θ0 + θ1τ11 + θ2τ12)zq1N1/2(θ0 + θ1τ12 + θ2τ22)zq2N2/2 ) (1 - θrH1)zq1N1/2(1 - θrH2)zq2N2/2
P Pi*
(41)
(
Q(N,P,T) ) QRQE exp -
PV RT
)
(42)
Combining this equation with eqs 24 and 33, we are able to derive the equations for all basic thermodynamic quantities of our interest. The equation of state of our system may be obtained by minimizing the free energy, with respect to N0, or to the reduced volume, V˜ , or
(∂G∂v˜ )
T,P,N
(∂ ln∂v˜Q)
) - kT
)0
T,P,N
(43)
which leads to the following equation of state for a binary mixture:
-β12 zq1N1/2
+ θ2e
P ˜i )
For the development of the equation-of-state model, it is convenient to work in the N, P, T ensemble. In this ensemble, the partition function of our system becomes
However, the surface fractions in our pseudo-ternary system obey the conservation equation:
θ0 + θ1 + θ2 ) θ0 + θr ) 1
T , T*ij
P ˜ T˜
+ ln(1 - F˜ ) -
()
(
l
)
z q F˜ - ln 1 - F˜ + F˜ r 2 r 2 Hi z ) 0 (44) θ r θi 2 i)1 1 - θrHi
∑
(33)
where The chemical potential of component 1 is obtained by
H1 ) 1 - Θ1τ11 - Θ2τ12 H2 ) 1 - Θ1τ12 - Θ2τ22
( )
µ1 ∂G ) kT ∂N1
(34)
and
Θ 1 ) 1 - Θ2 )
zq1N1 zq1N1 + zq2N2
(36)
while the average per segment interaction energies and the corresponding scaling temperatures are given by
z *ij ) - ij ) RT*ij 2
(37)
The following combining and mixing rules will be adopted for our binary mixture:
* )
∑i ∑j ΘiΘj*ij ) RT* ) P*V*
T,P,V˜ ,N2
( ) ∂ ln Q ∂ N1
(45)
T,P,V˜ ,N2
Combining eqs 42 and 45, we obtain
(35)
To proceed, some further definitions are needed. The segment fractions are defined as follows:
r1N1 r1N1 φ1 ) 1 - φ2 ) ) r1N1 + r2N2 rN
)-
(38)
*12 ) ξ12x*11*22
(39)
l ) x1l1 + x2l2
(40)
( ) () () [ ][
µ1 φ1 l - r1 + ln F˜ + r1(v˜ - 1) ln(1 - F˜ ) ) ln RT ω1r1 r q1 z q P ˜ V˜ r V˜ - 1 + ln 1 - F˜ + F˜ + r1 2 1 r1 r T˜ Θ1 - φ1 Θ1(τ11 - τ12) - θ0 H1 zq1 Θ2 ln(1 - θrH1) + θ2 + 2 1 - θ rH 1 Θ1 - φ1 Θ2(τ12 - τ22) - θ0 H2 Θ1 (46) 1 - θrH2
{[
(
()]
)]}
The terms in the first line of this equation come from QR, whereas all terms in the second line come from QE. The corresponding expression for the chemical potential of component 2 is obtained from eq 46 by interchanging the subscripts 1 and 2. An expression for the internal or potential energy of the system can be obtained as indicated by eq 22. In our case, we obtain
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[ (
) ( ( )
)
τ11* τ22* E 11 22 + Θ22 + ) qθr Θ12 N 1 - θrH1 1 - θrH2
Γ12 )
)]
τ12 τ12 Θ1Θ2 + * 1 - θrH1 1 - θrH2 12
[ (
) qθr Θ12
τ11*11 τ22*22 + Θ22 + 1 - θ rH 1 1 - θrH2 Γ12 + Γ21 *12 2Θ1Θ2 2
Γ12 )
()
[
()]
[
]
1 - τ11 q z z ln 1 - F˜ + F˜ - θr2 ) 0 (48) 2 r 2 1 - θr(1 - τ11) The equation for the chemical potential is (cf. eq 46)
() [
()
()]
q P ˜ V˜ z q r V˜ - 1 + ln 1 - F˜ + F˜ + r 2 r r T˜ zq ln[1 - θr(1 - τ11)] (49) 2 and the equation for the potential energy is (cf. eq 47)
-
[
]
θrτ11 E * )q N 1 - θr(1 - τ11)
(50)
The heat of vaporization is, then, given by
∆Hvap ) (E + PV)vapor - (E + PV)liquid ) ˜ V˜ ] - [-q*θrΓ11 + r*P ˜ V˜ ]liquid (51) [-q*θrΓ11 + r*P where the nonrandomness factor for the pure fluid is given by
Γ11 )
τ11 1 - θr(1 - τ11)
(23a)
) Γ21
1 ) Γ21 1 - θ1θ2[1 - (τ11τ22/τ122)] (Monte Carlo, Yan et al.6,15) (54)
Γ12 )
τ12 θ1 τ11 + θ2 τ22 + 2θ1θ2τ12 2
2
) Γ21
(ref 9)
(55)
To make the comparison on an equal basis, we also formulate the energetic factor QE of the partition function in ref 9 by counting intermolecular contacts rather than contact points, as we do in the present work. In other words, the appropriate central equation for ref 9, which corresponds to eq 55, is
QE ) (
∑ ∑θkθlτkl)zqN/2
(ref 9)
(56)
k)1 l)1
1 µ ) ln - l + ln F˜ + r(V˜ - 1) ln(1 - F˜ ) RT ωr
][
1 + x1 - 4θ1θ2[1 - (τ11τ22/τ122)]
(47)
From the aforementioned equations, we may easily obtain the corresponding equations for a pure fluid (pseudo-binary system). Thus, the equation of state for a pure fluid is (cf. eq 44)
l P ˜ + ln(1 - F˜ ) - F˜ T˜ r
2
(this work)
