4078
J . Phys. Chem. 1987, 91, 4078-4087
Decorated Lattice Gas Model for Supercritical Solubility G . C. Nielson and J. M. H. Levelt Sengers* Thermophysics Division, Center for Chemical Engineering, National Bureau of Standards, Gaithersburg, Maryland 20899 (Received: November 7, 1986)
We describe a self-consistent nonclassical model for supercritical solubility enhancement near the solvent's critical point. A Widom-Wheeler-Mermin decorated lattice gas transformation is used to obtain properties of a dilute supercritical solution
from known properties of the pure solvent. Phase equilibria between the solution and the additional solid or liquid phase are described. A semiquantitative representation of solubility data for three different solutesolvent pairs at several temperatures has been obtained. Infinite-dilution partial molal volumes based on parameter sets obtained from fits to solubility data do not agree very well with experiment.
I. Introduction Recent years have seen a renewal of interest in the phenomenon of supercritical solubility. Supercritical solubility is of interest from a theoretical standpoint, and the phenomenon has a significant practical value in industrial processes. In this paper we present a detailed description of a method for modeling the thermodynamic properties of these systems. A very brief description of this work has been given in a previous publication.' Supercritical solubility refers to the solubility of a liquid or a solid in a solvent that is in a supercritical state. Supercritical solvents have the advantage that solvent power can be modified by a change of pressure which, in certain applications, can be done more efficiently and with less damage to the solute than by boiling off an ordinary solvent. Our work refers to the case where the solubility is low and the solvent is near its critical point. When solute is added, a mixture critical line develops. When the solution is saturated, the critical line terminates at a critical end point which in the case of low solubility lies very close to the solvent's critical point. As discussed, for example, in ref 1 and 2, a dramatic increase of solubility is observed when the pressure P is increased. The solubility enhancement with pressure is given by (dx/dP),,,, where x is the mole fraction of the solute and u denotes that excess solute is present in an additional liquid or solid phase. The enhancement is governed by the relation ( a x / d P ) , , = ( V , - V2)/{(1 - x). (d2C/dx2),,} where V, is the molar volume of the solute in the additional phase, assumed, for simplicity, to be pure solute; V2 is the partial molar volume of the solute in the solution, and ( d 2 G / d ~ 2 ) is p Tthe second derivative of the Gibbs function which equals zero on the critical line. The enhancement therefore diverges at the critical end point. The amplitude of the enhancement can be very large in dilute solutions, because V2 diverges at the solvent's critical point,',3 which is close to the critical end point. Classical thermodynamic functions for mixtures such as proposed by van der Waals, generalized by engineers for the past century and systematized by van Konynenburg and S ~ o t tare ,~ capable of qualitatively accounting for the phenomenon of supercritical solubility enhancement in the case that the additional phase is a liquid. Since no known equations of state generate the solid phase with the fluid phase, the supercritical enhancement in the case of the presence of a solid has to be handled by using a separate formulation for the solid and then equating chemical potentials. Since the nonclassical critical isotherm x-P at T,,, the critical end point temperature, or the isobar x-T at P,,,the critical end point pressure, is much flatter than the classical one, it appears desirable to improve the description by using a nonclassical equation of state. (1) Levelt Sengers, J. M. H.; Morrison, G.; Nielson, G. C.; Chang, R. F.; Everhart, C. M. Int. J . Thermophys. 1986, 7 , 231. (2) Gitterman, M.; Procaccia, I. J . Chem. Phys. 1983, 78, 2648. (3) Rozen, A. M. Russ. J . Phys. Chem. (Engl. Transl.) 1976, 50, 837. (4) van Konynenburg, P. H.; Scott, R. L. Philos. Trans. R . SOC.London 1980, 298, 495.
Approximate nonclassical expressions have been suggested in the Russian literature3 and by Gitterman and collaborators2 that are analogous to those proposed in the 1950s for pure fluids, prior to the development of the scaling laws. These expressions impose the asymptotic form on the critical isochore and the critical isotherm and interpolate in between. Although such approximations yield correct answers in certain applications, they lead to nonanalyticities in the one-phase region and, if applied to mixtures, they miss known features of the critical-line b e h a v i ~ r . Another ~ suggestion has been to combine an accurate nonclassical description of the pure solvent with the classical idea of corresponding states for mixtures of constant composition.6 It can be shown that such an approach leads to undesirable internal inconsistencies at the solvent critical point.' Thus, there is no choice but to use an internally consistent fully nonclassical formulation. One such formulation is available in the Leung-Griffiths model and its variant^.*^^ The drawback of this model is a proliferation of constants and the need for a continuous critical line connecting the critical points of the two components. An alternative is the decorated lattice gas, introduced by Widom and co-workers10and used and reviewed by Wheeler and co-workers.11,12 The lattice gas model is closely related to the king model (Figure la). In a lattice gas, space is divided up into cells as illustrated in Figure l b for the square-planar lattice. The scale of the size of the cells is about the same as the scale of intermolecular distances. Molecules are free to move throughout the entire volume of the system; the intermolecular potential is determined by the manner in which the molecules occupy the cells. In a decorated lattice gas (Figure lc,d) molecules are allowed to occupy two distinct types of cells. The decorated lattice gas models make it possible to describe many systems which cannot be described by a simple lattice gas. The lattice gas models are mathematically isomorphic to the spin Ising model. This means that the lattice gas possesses the nonclassical critical behavior characteristic of the Ising-like universality class to which fluids belong. Traditional decorated lattice gas calculations have used the isomorphism provided by the decorated lattice gas to calculate properties of the fluid system being considered from known properties of the 3-dimensional Ising model reference system. However, the points in the supercritical solution which we wish to consider correspond to Ising model states of nonzero magnetic field. Since closed-form expressions for properties of the king (5) Griffiths, R. B.; Wheeler, J. C. Phys. Reo. A 1970, 2, 1047. (6) Hastings, J. R.; Levelt Sengers, J. M. H.; Balfour, F W. J . Chem. Thermodyn. 1980, 12, 1009. (7) Levelt Sengers, J. M. H.; Chang, R. F.; Morrison, G. In Equations of State-Theory and Applications; Chao, K. C., Robinson, Jr., R. L., Eds., American Chemical Society: Washington, DC, 1986; ACS Symp. Ser. No. 300, Chapter 5 , p 110. (8) Leung, S. S.; Griffiths, R. B. Phys. Reo. A 1973, 8, 2670. (9) Chang, R. F.; Doiron, T. Int. J . Thermophys. 1983, 4 , 337. (10) Widom, B. J . Chem. Phys. 1967, 46, 3324. (11) Wheeler, J. C. Annu. Reo. Phys. Chem. 1977, 28, 411. (12) Wheeler, J. C.; Andersen, G. R. J . Chem. Phys. 1980, 73, 5780.
