Critical Behavior in the Solubility of Ionic Compounds Gregory D. Gillispie North Dakota State University. Fargo. ND 58105 The dissolution of ionic compoundsin pure solvents shows a form of critical behavior when the ion activity coefficients are described by the Debye-Huckel limiting law (DHLL). For each electrolyte class there is a critical value of K,, the equilibrium activity product. Any electrolyte whose K, is greater than the critical value for its class is infinitely soluble. The critical activity coefficient is a universal constant, e-2, independent of electrolyte class, solvent, and temperature. However, the critical behavior is manifested only at concentrations greater than the range of applicability of the DHLL. A more accurate treatment of activitv coefficients eliminatescritical behavior for electrolytes yielding ionsofcharge 3 or less (for water at 25 "C as solvent). A 1-4electrolyte still retains vestiges of critical behavior, which could be manifested in either an unusually sharp temperature dependence of solubility or in a salt actually having a dual solubility. Origln of the Problem The process of an ionic compound dissolving in a pure solvent is a convenient way to introduce Debye-Huckel activity coefficients.' The activity coefficients depend on the (molality-scale) ionic strength, which is directly proportional to the stoichiometric molality of the electrolyte. In turn, the equilibrium solubility S., depends on the equilibrium activity coefficients. Thus, an iterative approach is necessary to find S, once a value for K. is specified. Iterative solutions for the equations that arise from this type of problem are easily carried out on hand-held cal~ulators.~ Consider the simplest case of a 1:l electrolyte. The equilibrium activity product, K, is written in terms of S,, and the mean ionic activity coefficient K , = (r+)'Siq
(2)
where I is the molality scale ionic strength of the solution s the solvent bulk dielecand A is a constant that d e ~ e n don tric constant and temperature but is independent of the nature of the ions dissolved in it. For a 1:l electrolyte the ionic strength a t equilibrium is equal to S,, and eqs 1and 2 can be arranged in the form log s,, = 112 I O ~ K+AS:? .
-
Graphical Demonslratlon of Critical Behavior Consider a salt dissolving in pure solvent to give u+ cations of charge z+ and v- anions of charge z- per formula unit of salt dissolving. The generalization of eq 3 is log S:f = (2v)-' log [KJV;]
(3)
A reasonable starting guess for S., is KiiZ,which corresponds to approximating the activity coefficients by unity. The pedagogical content of the exercise is enhanced if the chosen value for K. leads to a converged value of y, significantly different from unity; this emphasizes how much in error the simple expressions presented in general chemistry can be. Of course, if K, is too large, the solution to eq 3 corres~ondsto a situation in which the Debve-Huckel limiting law is of dubious validity. Other treatmints of ion activity coefficients, such as the excended Debve-Huckel or Davies equations; can then be introduced. A few years ago I assigned as a homework problem the case of a 2-2 electrolyte with K, = 4.70 X If the mean
+ (~lv)[1/2uz+lz~1]~~~S:f (4)
which uses the relationship
(1)
According to the Debye-Huckel limiting law (DHLL), the mean ionic activity coefficient is given by logy* = -Az+lz_lll"
activity coefficient is approximated by unity, the equilibrium solubility S, is 0.00686 molal and the ionic strength 0.0274 ( I = 4S., for a 2-2 electrolyte). This gives a first approximation of 0.460 for the activity coefficient, significantly different from unity as was desired. Clearly, the next approximations to S., and I will be greater than the first estimate, and y* drops below 0.460, making S,, and I still greater in the next iteration. I t was obvious that the converged solution would give a much greater S,, than the initial a ~ ~ r o x i m a t i o nProbablv . this convereed solution would be-well beyond the range (;f validity of the DHLL, hut the main point of the exercise would he hrouaht home to the students.-Secure in my intuition that the ite'ative process is convergent, I stopped my own analysis a t this point. A day or so later one of my students related his inability to arrive a t a mathematical solution. A quick glance at his work revealed no obvious flaw in his procedure and I told him I would look it over more carefully later. When "later" came, i t was much to my consternation to find I had the same trouble. The iterative solution always diverged in the fashion S., -. Whv the mathematical Drocess was a ~ ~ a r e n t l v pathdfogical (it skems that surely there must be a ioiubilit;) was mysterious, but the diveraence was unauestionable. The next approach was a graphicz solution of eq 3.