(Quasi-Chemical3,9,14) (53)
) ]
(
Γ12 )
τ12 τ12 , Γ21 ) θ1τ11 + θ2τ12 θ1τ12 + θ2τ22
(52)
In the case of hydrogen-bonded systems, the correction terms that must be added to the aforementioned equations are identical to those used in the NRHB model15 and, thus, are not reproduced here. The aforementioned formalism is sufficient for calculating the basic thermodynamic quantities of pure fluids and binary mixtures. Applications 1. Comparison of Nonrandomness Factors. Before making any calculations for basic thermodynamic quantities, it would be useful to compare the estimated nonrandomness factors (Γmi) by eq 23 and by the corresponding equations from the literature. For convenience, we reproduce here all compared equations for Γ12 for a binary mixture:
Figure 1a compares the estimated nonrandomness factors Γ12 for an equi-areal or equi-surface (θ1 ) θ2) binary mixture with the aforementioned four equations, which represent four different approaches to nonrandomness. As observed, the calculations by the present approach and that of ref 9 are identical to the Quasi-Chemical approach.3,9,14 In fact, in this particular case (equi-surface mixture, 11 ) 22) the nonrandomness factors, as calculated by eqs 23a and 55, meet all consistency and conservation requirements. This holds true not only for equimolar spherical molecules of the same diameter but also for all mixtures where the surface fractions of the two components are equal, or θ1 ) θ2. As already noted, however, the calculations of eqs 23 and 55 should not be taken literally as nonrandomness factors. Figure 1b compares the same factors for an equisurface mixture, but now 11/(kT) ) -0.5, 22/(kT) ) -1.5, and 12 varies between these two values. As observed, all four approaches are differentiated in this case, as they generally do for different compositions and/or interaction energies. In Figure 1b, we also observe the difference of the estimated Γ12 and Γ21 by the present approach (eq 23a) not only between themselves but also with the estimations by the other approaches. It is worth noting, however, that their average, 〈Γ12〉 ) (Γ12 + Γ21)/2, is similar to the estimations of the other three approaches, especially to the Quasi-Chemical approach. This is important for the calculation of basic thermodynamic quantities (cf. eq 47). 2. Application to Pure Fluids and Mixtures. We now will apply the new equation-of-state formalism for the description of thermodynamic properties of systems of real fluids. We will start with the volumetric behavior and the vapor-liquid equilibrium of pure fluids. For these calculations, we need the characteristic or scaling constants of pure fluids. Each fluid is fully characterized by three scaling constants: the interaction energy, *; the segmental volume, V*; and the specific hardcore volume, V*sp. Other than its molar mass, we will need its surface-to-volume ratio (s ) q/r), which is a geometric characteristic of each fluid, obtained from the UNIFAC tabulations.19 Following ref 14, we will consider in this work that the van der Waals or hard core volume per segment, V*, is the same for all fluids and equal to 9.75 cm3/mol. We will also consider
Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 7269
calculations with the present model. As observed, the model describes adequately the system over a large range of pressures but fails at the critical region, as do, essentially, all mean field equation-of-state models. In a similar manner, Figures 4 and 5 compare the experimental and calculated vapor pressures and orthobaric densities, respectively, for water. The agreement is rather very good. The hydrogen-bonding parameters used in the calculations of this work are reported in Table 2. The better description of the critical region of water is due to the incorporation of hydrogen bonding. However, this simple hydrogen-bonding picture is not sufficient to describe all anomalies known for water, especially the maximum in density near 277 K. Figure 6 compares the experimental24 and calculated PVT data for linear polyethylene. As observed, the agreement is rather very good. Of significant interest are the calculations of the nonrandomness factors that are reported in Figure 7. Note that eq 23, when applied to a pure fluid, gives