This article not subject to U S . Copyright. Published 1987 by the American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987 4079
Model for Supercritical Solubility
H
IOIO/-&
-J
a A
H
+J
-9
11. A Model for Supercritical Solubility In this section we describe our method of calculating properties of supercritical solutions. Since decorated lattice gas models play a fundamental role in this work, the reader who is unfamiliar with this topic may wish to refer to the review article by Wheeler.” Although the decorated lattice gas models which we use have been described before,’ we describe them here in some detail in order to clarify the description of our application of the models to supercritical solubility. The fact that the decorated lattice gas is mathematically equivalent to the spin Ising model is fundamental to our work. We therefore begin with a brief review of the Ising model and its relation to the simple lattice gas. We next describe the relationship between Mermin’s decorated lattice gas and the revised and extended scaled equation of state for fluids. We then consider the mixture. The decorated lattice gas model for the mixture and the description of the additional phase are considered. Finally, we describe details of the numerical calculation. A . The Simple Lattice Gas. Consider first the spin ’I2Ising model for magnetism. The model consists of spins arranged on the C sites of a regular lattice such as the square-planar lattice shown in Figure la. The spins are scalar variables that may take on only two values, plus or minus one, corresponding to the spin pointing up or down. The energy of a given configuration of up and down spins is given by 1314915
. . . ( a ) lsing
(b) Lattice Gas
(c) Mermin
-%a
-& - %B (d) Widom
Figure 1. (a) Sites in the spin
Ising model, (b) cells of the simple lattice gas, (c) Mermin decorated lattice gas, and (d) Widom decorated lattice gas. Energy parameters are as indicated.
E = -J(Ntt model in a nonzero magnetic field are not available, we use the revised and extended scaled equation of state which has been developed for describing near-critical pure fluids as our reference system.I3 Our model for supercritical solubility makes use of two decorated lattice gas systems. In order to make use of the revised and extended scaled equation of state of the solvent, we must reexpress this equation of state in the language of the lattice gas. Mermin’s decorated lattice gas model for pure fluids14 is, to leading order in the scaling fields, equivalent to the revised and extended scaled equation of state for fluids. W e therefore suppose that the pure solvent can be described by Mermin’s decorated lattice gas. The supercritical solution is described by a second decorated lattice gas system, a slightly modified version of the model of Bartis and Hall for two-component mixture^.'^ Since the two decorated lattice gas systems are mathematically isomorphic, properties of the mixture may be derived from those of the pure solvent. This mapping of properties of a dilute solution onto those of the pure solvent can be considered to be the nonclassical analogue of corresponding states. The decorated lattice gas can describe the supercritical mixture as well as its separation into a vapor and a liquid phase. It has, however, no capability for simultaneously describing the additional solid phase when the solute is present in excess. We introduce this solid phase by an artifact. We describe the variation with pressure and temperature of the chemical potential of the additional phase. The requirement that the chemical potential of the solute in solution equal the chemical potential of the additional phase then places a constraint on the mixture described by the decorated lattice gas and leads to the enhancement. An outline of the paper is as follows. The next section contains a description of the decorated lattice gas models which we use together with many details of the calculations. In section I11 we show that our model can be used to represent experimental data for several systems. We make some concluding remarks in section IV. Appendix A contains the details of the mapping of the properties of the solvent onto those of the solutions. Appendix B gives the algebraic expressions for the lattice gas equivalents of two parameters in the fluid model. (13) Levelt Sengers, J. M. H.; Sengers, J. V. In Perspectives in Statistical Physics; Raveche, H.J. Ed.; Wiley: Chichester, U.K., 1981; Chapter 4, p 103. Sengers, J. V.;Levelt Sengers, J. M. H. Annu. Rev. Phys. Chem. 1986, 37, 189. (14) Mermin, N. D. Phys. Rev. Lett. 1971, 26, 169, 957. (15) Bartis, J. D.; Hall, C. K. Physica 1975, 78, 1.
+ Nil - Ntl) - H(Nt - NJ)
(1)
where J is a coupling constant and H is the magnetic field multiplied by the magnetic moment per spin, and where N f t ,Ntl, and Nil denote the number of nearest-neighbor (adjacent) pairs of spins that are up-up, up-down, and down-down, respectively. The partition function is given by
z, = E’exp(-PE)
(2)
where C’ denotes summation over every assignment of up or down to each spin, and 8 = l / k T where k is Boltzmann’s constant and T is the absolute temperature. The free energy f of the system is related to ZI by
(3) The Ising model can be reinterpreted as a model for a pure fluid by centering cells of volume u,, on each site of the underlying lattice, as shown in Figure l b for the square-planar lattice. Molecules are free to move throughout the entire volume of the system; the cells simply provide a coordinate system for defining the intermolecular potential. A hard-core intermolecular repulsion insures that each cell is occupied by at most one molecule. Molecules which occupy adjacent cells are assigned an attractive energy, -e. Molecules which occupy nonadjacent cells do not interact. It can be shown]’ that the grand partition function for the simple lattice gas may be written as
E = C’zNexp(PtNI1)
(4)
where E’denotes summation over every assignment of filled or empty to each cell, N is the number of filled cells, N,, denotes the number of adjacent pairs of filled cells, and z is the dimensionless activity
Here M is the molecular mass and is the chemical potential. To obtain the correspondence between the simple lattice gas and the Ising model, identify filled cells of the lattice gas with down spins of the Ising model and identify empty cells of the lattice gas with up spins of the Ising model. Next, note the lattice identities q N = 2N11 + No1