I = 112S,,"~+l~~l
+
(5)
and the definitions v = v+ v- and (v*)" = [(u+)'+][(v-)"The right-hand side of eq 4 is a linear function of S,, , whereas the left-hand side is a monotonically increas~ng function of S$, so a plot of the form shown in Figure 1 is indicated. Changes in the value of K, move the straight line graph of the right-hand side up or down but with no change of slope. Depending on the value of K,, there are three classes of behavior: no point of intersection of the two curves, a single point of tangency, or two points of intersection. These three cases apparently correspond to no solubility (actually infinite solubility), a single value of solubility (as is expected), or two solubilities. For the 2-2 electrolyte example I had assigned to my class, the chosen value of K , was just large enough that there is no point of intersection; the critical value of K,, i.e., the one that gives the point of tangency, is 3.80 X l O W (vide infra). This analysis led to some interesting speculations. Suppose that the K, values for ionic compounds in a given electrolyte class were more or less randomly distributed. Then there would be a gap between the measured solubili-
,k
'
Levine, I. N. Physical Chemishy: McGraw-Hill: New York. 1978; Chapter 10. These equations can also be readily solved with Lotus 1-2-3 and oiher spreadsheet programs. Volume 67 Number 2
February 1990
143
of the solid salt is taken as unity. After algebraic manipulation the following expression is derived: ( ~ , , ~ r ) - ' ( a G l a S ) ~ , ,-h[K&,)"] =
+ v In?, + v InS
(8)
where M,,I is the mass of solvent in kilograms and S is to he interpreted as a molal concentration variable that increases as more salt enters into solution. Now a t equilibrium G is minimized so that if one sets the left-hand side of eq 8 to zero, eq 4 is recaptured as expected. However, aG/aS = 0 applies equally well to either a minimum or maximum in G. The apparent higher solubility solution t o e q 4 corresponds t o a maximum in the free energy profile and does not therefore represent an equilibrium situation.
Fngure 1 U.spncal melhod of mlul on to eq 4 Aosclssa is square r w t of S ,, The snalght loner rspressnl me rognlhana side of eq 4 for ddferenl va uer of K, Curve€ lone represents log S = low sol~ot9'' % . = ' hqh mlub!llty". Sc = "critical SOIU~III~Y.'
$2
.
ties for compounds with K. < K.,c and the very much greater solubilities for those with K. > K.c (assuming other factors will intervene to prevent truly infinite solubilities). Unfortunately, not enough data are available to test this hypothesis. Even more intriguing was the notion that a salt could have two distinct solubilities. The paper by Hugus and Hen@ in this Journal has its eenesis in similar circumstances. They showed that, in the progressive addition of one ionic solution to a fixed volume of another, there can he two points of incipient precipitation or dissolution provided the solution concentrations fall within a certain range fixed by the value of K,,. This ohservation was prompted by an exercise posed by one of the authors in a general chemistry class. Whereas the Hugus and Hentz result is arealone, the dual solubility finding above is chimerical, although the reason whv. mav. not be immediatelv obvious. Instructors might consider challenging their heit students to provide the explanation. I show below that the apparent higher solubilty does not represent an equilibrium situation. The critical solubilitv behavior is a consequence of extending the DHLL into a concentration region where the DHLL is no longer reliable. However, when a more accurate treatment of ion activity coefficients (i.e., one that covers a wider range of ionicstrength) is used, I find that in certain circumstances it actually is possible for a salt to have a dual solubility! Free Energy Solublllty Proflles T o see the fallacy of the dual solubility illusion in the example above, it is necessary to treat properly the free energy changes associated with the isothermal, isobaric process
for which the differential element of free energy can be written
re resents the chemical potential of the ith spewhere cies and dnpTis the change in the number of moles of that species in theath phase. Thechemical potentialsare written in terms of standard chemical potentials and activities, and the activities of the ionic species are represented as producs of molal concentrationsand activity coefficients;theactivity ~~~
~
~
~
-
Hugus, Z 2.; Hentz, F. C. J. Chem. Educ. 1985, 62, 645.
144
Journal of Chemical Education
Mathematical Analysis of Crltlcal Behavlor One can draw an analogy between the solubility behavior of ionic compounds and the P-V isotherms for gases described by the van der Waals equation of state. The free energy profile of a salt with K, > K,c is a monotonically decreasing function of Sjust as the pressure of a gas with T > Tc is a monotonically decreasing function of V. The free energy profile of adissolvingsalt with K, < K,,c as afunction of increasing S from S = 0 shows first a minimum, then a maximum, and then monotonically decreases. At K. = K,,c the free energy profile shows only an inflection point. (The free energy profiles, plots of G vs. S, are generated by numerically integrating eq 8; a spreadsheet program makes this easy.) Mathematically the critical pressure and volume are found from an equation of state by requring aPIaV = a2Pl aV = 0 for (P,V,T) = (Pc,Vc,Tc). We now apply a similar analysis to the solubility problem. The second derivative of G with respect to S is written in the form
and when the left-hand side is set equal to zero, an expression for the critical solubility is obtained Substitution of the expression for the solubility in eq 4, followed by rearrangement yields pK.,c, the negative log of the critical equilibrium activity product. Finally, the combination of eqs 5 and 10 with the DehyeHuckel limiting law expression yields the critical mean activity coefficient. Curiously, this turns out to he a constant independent of electrolyte, solvent, temperature, and anything else: Y+,C
= eC2 = 0.1353
(11)
The critical constants for the various electrolyte classes are summarized in the table. From the entries in the table it is clear that the critical region corresponds toconcentrations outside the range of validity of the Dehye-Hurkel limiting law. It is generally considered that the limiting law is reasonably reliable for ionic strengths up toabout 0.001 m. Even in the most .....- -favorable cases of the 3-3 and 4-4 electn)lvws, the critical ionic strength is still ahove 0.01 m (for water as solvent a t 298 K). Ion activity coefficients can be extended to much higher ionic strengths with the Davies equation. Unfortunately, the greater mathematical complexity of the Davies equation precludes an analytic approach to the type given ahove. However, a graphical analysis is still possible. The mean ion activity coefficients are written in the form
-
Fioure 2. Plot of ihe riaM hand side of so 13 for 1-1. 2-2. 3-3. and 4-4 electrolytes as a tunctlon of the logarnrhm of ionlc shength The A parameter value IS 0 509, appropriate for water at 25 ' C
Figure 3. eaphicsl so urim of eq 13 fa &4 electrolyte dissolving n water a1 25 ' C . The mree straight lines correspond to pK, = 8 4. 7 8, and 7.2 from top to bottom.
