J. Phys. Chem. 1986, 90. 5921-5927 is negligible!] Thus we arrive a t the heat capacity of CH4 due to its rattling motion in the cages by subtracting 1.5R from the total heat capacity contribution from CH4 to the hydrate. The resulting C, was then treated in terms of the quantum statistical model as described above to obtain ws and wL. The dvdwand d[s used for CHI are given in Table 111. The vibrational frequencies obtained for CH4are given in Table I11 and compared with those obtained from the M D simulation^.^^ The agreement is quite satisfactory. The literature values for AH(i+e) and ACp(i+e), and sometimes for Ap(i--e), reported in Table I1 are actually the parameters required to fit the ideal solid-solution model to the phase equilibrium results. The present results for each property were obtained by combining the results for eq 2 obtained from the ideal solid-solution model with the corresponding experimental results for the formation of hydrate from ice. Moreover, the results are based on xenon and krypton hydrates whose structures are now correctly known. Both xenon and krypton fit comfortably in the hydrate cages and thus the properties of the empty lattices reported are essentially those of the unperturbed lattices. Xenon and krypton hydrates are the systems which can be expected to be most adequately represented by the ideal solid-solution model. However, the use of a spherically symmetrical potential for describing the guest-host interactions is questionable. The 12-hedra, which are present in both structures, and the 16-hedra, which are present in structure I1 only, are nearly spherical whereas the 14-hedra, which are present in structure I only, deviate up to 14% from (41) JANAF Thermochemical Tables, 2nd 4.; National Bureau of Standards: Washington, DC, 1971.
5921
spherical symmetry.' John and Holder12 imposed the spherical potential over the potential obtained by discrete summation and arrived at a new set of z and a values for the various cages in the two structures. These values are given in parentheses in Table I. It was proposed4* that the use of this new set of values in the classical model should account for the departure from the spherical symmetry. The calculations for AC,(e+g+h) performed by using the parameters suggested by John and Holder yielded essentially the same results as obtained with the original parameters but yielded rather poorer values for Ap(i+e) and AH(i+e). The values of AH(i+e) and AC,,(i+e) for the two structures obtained are based on the assumption that the enthalpy of encagement and the heat capacity change are linearly dependent on 6, a direct consequence of assumption number 1 in the ideal solid solution model. This assumed ideality of the guest-host solution has never been verified experimentally. Xenon is enclathrated easily and its hydrates of different compositions can be prepared readily. Determinations of enthalpies of dissociation and heat capacities of xenon hydrates of different compositions would help in evaluating the extent of nonideality of these guest-host systems and thus in improving the estimates for the empty lattice properties reported here. Spectroscopic determinations of the vibrational frequencies of the guests dealt with in this work shall also be of interest. Registry No. Water, 7732-18-5. (42) John, V. T.; Holder, G. D. J. Phys. Chem. 1981,85, 1811. (43) Sortland. L. D.: Robinson. D. B. Can. J. Chem. E m . 1964. 42. 38. (44j Parrish, W. R.;Prausnitz,'J. M. Ind. Eng. Chem. Process d e s . be". 1972, 11, 26.
Behavior of Dilute Mixtures near the Solvent's Critical Point R. F. Chang* and J. M. H. Levelt Sengers Thermophysics Division, National Bureau of Standards, Gaithersburg, Maryland 20899 (Received: May 27, 1986)
In the limit of infinite dilution at the critical point of the solvent many thermodynamic properties such as excess properties and partial molar quantities exhibit remarkable anomalies. A striking effect is that finite properties such as the partial molar volume of the solvent exhibit dependence on the path of approach to the critical point. Using the Leung-Griffiths model of mixtures we are able to calculate these thermodynamic properties. The properties considered are partial molar volume, partial molar enthalpy, osmotic susceptibility, isothermal compressibility at constant composition, heat capacity at constant pressure and composition, partial molar heat capacity, osmotic coefficient, and activity coefficient. The Leung-Griffiths thermodynamic potential is nonclassical and is of a scaled form. By the use of the model, we are able to analyze the path dependence of many of these properties and to obtain their explicit x dependence (where x is the mole fraction of the solute) as well as asymptotic expressions along various paths leading to the pure solvent's critical point.