(6a)
q(C - N ) = 2Noo + NO’
(6b)
4080
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987
Here Noodenotes the number of adjacent pairs of empty cells and No, the number of adjacent pairs of cells in which one cell is empty
and one cell is filled. q is the coordination number of the lattice. Following Wheeler” we define the function X(X,{,C) = C’XN{NO]
(7)
For the two systems, X and f a r e chosen in such a way that it is possible to write E and ZI in terms of X. In the case of the magnetic system we rewrite Z , as 21= exp(PCH + qCPJ/2)NXJ9C)
@a)
X = exp(-2PH)
(8b)
f = exp(-2PJ)
(8c)
where
Nielson and Levelt Sengers The equations for X and { provide the means of mapping the decorated lattice gas onto the equivalent points in the Ising model or simple lattice gas. The interaction energies t and 4 are not necessarily independent quantities, which can be seen as follows. Values for fc, the value of {at the Curie point, are known for several 3-dimensional Ising systems.16 It is also known that along the coexistence curve X = 1. This property follows directly from eq 8b and the fact that the two-phase region is the fluid analogue of Ising model states of zero magnetic field. Equations 13b and 13c provide two relations between zA,&,and vA at the fluid critical point. Mulholland and Rehr” rearrange (1 3b) and (1 3c) so that zc, the value of zA at the critical point, is the independent parameter. Starting with a given value of z,, v,, the value of vA at the critical point, can be obtained from
In the case of the simple lattice gas we have
E
= X(X,f,C)
(9a)
where
E is then obtained from X = z exp(qpc/2)
f
= exp(-Pe/2)
(9b) (9c)
Thermodynamic points in the two systems which have the same values of X and f a r e equivalent. If we know X for one system, we can calculate it for the other. The pressure P is related to E in the thermodynamic limit by P 1 1 - = lim - In E = lim - In (10) kT v-m V c-- cu, B. Mermin’s Decorated Lattice Gas. The simple lattice gas model has some significant limitations. For example, it predicts a symmetry about the line p = p,: the sum of the densities of coexisting liquid and vapor phases is a constant. This symmetry follows directly from the symmetry of the king model with respect to the magnetic field. Since real fluids do not have this symmetry, it is desirable to use a model which does not possess this symmetry either. Mermin’s decorated lattice gas14 has the asymmetry of real fluids. In the decorated lattice gas (Figure IC) additional secondary cells are placed on the center of the line segments that connect centers of adjacent primary cells of the simple lattice gas. As was the case in the simple lattice gas, each cell can contain at most one molecule. Molecules which occupy adjacent primary cells contribute an energy - E , Molecules which occupy an adjacent pair consisting of a primary and secondary cell contribute an energy -9. Note that each secondary cell has two adjacent primary cells associated with it. If a secondary cell is occupied and both of the adjacent primary cells are occupied, an energy -29 is contributed. There is no interaction between nonadjacent cells, and secondary cells do not interact with one another. These latter restrictions are necessary in order to retain the isomorphism to the Ising model. The grand partition function of this system is
+
E(zA,T,C) = E’ZANtN’’(1 ZA)”(1
+ zAqA)No’(l+ ZAvA’)”’’
(11) where zA is the activity (of component A, in anticipation of introducing a second component, eq 34) and = vA = eBd (12)
1 &c
=
z,q1
+ z, + zcv,2)
Once vc and tCare known, t and 9 can be calculated and values of & and vA (eq 12) can be determined for any given temperature. Mulholland and Rehr17 have investigated the variation of the coexistence curve of Mermin’s model as a function of z,. Their work allows us to restrict z, to values which yield physically reasonable results. We have now described a decorated lattice gas model for a pure fluid and considered some of its properties. Equations 13 together with analogous equations which we will present for a mixture (eq 34 below) provide the means of mapping mixture properties onto those of the pure solvent. Before describing the mixture, we consider the revised and extended scaled equation of state and its relationship to Mermin’s decorated lattice gas. C. The Scaled Equation of State. The most accurate description of the critical-region thermodynamics of fluids is that of revised and extended scaling. ”Scaling” refers to the fact that the asymptotic behavior is that of the 3-D Ising universality class. “Revised” means that to lowest order liquid-vapor asymmetry is taken into account. Specifically, the chemical potential and temperature are linearly combined in the definition of the weak scaling variable; the structure of the scaled part of the revised scaled thermodynamic potential is the same as that of the onecomponent decorated lattice gas if the latter’s independent scaling variables are expanded linearly around the critical point. “Extended” refers to the incorporation of the first Wegner correction to scaling. For the revised and extended scaled thermodynamic potential of one-component fluids, closed-form expressions have been obtained13in terms of Schofield’s parametric variables. Since such expressions do not exist for the decorated lattice gas, we substitute the revised and extended scaled potential for the solvent of interest for the decorated lattice gas variables in the expressions we have obtained, retaining the structure of the mapping relations from two- to one-component decorated lattice gases. For the revised and extended scaled potential the reduced variables are defined as
Equation 11 can be rewritten as
E = (1 where
+ zA)9c~2x(X,f,c>
(13a)
where Pc, T,, and pc denote the values of pressure, temperature, and density at the fluid critical point. We will also use these reduced variables to describe the mixture. In all cases the critical parameters of the pure solvent will be used in the definitions of reduced quantities. Define the reduced energy and particle density (16) Domb, C. In Phase Transitions and Critical Phenomena;Domb, C., Green, M . S., Eds.; Academic: London, 1974; p 425. (17) Mulholland, G. W.; Rehr, J. J . J . Chem. Phys. 1974, 60, 1297.
