Henrv C. Thomas University of North Corolino at Chopel Hill Chapel Hill, 27514
1 I I
Wet Mica: Good Thermodynamics but Bad Statistical Mechanics An assembly of first approximations
I t is often necessary and convenient, though scarcely ever sufficient, to start a course in elementary statistical thermodynamics with a review of more or less pure thermodynamics. It is horing both to the lecturer and his audience, and therefore of little use, merely to go over the material to he found in most beginning texts on physical chemistry. Another approach which has been tried with some success is to work out a problem sufficiently complicated to illustrate a good many aspects of classical thermodynamics. The details of the calculations can be made assigned work, thus giving the lecturer opportunity to go more deeply into points usually not stressed in an elementary course. There follows an outline of such an experiment recently tried. It is a fact ( I ) that if one adds ground muscovite to water the pH of the suspension rises with increasing ratio weight mica to weight water until about 0.01, when the pH reaches 9.7 and thereafter remains constant. A freshman chemistry explanation of this is simple enough: the hydrogen ion concentration gets so low that it is no longer ahle to push the potassium off the mica surfaces. This much is surely correct but is somewhat qualitative. I t does, however, immediately lead to the presumption that the variance of the system (in the sense of the Phase Rule) has decreased by a t least unity and so the number of phases has similarly increased. It follows that at a given chemical potential, say for the acid in solution, there should he a sharp transition from a single potassium-hydrogen "surface solution" to two such solutions of different but fixed compositions. Further addition of mica to the water-mica slurry merely alters the proportions of these two phases. Can we make these ideas quantitative? The following discussion certainly accomplishes this much a t least. Whether it is correct for muscovite is quite another matter. First let us say something about muscovite, which is one of the mica family with the approximate composition KHzAIdSiO& plus important "impurities." The crystals are actually monoclinic but with an axial angle of very nearly 90". Most important for us, and for users of mica, it has perfect hasal cleavage. I t is almost certainly true that the potassium is present thanks to substitution of dipositive ions (Mgz+ or Fez+) for aluminum or tripositive ions (A13+ or Fe3+) for silicon. The net effect is a stack of slightly impure aluminosilicate sheets held together by potassium ions strongly held through an ionic hond to one sheet and loosely hound by van der Waals forces to the next. Hence the perfect basal cleavage; the van der Waals bonds are easily broken. Ground mica, then, consists largely of flat surfaces carrying localized negative charges, saturated, in muscovite, by potassium ions. Placed in water these surfaces hydrolyze; potassium goes into solution, hydrogen ion replaces it, and the solution becomes basic. Now let us try to he quantitative. In what follows there will he a series of questions to be answered or statements to be proved enclosed in numbered braces. These might be set as exercises for the students. 592 /Journal of Chemical Education
Consider a flat surface with localized negative sites saturated by a mixture of two kinds of positive, singly charged ions. Suppose this surface is immersed in a solution of these ions with a common cation. Such a system might be that produced by wetting muscovite with water. We shall a t once forget any attempt to he realistic and calculate details of an ionic atmosphere near a mica surface. This would carry us very far afield, and, while we might get a better description of mica, we certainly would not be able to give a brief review of thermodynamics. Our discussion will in essence he an enlargement in some respects of that given by Fowler (2) of critical adsorption phenomena. Having given up any attempt to calculate details of an ion atmosphere, which would demand analyzing the effect of forces varying as l l r z between charges and 112 hetween dipoles (long range forces), we have limited ourselves to the consideration of unspecified (and very short range) forces between the adsorbed ion and the surface. These forces will produce energies ascribable to the ions in some way dependent upon the composition of the surface. As we are free to do, let us suppose that these energies are measured from those of the ions in very dilute solution. Generally these energies will be negative quantities. 1. {Why are we free to do this? And why will these energies generally be negative quantities?l
Since we have supposed that these energies are not constant and since we refuse to analyze them in detail, we are forced to assign to them a t least the simplest forms which satisfy the Gihbs-Duhem equation. 2. lWhy?l 3. lDerive from first principles the Gibhs-Duhemequation.{
If the partial molecular values of the energies are denoted by El and Ez, we must have sdE,
+
(1
-
x)dE2 = 0
(1)
wherk x is the fraction of sites occupied by ions of kind 1 and 1 - x the fraction occupied by ions of kind 2. 4. {Whatmust be said about temperature and pressure if eqn. (1)is to he eorrect?l
This equation is, of course, identically satisfied if El and Ez are constants, a case leading to little of interest. Since we are considering the ion-site combination as a neutral species, we suppose that the partial energies can he expressed as truncated power series in the mole fractions. There is little justification for this supposition except that it is known to give a good approximation to the behavior of mixtures of electrically neutral liquids, the socalled regular solutions. We write