Introduction The thermodynamic behavior of dilute mixtures near the solvent's critical point has undergone a revival of interest in recent years. New experiments have revealed large anomalies in the partial molar volume of the solute,I4 apparent molar specific and apparent heats of dilution8 of dilute salt solutions (1) van Wasen, U.; Schneider, G. M. J. Phys. Chem. 1980, 84, 229. (2) Paulaitis, M. E.; Johnston, K. P.; Eckert, C. A. J .Phys. Chem. 1981, 85. 1770. (3) Eckert, C. A.; Ziger, D. H.; Johnson, K. P.; Ellison, T. K. Fluid Phase Equilib. 1983, 14, 167. (4) van Waser, U.; Schneider, G. M. Angew. Chem., Int. Ed. Engl. 1980,
-19- , -5 1. 5- .. (5) Smith-Magowan, D.; Wood, R. H.J . Chem. Thermodyn. 1981, 13,
1047. (6) Wood, R. H.; Quint, J. R. J . Chem. Thermodyn. 1982, 14, 1069. (7) Gates, J. A.; Wood, R. H.; Quint, J. R. J. Phys. Chem. 1982.86.4948.
in near-critical steam. Furthermore, extraordinarily large excess enthalpies of mixing have also been o b ~ e r v e d . ~ Many J~ of these anomalies, however, can be explained in terms of critical point phenomena."J2 For instance, the behavior of the partial molar volumes on the isotherm-isobar can be seen from the molar volume (8) Busey, R. H.; Holmes, €1. F.; Mesmer, R. E. J. Chem. Thermodyn. 19a4,16,343. (9) Christensen, J. J.; Walker, T. A. C.; Schofield, R. S.;Faus, P. W.; Harding, P. R.; Izatt, R. M. J . Chem. Thermodyn. 1984, 16, 445, and references therein. (10) Wormald, C. J. Ber. Bunsenges. Phys. Chem. 1984, 88, 826, and references therein. (1 1) Chang, R. F.; Morrison, G.; Levelt Sengers, J. M. H. J. Phys. Chem. 1984,88, 3389. (12) Levelt Sengers, J. M. H.; Chang, R. F.; Morrison, G. In Equations of S t a t e T h e o n e s and Applications; Chao, K. C., Robinson, R. L., Jr., Ms.; ACS Symposium Series 300; American Chemical Society: Washington, DC, 1986; p 110.
This article not subject to U S . Copyright. Published 1 9 8 6 by the American Chemical Society
5922
The Journal of Physical Chemistry, Vol. 90, No. 22, 1986
V I
Figure 1. Volume vs. composition diagram for a binary mixture near the solvent’s critical point when the solute is less volatile (a) and more volatile (b) than the solvent, respectively. ( u ) vs. composition ( x = mole fraction of solute) diagram of a near-critical fluid, the solvent, as shown in Figure 1. By the addition of the solute (assumed to be less volatile than the solvent) at a temperature, T,greater than the critical temperature of the solvent, T,, but far below that of the solute, the system must undergo a phase transition when x is greater than a certain value. Because the low volume liquid phase is richer in the less-volatile component (the solute), the slope of the tie lines must be negative, as shown in Figure l a , and the tie lines shrink to a critical point. The isotherm-isobar a t the mixture’s critical point is tangent to the coexistence curve and always remains in the one-phase region. When the temperature T approaches T,, the coexistence curve moves to the left and becomes tangent to the x = 0 axis at the solvent’s critical point. The critical isotherm-isobar remains tangent to the coexistence curve at the critical point and its tangent becomes vertical at that point. In the classical theory where one expands the molar Helmholtz free energy near the critical point in terms of ( u - u,), ( T - T,), and x , except fqr the ideal-mixing term, one finds, at T = T,, Iu - u,l a x1/2for the coexistence curve whereas lu - u,( 0: x1/3for the critical isotherm-isobar. Here u, is the critical molar volume of the solvent and the two vertical bars denote absolute value. One can guess that nonclassically the exponent for the coexistence curve should be the critical exponent 0 = 0.325 whereas the exponent for the critical isotherm-isobar should be 1/6 where 6 = 4.815 is another critical exponent. Consequently, the derivative of u with respect to x at constant temperature, T, and pressure, p , is proportional to and diverges to --m as x 0. Because the partial molar volume of the solute, B2, is expressed in terms B2 also diverges along the critical isotherm-isobar of (do/dx),,, as x1lb1. Additionally, the partial molar enthalpy can be expressed in terms of the partial molar volume and the same behavior is expected. The situation is essentially the same if the solute is more volatile than the solvent (Figure lb). A two-phase region exists only for T < T, but the isotherm-isobar at T = T, becomes vertical just as before and its slope diverges to +a. We have shown that higher derivatives, such as partial molar heat capacities, diverge much more strongly, but with the same sign as partial molar v o l ~ m e s . ~ l J ~ In recent heat capacity measurements of solutes in steam near its critical point, strong divergences of the partial molar volume and partial molar heat capacities of the solute have been reported, with negative sign for salt solutions5 and positive sign for argon
-