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987 4081
Model for Supercritical Solubility
U e =-
VP,’
is known as a function of from
p = -P Pc
z* = ZAo(T)eJ”fi
8 and 3 may be obtained from P via the relation dp = 8 d T +
p dp
(18)
Second derivatives of P yield the compressibility and specific heats. Now, rewrite (10) in terms of reduced variables
where w = kTC/v$,
(20)
As before, C i s the number of primary cells and uo is the volume of the primary cells in the underlying simple lattice gas. It will also be useful to rewrite the activity in terms of reduced variables
where s = Pc/kTcPc
In the revised and extended scaled equation of state, quantities are defined that are the pure-fluid analogues of the scaling variables uh = H/T and ut = 1/ T - 1/ Tc in the Ising model. Specifically Ap = p 0: u h
po(n
ut 0: A T
+ CAP;
AT =
T+1
(23)
Po( is the chemical potential at the liquid-vapor coexistence curve or the analytic continuation of this quantity into the onephase region. The “mixing parameter” c is a nonuniversal constant and-leads to the asymmetry between liquid and vapor phases. p o ( T ) is expanded in a Taylor series about the critical point: 3
Po(Q = P C + CPi(A?3’ i= I
the pi are constants which depend upon thtparticular fl_uid being described. The interrelation between AT, Ap, and P is described by a formulation of Wegner’s corrections to scaling’* in terms of the parametric equation of Sch0fie1d.I~ Schofield defined as independent parametric variables a ”distance” variable r and a “contour” variable 6 such that the anomalous critical behavior of thermodynamic properties is described by their r dependence only. The dependence on 0 is analytic. Thus, asymptotically near the critical point
(26)
Ap is calculated from the scaled equation for fluids; X and {are then calculated from (1 3b) and (1 3c). We have used two different methods to determine zA0(0. In the first, the condition that X be equal to unity on the coexistence curve, together with eq 13b, yields the lattice gas values of zA0(T) and therefore of the chemical potential at-coexistence. This procedure gives point-by-point values of zA0(7‘) and of the lattice gas equivalent of fi. We denote the latter by the symbol p, to distinguish it from its fluid counterpart. In the calculations of the supercritical solubility, we only require the values of the first derivative
but we prefer an algebraic rather than a numeric procedure. In Appendix B we derive the algebraic expression for the decorated lattice gas derivative; see eq B2. In cur first approach, we substitute (B2) for the derivative dpo/dT in the scaled equation of the fluid. This substitution leads to an inconsistency that we do not believe to be serious. Since, in the fluid model, the zero point of the chemical potential and its first derivative are arbitrary, the difference will appear only in the higher order terms. Since the modified fluid reference system still contains the correct singular scaling behavior, we expect that the singular properties of supercritical solubility will be described correctly. In the second method, howeve:, we use the fluid model to describe the variation of fi0 with T . It is necessary to subtract the contributions due to the internal modes of the solvent molecules from p o ( T ) since the decorated lattice gas transformations that we will use for mapping the mixture do not include internal degrees of freedom. In this second method, we therefore no longer use eq 13b, with X = 1, to calculate ~ ~ ‘ ( 7 ‘Instead, ). we obtain ~ ~ ‘ ( 7 ‘ ) directly from
(24)
pc and
zAo(n
Eo(n,
= zc expb[EO(?3 - /k1)[-1/m3’2
(28)
in (28), is not calculated from the published coefficients in (24). For PI, we accept the lattice of the expansion of gas value, and for fi2 and p 3 we subtract the internal-mode contributions:
po(n
~+6e(i- e2)
uh ut
7, zAat any state point is readily obtained
0:
r(1 - b2e2)
(25)
5 - 1 0 : F+6 etc. A complete set of relations, including the first Wegner correction to scaling, is given in ref 13. We have now described an equation of state for a near-critjcal pure fluid and shown the relationship between the quantity P of the equation of state and the quantity X of the lattice gas. However, we have seen that the variables X and { of the lattice gas provide the means of mapping points of one lattice gas system onto the equivalent points of another system. It is therefore necessary to compute h and {for a given point of the pure fluid-a point specified by the independent variables r and 6. D. Calculation ofX and {. The first step is to calculate z A o ( o , the activity of the fluid at the point from which Ap is measured, that is, the coexistence curve at T . Once the reference value zA0 (18) Wegner, F. Phys. Rev. 1972, 5 , 4529. (19) Schofield,P.Phys. Rev. Lett. 1969, 22, 606. Schofield, P.;Litster, J. D.;Ho,J. T.Phys. Rev. Lett. 1969, 23, 1098.
The sums in (29) and (30) are over the internal vibrational modes of the solvent molecule, and m is the number of rotational degrees of freedom per molecule. ui = hvi/kTc
(31)
where vi is the frequency of the ith vibrational mode and h is Planck‘s constant. In (29) and (30) the high-temperature limiting value of contributions from rotational motions has been used. We now write the argument in the square brackets of the exponential in (28):
EO(n - h, = PIAT + (,E2 - ~.L~’)(AT)’ + (fi3
- 13’)(An3 (32)
E . The Lattice Gas Model for the Solution. Our model of the mixture is very similar to that of Bartis and Hall,I5 a “hybrid” of the Mermin decorated lattice gas (section 1I.B and Figure IC) and of the Widom decorated lattice gas for a mixture of A (solvent) and B (solute) molecules (Figure Id). For details, see ref 11. Consider once again the construction of a decorated lattice gas as illustrated in Figure IC. As usual, each cell may be occupied by at most one molecule. The interaction between solvent mol-
4082
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987
ecules is exactly the same as that in Mermin’s model. Adjacent primary cells which both contain solvent molecules contribute an energy -e. The system’s energy is unaffected when a primary cell contains a solute molecule. A pair of adjacent primary and secondary cells which are both occupied by solvent molecules contributes an energy -+. If a secondary cell contains a solute molecule and one adjacent primary cell contains a solvent molecule, an energy -$lB is contributed. If a secondary cell contains a solute molecule and both adjacent primary cells contain solvent molecules, an energy -$2B is contributed. As before, nonadjacent cells do not interact and there is no interaction between secondary cells. Notice that there is no interaction between the solute molecules other than the hard-core repulsion. We expect that this approximation will not be too bad in the case of dilute solutions. Note also that the solute-solvent interaction need not be additive; in general f 2+lB
$2B
(33)
The grand partition function of the system is = (1
+
ZA’
+ zB’)4c/2x(X,{,C)
(34a)
where
where 6 and wA are as defined in (12), primes indicate mixture properties, and TnB
= exp(p#nB)
( n = 1, 2)
(35)
Equations 13 and 34 provide the means of mapping of solvent properties onto mixture properties. Appendix A contains explicit formulas for this mapping. As with the pure field, eq 19 provides the relation between E and the thermodynamic properties. We have discussed the need to subtract contributions from internal modes of the solvent molecules from the chemical potential; see eq 32. Questions can be raised about the need to add such contributions into mixture properties. In this paper we consider the pressure of the mixture and the densities of the two components. None of these properties depend directly on contributions from internal modes. However, the chemical potentials do contain such contributions. Since we subtract such contributions from the chemical potentials of both components, there is no need to consider contributions to mixture properties from internal modes. F. The Pure Solute Phase. Consider now the additional phas_e which is in equilibrium with the supercritical solution. Let T’ denote the temperature of the mixture. From an equation for zB’ that is completely analogous to (21) for zA, we readily see that the solute chemical potential pB’is given by