Γ11 )
τ11
, Γ01 )
1 - θr(1 - τ11)
1 , 1 - θr(1 - τ11) Γ00 ) Γ10 ) 1 (23b)
This equation indicates that the present approach cannot calculate the nonrandomness factors around a central empty site, because its interaction energy with any other site is equal to zero. Yet, it can calculate the nonrandom distribution of the empty site around a central molecular segment 1. In consistent expressions for local compositions, we would have Γ10 ) Γ01. Considering this equality to hold true in our case, and using the conservation equation (cf. eq 4), Figure 1. (a) Comparison of the calculated nonrandomness factor Γ12 in an equimolar binary mixture of spherical fluids. The lines are calculations with the indicated equations. The calculations were done with 11/(kT) ) 22/(kT) ) -1. (b) Comparison of the calculated nonrandomness factor Γ12 in an equimolar binary mixture of spherical molecules of the same diameter. The lines are calculations with the indicated equations. The calculations were performed with 11/(kT) ) -0.5 and 22/(kT) ) -1.5.
that the interaction energy * is a function of temperature14 and is given by
* ) *h + (T - 298.15)*s
(57)
In view of the temperature dependence indicated by eq 57, * in eq 51 and in the first terms in the parentheses should be replaced by * ) * h - 298.15* s. In most cases, this is a negligibly small correction. The specific hard-core volume will also be considered to be a weak function of temperature, or
V*sp ) V*sp,0 + (T - 298.15)V*sp,1
(58)
No optimization was made for the determination of V*sp,1. Its value was taken to be, more or less, characteristic for each homologous series. Table 1 reports the scaling constants of representative pure fluids of small and high molecular weight. The scaling constants in Table 1a were obtained by simultaneously correlating experimental22,23 vapor pressures, orthobaric densities, and heats of vaporization with a least-squares procedure. The scaling constants for polymers were obtained by a least-squares correlation of PVT data24 for pure polymers. Figures 2 and 3 show the experimental vapor pressures and orthobaric densities, respectively, for n-hexane, along with the
θ0Γ00 + θrΓ10 ) 1
(59)
we obtain
Γ00 )
θ0 - θr(1 - τ11) θ0(θ0 + θrτ11)
(60)
The lines for Γ00 in Figure 7 were obtained using eq 60. As observed in Figure 7, of all of the three nonrandomness factors, the most sensitive is Γ00, which indicates a tendency or preference of segregation of empty sites. In other words, a remarkable nonrandom distribution of free volume is calculated, even for this nonpolar polymer. Both temperature and pressure have a significant impact on Γ00. It is worth noting that the calculations in this figure are very similar to the corresponding calculations in ref 9. Let us now apply the model to mixtures. Figure 8 compares the experimental25 and calculated vapor-liquid equilibria (VLE) for the methane (1) + ethane (2) system at two temperatures. One single value for the interaction parameter ξ was used at both temperatures. The agreement is rather good. An analogous comparison is made in Figure 9, where the experimental26 and the calculated VLE for the carbon dioxide (1) + dichloromethane (2) system at 328.15 K are compared. As observed, the agreement is again rather satisfactory. A very similar picture is obtained for the carbon dioxide (1) + acetone (2) system at 328.15 K27 when ξ ) 1.032 is used for calculations. Tables 3-5 describe the VLE of binary systems, whereas Figures 10 and 11 illustrate the same representative results. As observed, the present model is at least as good as the NRHB