solution^.'^ A more remarkable finding has been that thermodynamic properties such as the partial molar volume of the solvent, although finite, exhibit path dependence of their limiting value as the solvent’s critical point is approached. The path dependence cannot be seen in a plot such as Figure 1, but it was noted experimentally ~~
~~~~~
~
(13) Levelt Sengers, J. M. H.; Everhart, C. M.; Morrison, G.; Pitzer, K. S . Chem. Eng. Commun., in press. (14) Biggerstaff, D. R.; White, D. E.; Wood, R. H. J . Phys. Chem. 1985, 89, 4378
Chang and Levelt Sengers by Krichevskii,15 formulated by RozenI6 in the framework of classical theory, and worked out by WheelerI7 on the basis of a nonclassical (scaled) model, the decorated lattice gas. The cause of this path dependence becomes obvious if one is to resolve the dilemma encolintered by the thermodynamic properties because of imposed contradictory conditions. As best exemplified by the osmotic susceptibility, ( d x / d p ) , , where p is the difference of chemical potentials of the solute and the solvent, p = p 2 - pl, the condition of criticality of a mixture that (ax/&),, diverge and the condition of infinite dilution that it vanish meet at the solvent’s critical point. The dilemma is resolved as one realizes that the first condition prevails on paths that stay close to the critical line whereas the other condition manifests itself on paths that dilute the mixture faster than the criticality is reached. It is the purpose of this paper to present a comprehensive nonclassical treatment of the limiting behavior of all thermodynamic properties of interest for a dilute binary mixture as the solvent’s critical point is approached along a variety of paths. The properties we have studied are partial molar volumes, enthalpies, and heat capacities of solvent and solute. In addition, we have analyzed the limiting behavior of the osmotic susceptibility, the isothermal compressibility a t constant x, and the heat capacity at constant p and x . We have also obtained an explicit expression for the osmotic coefficient as well as for the activity coefficient on the critical isotherm-isobar for dilute mixtures. The paths we have chosen explicitly to illustrate the path-dependent behavior are the critical line, the coexistence curve, the critical isochoreisotherm, the critical isochore-isobar, and the critical isothermisobar. We then develop a general method valid for any smooth path leading to the solvent’s critical point. The model of mixtures we adopt here is specifically designed for a binary fluid mixture near the line of vapor-liquid critical points and was initially proposed by Leung and Griffths18 for the mixtures of 3He and 4He. This model, henceforth referred to as the Leung-Griffiths model (LG model), has subsequently been applied to other binary m i x t ~ r e s l ~and - ~ is ~ in total compliance with the general universality theory of critical point phenomena postulated by Griffiths and Wheeler.23 The main assumption in their theory is that one may draw an analogy between the nonclassical critical phenomena found in pure fluids and that exhibited by mixtures and the analogy is most easily understood when the independent variables are chosen from among the field variables rather than density variables. Field variables take on the same value in coexisting phases whereas density variables take on different values in coexisting phases. The outline of this paper is as follows. We will present a brief summary of the LG model and define the symbols which deviate slightly from those defined in ref 18. Then we will derive the partial molar volumes and evaluate them in the limit of infinite dilution along a few chosen paths to demonstrate their path dependence. Next we will present a general analysis of the pathdependent behavior of partial molar volumes followed by an analysis of other properties mentioned before and a brief conclusion.
Leung-Griffiths Model For a detailed description of the model we refer readers to ref 18 and ref 22. Here we will present the general features of the model relevant to the subsequent analysis. The independent variables of the model are T, pl, and p 2 which are field variables and the thermodynamic potential, w , is chosen to be w =p/RT (1) (15) Krichevskii, I. R. Rum. J. Phys. Chem. 1967, 41, 1332. (16) Rozen, A. M. Russ. J. Phys. Chem. 1976, 50, 837. (17) Wheeler, J. C. Ber. Buiwenges. Phys. Chem. 1972, 76, 308. (18) Leung, S. S.; Griffiths, R. B. Phys. Rev. A 1973, 8, 2670. (19) Moldover, M. R.; Gallagher, J. S . AIChE J . 1978, 24, 267. (20) Rainwater, J. C.; Moldover, M. R. In Chemical Engineering at Supercritical Fluid Conditions, Paulaitis, M. E., Penninger, J. M. L., Gray, R. D., Jr., Davidson, P., Eds.; Ann Arbor Science: Ann Arbor, MI, 1983; p 199. (21) DArrigo, G.; Mistura, L.; Tartaglia, P. Phys. Rev.A 1975, I,?, 2587. (22) Chang, R. F.; Doiron, T. Int. J. Thermophys. 1983, 4, 337. (23) Griffiths, R. B.; Wheeler, J. C . Phys. Rev.A 1970, 2, 1047.