In (36) we choose the zero point of jiB’so that zB’ = z, when jiB’ = 0. In our description of supercritical solubility the variation of jiB’ with temperature and pressure is controlled by the variation of the chemical potential of the additional phase. Assume that the molar volume uB of the pure solute phase is a constant which is independent of temperature and pressure. The change in wB’ with pressure is ApB’
= vBAP
(37)
Here, it is convenient not to use reduced variables. The change of wB’ with temperature is
Nielson and Levelt Sengers to a final temperature T2. c1 is a constant, and c, is the specific heat of the pure solute. We assume that the difference of cv and cp is negligible and take cp from tabulated values. In many thermodynamic problems it is possible to eliminate the constant c1 by going to the ideal-gas limit. In our case, however, the limited range of temperature and density over which the revised and extended scaled thermodynamic potential is valid makes this impossible; c1 is therefore considered an adjustable parameter. Note that a change in the definition of zero point of jid (eq 36) will change the numerical value of c I . As explained above, we subtract contributions from the internal modes of the solute molecule from the experimental specific heat data used in eq 38. An equation similar to (29) is used for this purpose. G . Summary of Adjustable Parameters. We conclude the description of our model for supercritical solubility with a summary of the adjustable parameters. Three constants determine the intermolecular potential of the decorated lattice gas model: z, determines the intermolecular potential for Mermin’s decorated lattice gas; qlBand qZBdetermine the interaction between solute and solvent molecules. The size of the cells of the lattice is related to w (eq 20). We can choose among the four 3-dimensional lattices for which tcis known for the spin Ising model. c1 controls aspects of the behavior of the chemical potential of the solute (38). We describe the selection of these parameters in section 111. H . Numerical Calculation of Mixture Properties. Recall that a point in the mixtu_reis-most easily specified by the independent variables r, 8, and T’. T’is the temperature of the mixture. In this subsection we outline the num_erical calculation of mixture properties for a given point ( r , 8, _T?. For given values of r, 8, and T’, the numerical calculation involves the following steps. First, calculate the properties of the reference system specified by the revised and extended scaled equation of state. Next, calculate X and 5; as described in subsection D above. Next, use the isomorphism between eq 13 and eq 34 to calculate the activities zA’ and zB’of the mixture. Finally, use equations from Appendix A to obtain thermodynamic properties of the mixture. Newton’s-method provides an economical, accurate means of finding if we require that X = 1 along the coexistence curve and use eq 13b to determine zA0. Newton’s method is also used in the determination of zA’ and zgl. Questions can be raised about the existence of multiple solutions to the equations. In the case of the numerical solution of (13b) for zA0the restrictions on the range of z , provided by Mermin and Rehr allow us to avoid this problem. The situation for the solution for zA’ and zB’ is not so clear. Bartis and Hall have shown that our lattice gas model can describe systems which have two critical points at a given temperature. However, it seems unlikely that such phenomena appear in the present work. The values for adjustable model parameters used by Bartis and Hall are very different from those we use. Also, we expect that the existence of multiple solutions would cause Newton’s method to converge to different solutions depending on the choice of starting point for the iteration. In our numerical calculations, the Newton’s method solution for zA’ and zB’ always converged. We would also expect that a system with multiple critical points at a given temperature would have isotherms very different from those shown in our results. Finally, a few test calculations in which all roots of a polynomial were found showed that only one root was physically acceptable. I . Calculation of an Isotherm. An isotherm consists of those points for which the chemical potential of the solute in solution is the same as the chemical potential in the additional solute phase. We begin by calculating pB)(ce) and Pe, the values of the solute chemical potential and the pressure at the critical end point. This calculation is straightforward in practice since we know that at the pure solvent critical point r = 0 and the value of 0 is irrelevant. The calculation of pB’(ce) provides the starting point from which changes in pB‘can be calculated in a consistent manner. The next step is to calculate
zA0(n
a( pq
In (38) the temperature changes from an initial temperature T I
pg’( F’,pce)
(39)
where p’denotes the temperature of the isotherm. a( T? is de-
Model for Supercritical Solubility m
-
(01
k9
/'
. z-
&/:/'
E U mh ?
a.
,
%1
A/"
n
/
, MOLE % SOLUTE
MOLE % SOLUTE
?
LA1 A 0
?
O
%-
I
I
MOLE % SOLUTE
Figure 2. Schematic of behavior that is found when the decorated lattice gas is used outside the region of validity.
termined by adding pd(ce) and the quantity Apg' calculated in eq 3-8. Next, select a set of points ((r,0)) and for each point ( r l , 01, T')calculate properties of the mixture. For each point i we can now obtain the quantity di = (r(T")
+ ug(P(i) - Pee) - pg'(i)
A
(40)
where ps'(i) and P ( i ) denote the solute chemical potential and the pressure a t the ith point. A point lies on the isotherm when d = 0. Now, for some point d is positive and for others d is negative. If this condition is not satisfied, a better set ((r,0)) must be found. Suppose that di is positive and dj is negative. Then somewhere on the line connecting (ri,Si) to (rj,Oj) there is a point of the isotherm. The method of bisection is used to find all such points.
\
A
0.0
0.1
0.2
0 I (N)/CI 1 0 I ( N i , ) / C I q/2
( i , j = 0, 1)
(41) (42)
In our calculations it was found that (41) and (42) are more restrictive than the requirement that the scaled thermodynamic potential be used only within its range of validity. Figure 2 illustrates the type of behavior that was obtained when the pure-solvent representation was used to calculate Po, and (41) and (42)were not satisfied. The solubility of the solute does not always increase as the pressure is increased. While this behavior does not violate any fundamental physical principles, it is highly unlikely that it occurs in the experimental systems being considered. In all plots Figure 3-9, the calculated curves have been restricted to the range where (41) and (42) are satisfied. In general, the Cestriction was more serious when the fluid chemical potential go(7') was used than when we used the lattice gas po( 2").
'\
0.5
0.4
0.6
MOLE % SOLUTE
9
111. Calculations and Results In this section we describe the use of our model to represent observations for three systems: naphthalene dissolved in supercritical CzH4,hexachloroethane dissolved in supercritical CzH4, and naphthalene dissolved in supercritical C 0 2 . A . Range of the Calculations. We have mentioned that the limited range of temperature and density over which the scaled thermodynamic potential is valid implies a restriction on the thermodynamic states for which the decorated lattice gas is valid. Additional restrictions arise from a consideration of the quantities N, Noo,Nol,and N l , of the decorated lattice gas. In our calculation (see eq A8) we obtain the ensemble average of these quantities. These quantities must satisfy the conditions
0.3
\
IRI
A
A \
\\ \
A
0.0
l
I
0.1
0.2
,
0.3
A
I
0.4
A I
I
0.5
0.6
MOLE % SOLUTE
Figure 4. u-x isotherms at (A) 12 and (B) 25 OC for naphthalene dissolved in ethylene. Symbols are as in Figure 3.