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Table 1. Characteristic Constants of Pure Fluids and Pure Fluids/Polymers fluid/polymer N2 O2 CO CO2 methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-decane n-dodecane n-hexadecane n-octadecane n-pentacosane n-octacosane n-triacontane n-dotriacontane n-hexatriacontane 2,2,4-trimethyl pentane cyclohexane benzene toluene acetone n-butyl acetate diethyl ether CCl4 CHCl3 CH2Cl2 methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol 1-octanol 1-decanol diethylamine n-butylamine NH3 water
*h ) RT* (J mol-1) 813 1092 1049 3367 1833 2915 3404 3656 3818 3940 4029 4098 4204 4260 4338 4364 4444 4416 4370 4231 4302 3799 4558 5152 5065 4908 4491 3967 4676 4970 5155 4279 4172 4244 4309 4350 4380 4461 4483 4118 4364 4334 6622
V*sp,0 (cm3 g-1)
*s (J mol-1 K-1) (a) Characteristic Constants of Pure -3.420 -5.570 -2.660 -5.724 -1.773 -1.423 -0.630 0.235 0.954 1.146 1.169 1.365 1.459 1.661 1.835 1.864 1.917 2.094 2.286 2.605 2.520 1.919 0.719 -0.693 -0.334 -1.492 0.434 -0.523 0.627 -0.638 -1.717 0.260 -0.125 0.504 0.820 1.218 1.414 1.486 1.521 0.624 0.630 -2.632 -6.954
V*sp,1 (cm3 g-1 K-1)
s ) q/ra
0.0002 0.0002 0.0002 0.0005 0.0002 0.0000 -0.0001 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0003 0.000 0.0002 0.0002 0.0002 -0.0001 -0.0001 0.000 0.0001 0.0002 0.0001 0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 0.0007 0.0001
0.931 0.945 0.935 0.909 0.961 0.941 0.903 0.881 0.867 0.857 0.850 0.844 0.836 0.83 0.823 0.821 0.815 0.814 0.813 0.812 0.811 0.857 0.800 0.753 0.757 0.908 0.869 0.888 0.858 0.840 0.881 0.941 0.903 0.881 0.867 0.857 0.850 0.839 0.833 0.896 0.874 1.039 0.861
Fluidsb 1.151 0.841 1.182 0.793 2.128 1.562 1.398 1.315 1.266 1.233 1.210 1.936 1.163 1.147 1.125 1.117 1.100 1.092 1.079 1.074 1.070 1.161 1.116 1.034 1.039 1.102 0.978 1.122 0.546 0.589 0.662 1.088 1.083 1.071 1.070 1.067 1.061 1.062 1.063 1.183 1.154 1.415 0.971
(b) Characteristic Constants of Pure Fluids/Polymersc polyethylene linear branched polypropylene polyisobutylene polystyrene poly(dimethyl siloxane) poly(vinyl chloride) polyacrylonitrile poly(methyl methacrylate) poly(-caprolactone) poly(vinyl acetate) poly(ethylene oxide) poly(propylene oxide) poly(phenylene ether)
5557 5538 5117 6099 5903 4040 8681 15467 7603 6774 5814 7249 5510 6090
2.335 2.613 3.727 4.177 1.695 3.753 -6.189 -12.325 -0.107 -0.727 1.648 -4.151 1.248 0.509
1.095 1.096 1.077 1.032 0.896 0.912 0.691 0.844 0.796 0.875 0.783 0.850 0.926 0.833
0.800 0.800 0.799 0.839 0.667 0.744 0.780 0.887 0.843 0.818 0.825 0.829 0.820 0.704
a From UNIFAC19 and NRHB.15 b For hydrogen-bonded fluids, the parameters for the hydrogen-bonding interactions (Table 2) are needed also. c For all polymers, we have V* sp) V* sp,0 + 0.0001(T - 298.15) - 0.0001P (where P is given in MPa).
model, which accounts for nonrandomness through the QuasiChemical approximation. This holds true for pure fluids as well. Discussion and Conclusions References 9 and 15, along with the present work, may be considered as a set of three alternative equation-of-state approaches that attempt to incorporate nonrandomness in their formalism. In ref 15, the Quasi-Chemical approach3 was used
to account for the nonrandom distribution of free volume only (the molecular species were assumed to be distributed randomly). In ref 9, a new theoretically sound approach was presented which could account for the nonrandom distribution of all species, including the empty sites or the free-volume. In contrast with the Quasi-Chemical approach,3 the approach of the present work leads to simple analytical equations for the nonrandomness factors in systems of any number of compo-
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Figure 2. Experimental22 (symbols) and calculated (line) vapor pressures of n-hexane.