E
=
E
-
El*+
I
-
)
+
1
- z )
(2)
5. {Showthat if these two expressions are to satisfy the GihbsDuhem equation we must have the following relations: a, = az = 0 and bl = bz. Add another term to each of the series (eqns. (2) and (3)) and deduce the necessary relations.1
We first investigate the condition for equilihrium hetween our somewhat synthetic mica and a solution in which it is immersed. This condition is given by the equality of the chemical potentials ,,,surf
Replacing b~ and bz by w, we must have
E,
= E,,
E,* + w(1 - x)' E, = E,,- E,* + wx2 -
p,ao 8"
(15)
(4) (5)
Here El1 and Ezl are the energies of surfaces fully loaded with the corresponding ions and &*, Ez* are the energies of these ions in "infinitely dilute" aqueous solution. The fact that we cannot measure these energies individually need not disturh us, a s will he seen. This gives us for the total energy of a surface carrying nl adsorbed ions of kind 1and nz of kind 2
measured from the energy of the aggregate of ions in dilute solution. 6 {Review the i u h p t of partial mdal quant~tresand Euler'i themem on homugenews functiunq of various drgrees. Show that tf L.'aufthe fmt degree in ns and m then r q n . ( 6 1muct bc true I
We now make an approximation which may indeed he quite rough hut which certainly would he quite difficult to improve significantly. We suppose that we can calculate the Helmholtz free energy, A = E - TS, of the surface by using the ideal entropy of mixing for the particles on the surface, again measuring from dilute solution
n2(Z1, -
=
Si *)
10. {Derivethese from Gibhs' fundamental criterion (6E),,,,,
=
0 and in the process review thoroughly Lagrange's method of un-
determined multip1ien.l If eqns. (15) and (16) refer to charged ions in the solution, as we suppose, we have here the chance of getting into a pointless difficulty (pointless in this context) about the impossibility of measuring single ion chemical potentials. We avoid this by the following gambit. Suppose that species 3 is the anion in solution and that all ions are singly charged. Then the difference between eqns. (15) and (16) can he written - ,,9'"'' = (,;"I" + ,PI") - @*'"I" + ,,3'01") (17) PIsYrr Writing p = p* + k T In yc for the ionic chemical potentials in solution, in which y is the usual activity coefficient which goes to unity a s c goes to zero, we get the "equilibrium constant" expression xf,.az
=
(1- x ) f * . a ,
exp
{(F:
- TS,,) -
,
-T
f?T
+*
-*
It is seen that k T In K(T) is the difference between the free energies of forming the two pure surfaces (per ion) when the ions are taken from very dilute solution. In eqn. (18) we have written for brevity
(7)
7. IHere indeed lies a story! Review the laws of ideal gases and in particular Dalton's law of partial pressures. Really pure thermodynamics couldn't produce such an explicit formula as eqn. (7). What additional information is necessary in order to show it to be true for ideal gases?/ We then get for the Helmholtz free energy of the surface 11. {Review the thermodynamics of electrolytic solutions. Get eqn. (18)from eqn. (171.1
In order to discuss the equilihrium and stability of the surface we need the chemical potentials of its component species as given by the prescriptions
in which pl* = &* - T&*, p2* = Ez* the reference terms we find
- T S ~ * Dropping .
Since our expression for AbUrris given both in terms of x and of n l and nz, it is convenient to have the formulas
Equation (18) might actually he roughly applicable to mica suspended in an aqueous solution provided the mica remained in a single homogeneous phase. By experiment ( I ) and, as we shall see, according to our rough theory it does not necessarily do so. The condition for chemical stability is (dpl/ax) > 0. 12. IReview the conditions for the mechanical, thermal, and chemical stability of a thermodynamic system./ From eqn. (11) we get
The surface will he unstable if (apl/iix) < 0 or w > [kT/ 2x(1 - x ) ] . If w is positive there will therefore be a region in which the surface is unstable. For the transition from stahility to instability = ,!+ !
ax
8. [Derivethese.{ 9. {Do the differentiations which produce eqns. (11) and (12).
Verify that the results satisfy the appropriate Gibbs-Duhem equation, which at least shows that our expressions are thermodynamically "legal" even if they don't completely describe wet mica.1
and
kT x(l - X I = 2u;
(24) (251
Since x (1 - x) < 1, we must have w > (kT/2) which implies that there must he some critical temperature above which the surface is always stable Volume 50. Number 9, September 1973 / 593
To find the value of this temperature we notice that a t the critical point we must have both (apl/dx) = 0 and (azpl/ax2) = o. 13. {Give a reason why the second of these relations must be true.1
These give
and
kT
= 2w xi
from which we find that
Our description of mica may be illustrated in terms of what might be called an exchange adsorption isotherm. From eqns. ( l l ) , (12), (15) and (16) we get
Here we have put yl = 1 and T = r8 = (rw/2k). The last term on the.right is a constant to be determined by experiment. Plots of this isotherm for various values of r are given in the figure. Equation (18) can he written, for dilute solutions
In the hydrolysis reaction KMi
+
H,O
=HMi +
OH-
+
K'
we have fraction HMi = s = [OH-] = K , I o [ ~ + ]
where a is the "exchange capacity'' of the mica suspended in a liter of solution: This latter essential piece of information is not available. Among other things it would depend on the fineness of the mica flakes. Thus
E
- 2pH +
2wK.