B. Descriptions of the Pure Substances. The scaled thermodynamic potential parameters of ref 20 were used to describe near-critical CzH4, and those of ref 21 were used to describe near-critical COP Contributions to pz and p 3 due to internal modes were subtracted, as given in ref 22 for C2H4and in ref 23 for C 0 2 . For both solutes considered, data were available which allowed us to treat the variation of pg' with temperature and pressure by (20) Levelt Sengers, J. M. H.; Olchowy, G. A,; Kamgar-Parsi, B.; Sengers, J. V. "A Thermodynamic Surface for the Critical Region of Ethylene", NBS Technical Note 1189; Government Printing Office: Washington, DC, 1984. (21) Albright, P. C. private communication. Also, second reference in footnote 13. (22) Jahangiri, M. Ph.D. Thesis, University of Idaho, 1984. (23) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976.
4084
Nielson and Levelt Sengers
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987 9 ah
0
0
,Y
IB:
I81
a 9 0.0
0.1
0.2
9
?1
0.3
0.5
0.4
0.6
MOLE 7. SOLUTE
A
A
0 0
a 0.0
0.2
0
0
ifl'
\:
dE z
;-
I
0.2
,
~
IRI
,
l
0 0
1
0.5
0.4
0.3
0 9
,
I
0.1
: .O
0.8
NOLE % SOLUTE
A
0.0
0.6
0.4
0.6
0.2
0.0
MOLE % SOLUTE
0.4
0.6
0.5
1 .o
HOLE % SOLUTE
Figure 5. P-x isotherms for naphthalene dissolved in ethylene: (A) pure fluid Eo and pB' corrected by (37) and (38) and (B) lattice gas ii,,and pB' from (43); (-) predicted 16.5 "C isotherm; (- - -) predicted 20 " C isotherm; (-.-) predicted 23.5 " C isotherm; (a)observed 16.5 "C isotherm; (0)observed 20 " C isotherm; (A) observed 23.5 " C isotherm. The predicted curves result from parameters obtained from fitting the data in Figures 3 and 4. Experimental data from ref 33.
Figure 7. u-x isotherms for hexachloroethane dissolved in ethylene. Fitted curves based on (A) pure fluid Eo and fig' corrected by (37) and (38) and (B) lattice gas po and pB' corrected by (43). Symbols are as in Figure 5. Experimental data from ref 33. 0
0
n
IBJ
/ A
~
3
c C 0.2
0.0
.i
0.4
0.6
0.8
1:.
3.8
I .3
VOLE % SOLUTE
0
0 0 C
Y 7
0
0.0
0
0.2
0.4
0.6
VOLE % SOLUTE L I
- %%
c.4
C 6
:B
YCLE i. S3-JT7 Figure 6. P-x isotherms for hexachloroethane dissolved in ethylene. Fitted curves based on (A) pure fluid Eo and p ~ corrected ' by (37) and
(38) and (B) lattice gas Po and pB' corrected by (43). Symbols are as in Figure 5 . Experimental data from ref 33.
means of eq 37 and 38. The molar volumes of pure naphthalene and pure hexachloroethane have been measured a t temperatures near those of interest here.24 Equation 38 was evaluated by using experimental specific heat data for the solute of interestz5vz6which had been corrected by removing contributions from internal (24) CRC Handbook of Chemistry and Physics; Weast, R. C . , Astle, M. J., Eds.; CRC: Boca Raton, FL, 1981. (25) Timmermans,J. Physico-Chemical Constants of Pure Organic Compounds; Elsevier: Amsterdam, 1965, p 132. (26) Seki, S.; Momotani, M. Bull. Chem. SOC.Jpn. 1950, 23, 30.
Figure 8. P-x isotherms for naphthalene dissolved in CO,. Fitted curves based on (A) pure fluid Po and pB' corrected by (37) and (38) and (B) lattice gas Eo and peg/ corrected by (43): (-) fitted 35 "C isotherm; (-- -) fitted 45 " C isotherm; (0) observed 35 " C isotherm; (0)observed 45 "C isotherm. Experimental data from ref 32.
mode^.^'^^^ In the case of hexachloroethane there is substantial anharmonicity in the internal torsion mode.28 In the present work the approximate harmonic frequency given in ref 28 for this mode was used. The integral in (38) was evaluated by using an interpolation package written by Amos.29 The value of the integral and the value of c , depend on the choice for the zero of entropy. In naphthalene the entropy was taken to be 0 a t 0 O C . For (27) McFee, D. G.; Lielmezs, J. Thermochim. Acta 1979, 30, 173. (28) Chao, J.; Rcdgers, A. S.; Wilhoit, R. C.; Zwolinski, B. J. J . Phys. Chem. Re$ Data 1974, 3, 141. (29) Amos, D. E . SAND (Sandin Narl. Lab.) 1979, 78, 1968; 1979, 79, 1825.
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987 4085
Model for Supercritical Solubility
TABLE I: Parameters Used To Represent the Experimental Solubility Data lattice gas d,iio/dT (Appendix B)
solute solvent
ClOH8
CIOH8
co2
C2H4
6
4-Tce
6
‘$lEJ/b
-0.995 -3.3 3.7472 -2.4
#2B/6
1.6
-0.9889 -3.1 4.2707 -3 1.85
11
-4.5
In zc W
C1
Cl’
pure solvent dpo/dp (eq 32) ClOH8
c2c16
C,oHs
C2H4
C2H.4
co2
-0.995 -3.3 3.7472 -3 1.3 5.5 10.5
-0.98 89 -3.15 4.1783 -2 1.75 19.5 28
12 -0.988 -3.1 3.2934 -4.5 1.5
-5
6
6
c2c16 C2H4
12 0.988 -3.1 3.2934 -3.5 1.55 13.2 21
OPure solvent parameters: C2H4,ref 20; C02, ref 21. in Figures 3 and 4. Figure 5 shows the prediction made by the model for the isotherms at 16.5, 20, and 23.5 OC. It was found that the choice of the method for calculating ,Go (see section 1I.D) had a significant effect on the density range over which the lattice gas calculations could be made. The range is larger when Do is calculated from the lattice gas than it is when Po for ethylene is used, It was found that contributions from (37) and the integral in (38) can be neglected if the variation of gB’with temperature is taken to be
u0 00 3
E.0
All,’ = cI’AT 0.2
3.4
0.6
3 0
Figure 9. u-x isotherms for naphthalene dissolved in Cot. Fitted curves based on (A) pure field po and pgl corrected by (37) and (38) and (B) lattice gas po and pB’ corrected by (43). Symbols are as in Figure 8.