Figure 3. Experimental23 (symbols) and calculated (lines) orthobaric densities of n-hexane.
Figure 4. Experimental22 (symbols) and calculated (line) vapor pressures of water.
nents. From the comparison that was made, the performance of the present model is at least as good as that of the NRHB model,15,26,30 which incorporates the Quasi-Chemical approach for nonrandomness. The approach of the present work accounts for the nonrandom distribution of all molecular species but only indirectly for the distribution of the free volume. The key equation is eq 18 for the energetic factor of the partition function that is dependent on the intermolecular interactions. The equations for the nonrandomness factors (eqs 23) are derived from eq 18 and remain analytical in systems of any number of components. In the limit of infinite coordination number z, the classical Flory-Huggings expression should replace eq 24 and
Figure 5. Experimental23 (symbols) and calculated (lines) orthobaric densities of water.
Figure 6. Experimental24 (symbols) and calculated (lines) specific volumes of linear polyethylene.
Figure 7. Estimated nonrandomness factors Γ00, Γ11, and Γ10 in polyethylene, as a function of temperature and at three pressures. Table 2. Hydrogen-Bonding Parameters interaction
EH (J mol-1)
SH (J K-1 mol-1)
OH ‚ ‚ ‚ OH water ‚ ‚ ‚ water NH ‚ ‚ ‚ NH
-25100 -15000 -12277
-26.5 -15.2 -26.5
that would lead to a Sanchez-Lacombe lattice-fluid model12 of mixtures that accounts for nonrandomness. A similar equation to eq 18 was already available in the literature,10,11 although serving an entirely different purpose. Our approach resides on a much broader spectrum of
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Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 Table 3. Comparison of Correlations of Vapor-Liquid Equilibria (VLE) of Methane with Various n-Alkanes NRHB Model
Figure 8. Experimental25 (symbols) and calculated (line) vapor pressures of the mixture of methane (1) + ethane (2) at 172.04 K (circles) and at 199.92 K (squares). A value of ξ ) 1.008 was used in the calculations at both temperatures.
Present Model
% AADa in P
% AADa in P
temperature, T (K)
ξ
172.04 199.92
1.014 1.017
Methane-Ethaneb 0.8 1.003 0.9 1.014
1.2 1.4
310.9 338.7 377.6
1.160 1.166 1.180
Methane-n-Hexanec 3.1 1.046 2.7 1.058 4.3 1.087
1.3 1.1 1.9
323.2 373.2
Methane-n-Dodecanec 1.226 3.0 1.038 1.258 2.4 1.075
0.4 0.3
total
2.5
1.1
ξ
a
Average absolute deviation. b High-pressure VLE data.25,31 pressure solubility data.32
c
High-
Table 4. Comparison of Correlations of Vapor-Liquid Equilibria (VLE) of CO2 with n-Alkanesa NRHB Model temperature range (K)
Figure 9. Experimental26 (symbols) and calculated (line) vapor pressures of the mixture of carbon dioxide (1) + dichloromethane (2) at 328.15 K. ξ ) 0.967 was used for the calculations.
% ξ
AADb in P
Present Model AADb
% in Υ1
CO2-Ethane 2.8
ξ
% AAD % AADb in P in Υ1
230.0-270.0 0.903
1.2
0.892
2.0
3.3
230.0-270.0 0.912
CO2-n-Propane 3.7 3.0 0.880
6.0
3.1
319.3-344.3 0.922
CO2-n-Butane 3.8 9.6 0.931
2.0
5.1
310.2-363.2 0.965
CO2-n-Pentane 3.0 8.9 0.906
5.3
4.2
total
2.9
3.8
3.9
a
High-pressure VLE
6.1
data.28,29,33,34 b
Average absolute deviation.
Table 5. Comparison of Correlations of Vapor-Liquid Equilibria (VLE) of Binary Mixtures Containing One Self-Associating Fluid NRHB Model temperature range (K)
ξ
Present Model
% AADa % AADa in P in Υ1
ξ
% AADa % AADa in P in Υ1
313.2-328.2 1.036
2.5
CO2-Ethanolb 1.0 0.960
4.9
1.1
313.4-333.4 1.044
CO2-1-Propanolb 4.1 0.9 0.961
4.9
0.9
314.8-337.4 1.059
CO2-1-Butanolb 7.6 0.6 0.958
7.0
0.7
314.7-337.5 1.079
CO2-1-Pentanolb 9.2 0.5 0.948
6.9
0.6
Figure 10. Vapor-liquid equilibrium (VLE) for the CO2 (1)-ethane (2) system at 230 and 270 K.