10 pH
+ constant
(29) (30)
if [H+]>> (K,/a). The results of Garrels and Howard ( I ) . for low values of [K+], where presumably we are dealingkith a mixed H+ - K+ mica in a single phase, do indeed bear a more than entirely qualitative resemblance to the requirements of eqn. (29). The slope of the log [K+i versus pH plot is a little greater than -2. It is just this region which cannot be explained for a two-phase mica mixture, as was pointed ,out by the authors. Thus our review of thermodynamics has the added attraction of not being entirely a fiction. Appendix
Suggestions for the Questions 1. We can measure only energy differences, so where we put the zero is immaterial. Provided we do not shift the zero in the middle of a calculation, differences will be fixed. If mica is stable in water, and i t is, its energy of formation as specified will be negative unless there is an over-riding entropy effect.
594 /Journal of Chemical Education
Adsorption imtherms as functions of T = r0.
2. Otherwise the discussion would be thermodynamically inconsistent, and such consistency is a minimum requirement for any theory of macroscopic systems. 3. Start with d E = TdS - W V + Zrjdn, and increase the size of the system a t constant temperature, pressure, and all chemical patintials, getting E = TS - P V + Zpin,. Differentiate this completely and subtract the starting equation. 4. They must be held fixed. 5. It will he enough to indicate the answer to the first part. From eqn. (1) we must have d d x-(a,(l-x) dx bl(l-x)'I + (1-x)-(a,x b,x2/ = 0 dx Carry out the differentiations, arrange in powers of x, and equate the coefficients of each Dower to zero. If this sounds susoicious. rawider: suppose u e h a w that oa + n ~ xT a d T a , r 3 = 0 fur nll r bttwecn and irlrludinr 0 a n d 1 Put r = 0,then oo = 0. nilwrenriare once nirh respect to x and put x = 11. then a , = 0 And so on. 6. See Denbigh (3), p. 99: Kirkwood and Oppenheim (51, p. 9; Klotz ( 6 ) ,p. 243; see especially Wall (7), p. 177. 7. See Denbigh (3), p. 118; Kirkwood and Oppenheim (5), p. 97; Wall (7),p. 120,348, 350.
+
+
I-x etc. n, n2 9. The only thing new here is to show that xd ( k T In x ) + (1 - X ) d {kTln(1 - x)l = 0, which itis. 10. See Denhigh (3), p. 182 ff; for those who enjoy very difficult reading, see Gibbs ( 4 ) . p. 62 et seq.; also Kirkwood and Oppenheim (5),p. 116ff; Wall ( 7 ) , p. 211 ff. 11. See Denbigh (3), p. 302; Kirkwaod and Oppenheim (5), p. 189ff; Wall (7). D. 399. Equation (18j comes from eqns. (11). (12) and (17) and p 1 n = #I* k T l n ye by straight-forward substitutions. 12. Those who like atrociously difficult reading may try the source, Gibhs ( 4 ) . p. 70 et seq.; but i t might be better to settle for Kirkwwd and Oppenheim (51, p. 59. Most texts seem to avoid the subject. 13. The argument is a generalization of the discussion of the critical point of a van der Waals gas, to be found in nearly every texthook on physical chemistry.
--
+
+
a.
Willard, "Collsetd Works. Thermcdynamiea," Vol. I. Yale Univusity
Literature Cited
(41 Gibbs,
(1) Garreis. R. M.. and Horard, Petor, Pm.Silfh Nati. Clay Conference, Pexamon PI-, New York 1359, pp. 68-88. (2) Fowler. R. H., "StatisticalMechanii.). CsmbridgePress. 1936, pp. 82-36,
(6) Klotz. Irving M.. "Chemiesl Thermcdynsmies. Basic Theory and Methads," revied
(sE y b i ~ h Kenneth, , "The
Re% New Haven, 1948. (5) ~ i i k d John . 0.. and Oppnhcim, Irwin, "Chemical Thcrmcdynamiea,"
Ptineiplol of Chemical Equilibrium," Cambridge Uni~er-
a t t y Prrss. New York, 1968.
McGraw-Hill Bmk Company, Ine., NavYork, 1961. 4..W.A. Benjamin. Inc. New Yark. 1964. (7) Wall. Fredetiek T.. "Chemical Thermodynamics." W. H. Freeman and Company,
San Raneiseo, 1965.
Volume 50, Number 9, September 1973 / 595