hexachloroethane it was taken to be 0 at 10 OC. C. Choice of Model Parameters. Table I displays the parameters used to represent solubility data for the three systems considered. The lattice gas parameters were selected by trial and error. The lattice gas equations are sufficiently unstable that a standard nonlinear least-squares algorithm is expected to fail. Initially, an attempt was made to choose z, so that the value of F calculated from (B5) agreed with the value from the scaled thermodynamic potential. Although it was always possible to choose z, in this way, the optimal choice proved to be different from that made by this method. w was chosen such that half of the sites in the isomorphic simple lattice gas were filled at the solvent critical point. Since the computer time required to generate an isotherm was fairly modest, it was possible to scan a large number of parameter sets in this process of choosing the model parameters. For each system an effort was made to find a parameter set which predicted the correct critical end point. In general, it was impossible to find parameter sets which reproduced the isotherms in a correct manner while also reproducing the rate of increase of pressure along the critical line. In the parameter sets of Table I the predicted critical end point pressure is, for those cases for which it is known experimentally, lower than that actually observed, so that the initial P-x slope of the critical line may be low by as much as a factor of 2 . D. Modeling of Solubility. Figures 3-5 display the representation of P-x and u-x isotherms for naphthalene dissolved in supercritical ethylene. The model parameters were selected by optimizing the fit to the observed isotherms at 12 and 25 OC shown
(43)
where cl’ is an adjustable parameter and the influence of pressure on the additional phase is neglected. Note, however, that cl’ is not equal to c1 in (38), since cl’ contains contributions from the integral in (38). Figures 6-9 display fits to the measured solubility of hexachloroethane dissolved in C2H4 and naphthalene dissolved in COz. The results shown in Figures 3-9 show that our model for supercritical solubility can provide a semiquantitative representation of experimental solubility data. E. Infinite-Dilution Partial Molal Volumes. Calculations were made of the infinite-dilution partial molal volumes for the systems naphthalene dissolved in C2H4and naphthalene dissolved in C 0 2 . While the qualitative behavior of the partial molar volume as a function of solvent density agreed with e ~ p e r i m e n tthe , ~ ~quantitative agreement was poor. The magnitudes of the partial molal volumes were overpredicted by as much as 50%, which means that we have not fitted all critical end point parameters a ~ c u r a t e l y . ’ ~ ~ ~ It is likely that a more satisfactory fit could be obtained by readjusting the model parameters. Such a fit was not attempted in this work because it would degrade the fit to the other properties.
IV. Discussion We have developed a nonclassical model for dilute near-critical solutions. A relatively small set of parameters can represent a significant body of thermophysical data for supercritical solutions. A goal of this work was to obtain a method for mapping properties of a pure near-critical solvent onto a dilute solution. In the case of the decorated lattice gas model studied in this work, this was possible for a rather limited range of thermodynamic states. In order to increase the range over which mixture properties can be calculated, it was necessary to modify the equation of state that described the pure solvent. A second limitation of the model of this work was the inability to simultaneously fit isotherms and the critical end point. At the solvent critical point, the partial molal volume of the solute can be expressed exactly in terms of the compressibility of the pure (30) Eckert, C. A,; Ziger, D. H.; Johnston, K. P.; Ellison, T. K. Fluid Phase Equilib. 1983, 14, 167. (31) Diepen, G . A. M.; Scheffer, F. E. C. J . A m . Chem. SOC.1948, 70, 4085. (32) Tsekhanskaya, Yu. V.; Iomtev, M. B.; Mushkina, E. V. Russ. J . Phys. Chem. (Engl. Trans!.) 1964, 38, 1172. (33) van Gunst, G. A. Ph.D. Thesis, Technical University of Delft, Netherlands, 1950. (34) Chang, R. F.; Morrison, G.; Levelt Sengers, J. M . H. J . Phys. Chem. 1984, 88, 3389.
4086
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987
solvent, the critical slope of the solvent vapor pressure curve, and the initial slopes of the critical line.34 Since our modifications of the scaled thermodynamic potential do not affect the puresolvent properties which enter into this relation, it is evident that the failure to obtain a good prediction of the infinite-dilution partial molal volumes results from the failure to adequately represent the critical line behavior. Finally, the physical reasonableness of values for and $2B shown in Table I can be questioned, especially the repulsive character of These choices were necessary in order to obtain a qualitatively correct critical line. The opposite signs of JilBand $29 may cause the unrealistic shape of the isotherms shown in Figure 2. The lattice gas model cannot be expected to give a very realistic representation of a fluid mixture. Recently, Gilbert and E ~ k e r have t ~ ~independently developed a decorated lattice gas model for supercritical solubility. The details of their model are quite different from those of the present work, but the basic philosophy is similar. Gilbert and Eckert employ a simpler nonclassical description of the pure solvent than that used here; they do not treat the variation of the chemical potential of the additional phase with temperature and pressure, and they interpret Mermin's decorated lattice gas in a manner that is quite different from our interpretation. Gilbert and Eckert have successfully fitted infinite-dilution partial molal volume and density-mole fraction isotherms. They have not yet published any fits of pressure-mole fraction isotherms. It will be interesting to see whether fits to pressure-mole fraction isotherms with the Gilbert-Eckert model will avoid the difficulties which we have encountered. The results of the present work indicate that it is possible to devise equations that represent properties of a near-critical mixture with fewer parameters than are required in the Leung-Griffiths model. It is hoped that the experience gained in the present work will be useful in the development of alternative nonclassical models for fluid mixtures.
Nielson and Levelt Sengers
From eq 19 and A3 w,e readily obtain the expression for the pressure of the mixture P'in terms of that of the pure solvent: P I =
of
(wq/2)(ln (1
+ zl' + z2')
-
In (1
+ z l ) )+
(A6)
To facilitate the presentation of equations for first derivatives P I , define the column vectors v=(f),
vo=($),
v'=(:j
P2I
(A7)
Here PI'and jj2/ are the reduced densities of the solvent and solute, respectively. In (A7)
where the angle brackets denote an ensemble average and T*
1 = lim - In X(A,l;C) c-c=
c
Mapping from 1 to 2 components can be described by the equations
+ B(')) (A10) Here A is a ( m + 1) X 2 matrix and B an ( m + 1)-element column
YO
= [A~')]-'(v- wB('))/w,
V'
= w(A(')v0
vector. To simplify the equations for the elements of A and B, define the functions
Acknowledgment. We have had helpful discussions with Profs.