The Journal of Physical Chemistry, Vol. 90, No. 22, 1986 5923
Thermodynamics of Dilute Mixtures
h = nrAO(l - 02)
(10)
t = r(l - h2O2)/RT,
(11)
Here T, is the critical temperature of the solvent and In 8
62
= (6 - 3)/(6 - 1)(1 - 2p)
(12)
+
0
1
(1
Figure 2. Schematic drawing of the field variables {, T , and h in relation to the coexistence surface and critical line.
where R is the gas constant. However, the independent variables are transformed to three others of a more convenient form
P = c2 exP(P2/RT)/e
(2)
= B,({) - B
(3)
h = In 8 - H ( { , T )
(4)
T
where 0=
c1 exP(Pl/RT) + c2 exph2/RT) B = l/RT
(5)
P = P1
(6)
Here C 1and C2 are two positive constants. The variable {is so constructed that it is 0 if the mixture is pure component 1 and unity if the mixture is pure component 2. B,(O is the critical value of l/(RT) as a function of {. H ( { , r ) is the value of In 8 on the liquid-vapor coexistence surface and its smooth extension as a function of { and T . It is assumed that B,({) and H({,T) are analytic functions and can be expanded into a power series of respective independent variables. Generally, each of the two components in a binary mixture plays an equivalent role. However, in a dilute mixture, we designate the solvent as component 1 and the solute as component 2. The field variable { is defined in the direction of critical line and plays a composition-like role. In analogy with pure fluids, the variables T and h are known as the weak and the strong variable, respectively. They are defined in the direction parallel to and that intersecting the coexistence surface, respectively, as shown in Figure 2. They are so named because T specifies the weak direction while h specifies the strong direction in the framework of the universality theory of Griffiths and Wheeler.23 The central theme of the LG model is that mixtures are like pure fluids when described in the constant-{ plane and the properties of mixtures can be interpolated from those of the two pure com’ ponents constituting the mixtures. The potential w, now considered as a function of {, r , and h, consists of two parts: a regular part, wr({,T,h),and a singular part, os({ , ~ , h )which , contains in its derivatives the divergences at the critical line and the discontinuities of density variables at the coexistence surface. It is assumed that the regular part is an analytic function of {, T , and h and can be expanded into a power series whereas the singular part takes on the form w,(S;T,h) = q(t).rr(t,h)
The critical indices cy, p, y, 6, and A = pS = y = 2 - cy - p are in their customary notations. The functions q( {) and 1( {) are smooth functions of {but equal to unity at { = 0. The variable r is a measure of the distance of a given state from criticality and vanishes at a critical point while 0 measures the distance along a contour of constant r. The singular behavior of a certain thermodynamic property at a critical point is determined by how the property is expressed in terms of r. The function p ( 0 ) is a quartic function of 0 with only even powers and its exact form can be found in ref 18. Although this model has chosen Schofield’s “linear model” for us,there is nothing in the assumption of the model to preclude the use of other equivalent functional form. The choice is made because there is no known closed-form expression for ir(7,h) in terms o f t and h that is free of nonanalyticities in the one-phase region. As a shorthand notation, partial derivatives with respect to one of the independent variables while the other two are held fixed are denoted by subscripts. For instance, wf = (dw/d{),hand w,, = (d2U/aT2)fh. Then it has been shown that1*
(7)
PI
+ P2 = W h
= (1 - {)P PZ
= {P +
-
(13)
- OQ
(14)
OQ
(15)
with
Q = (Wf + Bpw,) - P ( H+~BfHr)
(16)
where p is the total density (in moles per unit volume), p1 and p2 are the density of the solvent and the solute, respectively. The mole fraction of the solute is then x = P ~ / P=
l+
-~ Q / P
(17)
The definition of x here is different from that in ref 18 where x is assigned to the first component which would be the solvent. As mentioned before, the derivatives of w, exhibit discontinuities of density variables at the coexistence surface and divergences at the critical line. These singular properties come from the derivatives of ir. The second derivative of the thermodynamic potential with respect to T , w,,, is asymptotically proportional to F , which diverges weakly at the critical line where r = 0 while the second derivative with respect to h, whh, is asymptotically proportional to r 7 , which diverges strongly at the critical line. The derivative with respect to r and h, W r h , is asymptotically proportional to The detailed form of these derivatives can also be found in ref 18.