Crltlcal Solublllty Data lor Electrolyte Clasws
and 8.4. The lower and the higher values each admit only one solution (againafreeenergy minimum). Note, though, that a difference of only 1.2 pK, units leads to an ionic strength difference (and corresponding soluhility difference, of more than two orders of magn~tude!The soluhility is an exceedingly sensitive function of K, in this range. Actually asimilar sensiti\,itv i3 oresent for the 3-3 and 4-2 electrolvtes, albeit to a significantly lesser extent. Over a very narrow range of pK, there are three solutions LO eq 13 (see middle straight line of Fig. 3). The solution at intermediate ionic strength is obviously a free energy maximum. hut the other two solutions do reoresent localminima ~~~~~~~~, in the freeenergy. It appears that a dual solubility isactually oossihle! It should be recoenized that this behavior is occur&in an i ~ n i c s t r e n ~ t h r e i i m ewhich in theDavies equation is reasonahlv accurate. How might one experimentally search for confirmation? We earlier drew the analogy hetween critical behavior in gases and critical solubility behavior. The analogy is imperfect since the controlling variable in the solubility case is K., which cannot he adjusted at will in the same way that temperature can be for measuring P-V isotherms. Also, the A narameter of the activitv coefficient exoression is itself a Tdependent quantity. No doubt, however, conditions obtain where the T dependence of K, is significantly greater than that of the A parameter. In such a case, temperature changes could bring a salt from the subcritical to the supercritical soluhility region (or vice versa). The experimental manifestation would he a soluhility which increases unusually rapidly with T. Of course, one would also need the pK, value to fall in the proper range. hea analysis h&e has ignored ion pairing, which is expected to be substantial for highly charged ions in aqueous solution. This does not. howwer.. neeate the oossibilitv of dual affects the relationihip betwken ionic soluhility. Ion streneth and analvtical concentration but the activitv roduct and activity coefficient expressions leading up & kq 13 do not require modification. On the other hand, almost any solvent other than water has too small a n A parameter for critical behavior even for a 4-4 electrolyte (of which examples are not in abundance).
Electrolyte Type 1:l 2:l 22 3:l 3:2 33 4:4
+ 1 2 4 3 6 9 16
v
(u#
2 3 2 4 5 2 2
1 4 1 27 108 1 1
&(m)
PKG
2.911 0.243 0.0454 0.0539 0.00539 0.00399 0.000711
0.81 3.84 4.42 7.12 13.65 6.54 8.03
~~
and these are substituted into eq 8. We are interested in the extrema of G and so aG1aS is aeain set eaual to zero. After substitution of eq 5 to express S in terms of 1 and further algebraic manipulation, the following equation is obtained:
where
Equation 13 is t o he compared with eq 4, and the graphical analysis proceeds in a fashion analogous to that in Figure 1. The rhs of eq 13 has the same shape for every electrolyte class, but the charges on the ions enter in multiplicative fashion, as is illustrated in Figure 2. If the abscissa is written as log I, the graphical solution is easily demonstrated. The lhs of eq 13 appears as a straight line of unit slope while different choices for K, move this diagonal line up and down. Intersections of the straight line with the plot of the rhs of eq 13 correspond to extrema (either maxima or minima) in the free enerw. .., For the less highly charged electrolyte classes, the derivativeof the rhs of eo 13 with resoect to log I isalwavs less than unity so there can be only onesolution of eq 13;correspondingly, there is a single extremum (a minimum) in the free energy profile and solubility is a well defined concept. For the 4 4 electrolyte, however, multiple extrema in G are possible provided K. exceeds a critical value (near pK, = 7.8) and the A parameter is as high as for water at 25 OC. The three straight lines in Figure 3 correspond to pK, = 7.2,7.8,
~
-
Acknowledgment
I wish to thank my colleaeues for their patience and willingness to entertain my early speculatiok on why a salt might have two solubilities. An anonymous referee provided two very careful readings of the manuscript and detected several typographical errors. Volume 67 Number 2
February 1990
145