B. Widom and J. C. Wheeler and with Drs. G. W. Mulholland, P. C. Albright, and J. Fox. We are indebted to a referee for incisive comments and criticism. Appendix A In this Appendix we present equations which describe the relationship between the thermodynamic properties of the pure solvent and the properties of isomorphic points in the mixture. These equations are written in a form convenient for use in computer coding. The equations can be derived from eq 13, 18, 19, and 34. In the following, we will sometimes use primed variables to denote quantities which describe the mixture. It is convenient to use a slightly different notation from that of section 11. Rather than using the subscripts A and B to denote the two components of the mixture, we use the subscripts 1 and 2 for this purpose. Let -$, be the interaction associated with occupation of a secondary cell by a molecule of type 01 (01 = 1, 2), where the two adjacent primary cells contain a total of n solvent molecules. Then $01 $02
= 0,
$11
= 4,
$21
= 24
(All
= 0 and we have already introduced the interaction parameters and 422(=$2B) in section 1I.E. Define
The derivation of the following equations becomes clearer if we write eq 13 and 34 in common notation. Consider a system with m components ( m = 1 or 2). To simplify the notation, we instead of writing CE,. use the shorthand
x,
(35) Gilbert, S.W.; Eckert, C . A. In Proceedings of the 4th International Conference on Fluid Properties and Phase Equilibria f o r Chemical Process Design, Denmark, 1986; Fluid Phase Equilib. 1986, 30, 41.
In the following equations for the elements of A and B we do not explicitly show the dependence of J,,(m) and K,,(m) on m; the symbol 6 stands for the Kronecker delta function.
-39cJJ0, 4F
B,(m) = -
Appendix B This Appendix contains expressions derived from Mermin's decorated lattice gas model which correspond to quantities in the revised and extended scaled equation of state. As usual, a horizontal bar will be used to distinguish quantities obtained from the decorated lattice gas from their analogues in the scaled equation of state. The significance of these expressions has already been considered. The variation of po with temperature is obtained by noting that
J. Phys. Chem. 1987, 91, 4087-4091
In (Bl) the value of X from eq 13b is used. After some algebra we obtain
4087
definition of the temperature-like scaling field, also has a lattice gas analogue. This becomes evident by comparing eq 13c for the temperature-like scaling field of the Mermin model with eq 8c for the simple lattice gas and noting that (13c) contains the activity as well as the temperature. From the definition of the mixing parameter (eq 23) we find, for the decorated lattice gas
Here
t, =
ZA?An
1 + ZA?An
where (13c) is used for f. After some algebra we obtain
( n = 0, 1, 2)
At the critical point, eq B2 becomes the lattice gas analogue of the parameter p, (eq 24) of the scaled equation. The mixing parameter c of the fluid (eq 23), which indicates the strength of the contribution of the chemical potential in the
In (B5) t,C denotes the quantity defined in (B3) evaluated at the solvent critical point.
Electrical Conductivity, Viscosity, and Density of a Two-Component Ionic System at I t s Critical Point Donald R. Schreiber, M. Conceicao P. de Lima,+ and Kenneth S. Pitzer* Department of Chemistry and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 (Received: November 14, 1986)
The density, viscosity, and electrical conductivity of a system composed of a fused salt in a low dielectric constant solvent were examined at a temperature just above the liquid-liquid critical point. The system examined was tetra-n-butylammonium picrate (TNBAP) in 1-chloroheptane. The conductivity decreased to a minimum near 0.001 mol/dm3 and then increased rapidly through the region of the critical concentration. The data in the region of the minimum were fit by using the method of Fuoss and Kraus to determine the ion-pair and ion-triplet dissociation constants. An extrapolation of these equilibria toward the critical region indicated a rapidly increasing importance of ion triplets at the expense of the neutral ion pairs. The conductivity results were also compared to a recent cluster-equilibriumtreatment of the restrictive primitive model (RPM). This treatment of the RPM also shows an increasing importance of charged clusters, at the expense of neutral clusters, as the concentration approaches that of the critical point. The dielectric constant of 1-chloroheptane was also measured over a range of temperature.
Introduction While the region of the liquid-liquid critical point for partially miscible mixtures has been extensively studied for systems of neutral molecules,' little is known about ionic systems at their critical points. Recently we reported the liquid-liquid phase diagram including the critical point for the tetra-n-butylammonium picrate (TNBAP)-1-chloroheptane system (T, = 414.4 K, x, = 0.085, c, = 0.43 m ~ l / d m ~ )This . ~ system presented us with an opportunity to examine for the first time several other properties of a two-component ionic system near its critical point. In this paper we present the results of an examination of the electrical conductivity, viscosity, and density of this system. Values are reported over a range of concentrations, including the critical composition, a t a temperature just above the critical point and at a somewhat higher temperature. The dielectric constant of I-chloroheptane was also measured as it was not available at elevated temperatures. Mixtures of neutral-molecule liquids in the region of their critical points have been investigated and it was found that the shape of the two-phase curve deviates from that predicted by classical theory. Similiar deviations have been seen in nonionic liquid-vapor and magnetic systems. These systems are characterized by fluctuations in composition, density, or magnetism that become nearly macroscopic as the critical point is a p p r ~ a c h e d . ~ 'Permanent address: Department of Chemistry, University of Coimbra, 300 Coimbra, Portugal.
0022-3654187 12091-4087$01.50/0 , I
,
The deviation from classical theory occurs when the range of these fluctuations becomes longer than the range of interparticle interactions. Modern critical point theory takes into account these fluctuations and can therefore account for the shape of the phase separation curve in the region of a critical point.3 A recent review4 discusses these characteristics of the two-phase curves as well as certain other properties showing anomalous behavior near the critical point. Unlike the results seen with neutral molecules, our earlier results indicated classical behavior for the ionic system. These results suggested that the interparticle forces for the TNBAP-chloroheptane system near its critical p i n t were of much longer range than those for typical neutral molecules. This conclusion seems reasonable since ionic forces are longer range ( l / r ) than those of neutral molecules ( l/r6). In an attempt to determine the nature of the solution in the region of the critical point, the electrical conductivity, viscosity, and density were examined. The conductivity of a solution is dependent on the concentration and mobility of the ions present in solution. Reduction of the number of ions present via formation of neutral ion pairs has been shown to reduce the equivalent condu~tance.~,~ Previous studies's8 have shown that TNBAP forms (1) Heller, P. Rep. Prog. Phys. 1967, 30, 7 3 1 . (2) Pitzer, K. S.;de Lima, M. C. P.; Schreiber, D.R.J. Phys. Chem. 1985, 89, 1854. (3) Fisher, M. E. Rep. Prog. Phys. 1967, 30, 615. (4) Greer, S. C.; Moldover, M. R. Annu. Reu. Phys. Chem. 1981, 32,233.
0 1987 American Chemical Society