298.15
1.046
Ethane-Methanolc 3.5 1.5 0.988
2.8
1.3
ideas3-11,14-21,39 and its novelty is the derivation from the sum over states (cf. eq 1) of eq 18 as an energetic or thermal partition function of multicomponent systems and, from it, the derivation of eqs 23 for the local composition or nonrandomness factors. Equations similar to eqs 23 were obtained by Tao39 who adopted the ad-hoc arguments that lead to the Wilson-type expressions.7 In essence, eqs 23 are also of the Wilson-type, and the procedure for the derivation of eq 18 is essentially a multifluid approach (a two-fluid approach for a binary mixture), because the summation was independent for each type of molecules (cf. eqs 7 and 8). It is this multifluid picture that leads to the Wilsontype4,7 eqs 23 for the nonrandomness factors, with different expressions for Γij and Γji when j * i. Thus, the Γ parameters
313.4-333.4 1.038
Ethane-Ethanolb 8.7 1.9 0.988
6.3
1.7
313.4-333.4 1.053
Ethane-1-Propanolb 5.6 2.7 0.997
3.3
0.3
total
5.9
5.1
a
1.3
Average absolute deviation. b High-pressure VLE data.26,35-37 pressure VLE and LLE data.38
0.9 c
High-
neither in the present approach nor in that of ref 9 should be taken literally as consistent nonrandomness factors. As previously noted repeatedly, this in no way diminishes the applicability of the formalism in both approaches. Disregarding, however, the aforementioned inconsistency, the approach of the present work is simpler and, at least, as good in performance as the approach in ref 9.
Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 7273
scripts, that had not been previously noted/corrected. The corrected version of the paper was published on the Web September 7, 2006. Literature Cited
Figure 11. Vapor-liquid equilibrium (VLE) for the CO2 (1)-propane (2) system at 270 K.
Strictly speaking, in a binary mixture, when N12 is given, the N11 and N22 are also given (cf. conservation eqs 4). One might then perform the summation in eq 7, with respect to the less abundant component in the system only. If component 2 were the less abundant, such a summation would lead, in the zeroth approximation (Γ ) 1) and after normalization, to the following equation for QE:
QE ) τ11z(N1-N2)/2
(
)
τ122 + τ11τ22 2
zN2/2
(61)
As observed, this equation is asymmetric and essentially offers no advantage over eq 18. In fact, for a real pure fluid, replacing index 2 by index 0 in eq 61 and applying the procedure of eq 22, we would obtain an expression for Γ11 that would give results comparable to those obtained by eqs 23. Thus, although approximate, eq 18 is the preferable equation for QE. As already mentioned, the present approach cannot calculate the nonrandom distribution around an empty site. However, by adopting the consistency and conservation equations, the new approach is enabled to make these calculations and the results are similar to those of ref 9. Again, a significant degree of nonrandom distribution of free volume is found, even in systems of nonpolar fluids such as the polyethylene melt, and a significant influence of temperature and pressure variations. The strength of the present approach is its simplicity. This is essential when developing group-contribution formalisms. Real fluids may consist of a large number of groups. As it was shown recently,40 the COSMOtherm or COSMO-RS model,41 apart from its quantum-mechanics component, is essentially a groupcontribution model where nonrandomness is taken into consideration via Guggenheim’s Quasi-Chemical approach.3 In this model, each fluid is decomposed into a very large number of “functional” groups. In such cases, it is of great significance to have simple analytical expressions that account for nonrandomness, not only for the speed of calculations but also for obtaining analytical expressions for higher-order derivatives of free energy (e.g., spinodal lines, speed of sound, heat capacity, etc.). The present model offers these features and advantages. Note Added after ASAP Publication. Equation 46 in the version of this paper that was published on the Web August 25, 2006 had an error in it involving one of the operators. The equation has been revised, and the corrected version of this paper was posted to the Web August 29, 2006. In addition, the version of this paper that was posted August 29, 2006 also required numerous changes, primarily involving symbols and super-
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ReceiVed for reView April 20, 2006 ReVised manuscript receiVed July 24, 2006 Accepted July 31, 2006 IE060490P