Partial Molar Volume The partial molar volumes of the solvent and the solute are defined as follows: o1 = u - x(au/ax),,, (18) 02
=u
+ (1 - x)(au/ax),,
(19)
where u = p-’ is the molar volume. It can also be shown that
where T
= l({)r
(8)
and n(r,h) is the Schofield “linear model”24which is the same as for pure fluids and is given in a parametric form ir(T,h)
= ?-y(O)
with (24) Schofield, P. P h p . Rev. Lett. 1969, 22, 606.
(9)
(au / ax),, = ~ K T A/ax>”, ~P
(20)
where K T x is the isothermal compressibility at constant composition which diverges weakly near the critical line. However, ( a u / a ~ ) , , ~ remains finite on the critical line because (dp/ax),, goes to zero there weakly. An explicit calculation of (du/ax),, using the LG model is also informative of its behavior near the pure solvent’s critical point. For the calculation it is convenient to convert the variables u, p , and T into p, w, and B which are natural to the model and then calculate the partial derivative by using the
5924 The Journal of Physical Chemistry, Vol. 90, No. 22, 1986
where { =
technique of Jacobians. Then we have = -p-2(dp/dx),,
(du/ax),,
= -p-td(p,w,B)/d(x,w,B) = p-2K/G ( 2 1 )
where K and G are the Jacobians given by K
d ( ~ ~ , B ) / d U , 7 ,=h )(wf+ Bp,)%h - P ( w g
+ Bpw,h) (22)
G
d(x,w,B)/a({,r,h) (P
+ Q) +
- {)P-2(wf
'
-
-
lim ( d u / d ~ ) , = ~ v K , lim ( d p / d ~ ) , ~ x-0
(24)
where KT is the isothermal compressibility of the pure solvent which diverges strongly near the critical point. Then (18) and (19) yield respectively fit"
E
lim 0, = u
lim O2 = u[l
x-0
+ KT lim ( d p / d ~ ) , ~ ]
(26)
X-0
If the critical point of the solvent is now approached, 0,- becomes the critical molar volume, uc, as one might expect whereas f i 2 diverges as KT to +-m or --m depending on the sign of limxd0 ( d p / d ~ ) at , ~ the critical point. The quantity ( d p / d ~ ) can , ~ also be evaluated from the LG model: (dP/dX),T = B'(dw/ax),B = Bl[d(w,p,B)/a(x,p,B)I = K / F B (27) where K is given in (22) and F
d(x,p9B)/a({,7,h)
(1 + Q / P ) w h h + K1 - {)p-'(B{)2[w&hh
- (wdz)*] (28)
Only the two most dominant terms in F a r e given in (28) but the complete expression for F is also given in eq 2.31 of ref 18. Equation 17 shows that taking the limit x 0 is the same as taking { 0 and we have
-
-
lim ( ~ P / ~ x ) , T B 1 ( w f + B P r ) / ( l + Q/P)lr=o
X-0
2
= h = 0. Defining A to be this limit we have
Vim (dp/ax),,lC = [ B ' ( w f + B p 7 ) / ( l + Q/P)I" X - 4
(30)
where the superscript c denotes that the expression is evaluated at the solvent's critical point. Because the neglected terms in (29) are proportional to r y , the expression becomes exact when evaluated at the critical point where r = 0. It can also be shown that the right-hand side of (30) is related to the initial slopes of the critical line and the vapor pressure curve
where IcL and lo, denote that the derivatives are taken along the critical line of the mixture and coexistence curve of the pure solvent, respectively. Depending on the relative magnitudes as well as the signs of the slopes, A can be either positive or negative leading 02" to diverge to +m or --CD at the solvent's critical point. Krichevskii stressed the importance of (3 1) to the understanding of dilute-mixture effects.I5 Path 2. On the critical line we have r = 7 = h = 0. Therefore from (21) to (23) we have (du/ax)pT/CL = [((I - ()(wf
(29)
Equation 29 is approximate because only the dominant term in K containing Wh), is retained. The notation means the expression is evaluated for { = 0. The sign of 6," at the critical point can be obtained by evaluating (29) at the solvent's critical point
(32)
+Bpr)l-lIr=i=h=O
As x approaches zero along the critical line { also goes to zero and x becomes approximately proportional to { when {
