Feb., 1955
INTERACTION BETWEEN SOLUTES IN DILUTETERNARY SOLUTIONS
BOILINQ POINTELEVATIONS OF WATER HELDIN Relative vapor pressure (2)
0.970 ,865 .756 .557 ,404 ,220 ,082
Corresponding capillary radius,
Relative vapor pressure
A. 338.6 70.5 36.8 17.5 11.3 6.8 4.1
THE
TABLEI VARIOUSPOROUS BODIES AT DIFFERENT RELATIVE VAPORPREBSURES
(z')
Theor.,b.p. elevation, AT
0 994 ,910 ,836 .708 .574 ,414 ,259
0.2 2.7 5.2 10.3 16.7 27.6 45.4
The elevation in boiling point ( A T ) for a given value of relative vapor pressure can be obtained from the Clausius-Clapeyron equation provided A T is small so that the heat of vaporization ( A H ) can be taken as constant. However, for a large range of A T , AH is no longer constant. From the variations of AH with temperature in the case of water, the following relationship can be deduced
-
-log102 = 1.523 X X AT Po 0.501 X X AT2 -1.201 X
161
X A T r (2)
Substituting the various values of z' for p / p o , the corresponding values of A T were calculated. Results and Discussion The boiling point elevations of water held in various adsorbents a t different relative vapor pressures are given in Table I. It is interesting to note that the values obtained a t a given relative vapor pressure are almost identical in the case of all the adsorbents although they were of different nature and were known to differ appreciably from one another in their moisture content-vapor pressure
Determined boiling point elevations in the case of Silica Sugar BenClaytonite loam gel charcoal
1.0 3.0 6.7 10.0 15.5 23.5 35.0
1 .o 3.0 5.7 9.8 11.5 20.0 35 0
1.0 4.1 4.7 11.2 15.0 22.0 35.5
1.0 3.5 5.5 10.3 13.5 24.0 35.0
relationships. This shows that the boiling point elevation of the sorbed water, just like its freezing point depression, depends largely on the radius of the capillary pores in which it is condensed. The theoretical values are also included in Table I. The agreement between the determined and the calculated values is seen to be quite close up to the capillary pores of radius 17.5 8. in the case of all the adsorbents and except for sugar charcoal it is not so b%d even in the case of the nest lower size, viz., 11.3 A. The validity of the agreement decreases as we approach finer capillary sizes. This shows that the Kelvin equation is probably not applicable in the case of capillaries a few molecular diameters in width.2 The probable reasons for this have been already discussed.' However, the fact that the results follow the Kelvin equation for an appreciable range of vapor pressure shows that probably beyond a monolayer all liquid held up in a porous body is largely due to capillary condensation. ('2) C. Pierce, J. W. Wiley and R. N. Smith, THIS JOURNAL, 63, 669 (1049).
ON THE THERMODYNAMICS OF THE INTERACTION BETWEEN THE SOLUTES IN DILUTE TERNARY SOLUTIONS1 BY HAROLD L. FRIEDMAN Contribution from the Department of Chemistry, University of Southern California, Los Angeles, Cal. Received August 8 , 1964
In widely different ternary solutions it is found that the chemical potential, P I , of one solute depends lincarly upon tho molality, me, of the other if ml and m2 are small. The limiting value of ( & ~ l / b r nas~ ml ) ~ and ~ m2 approach zero equals (dp2/dml),z in the same limit and is the simplest solute I-solute 2 interaction coefficient in such a system. This coefficient may be determined by measurements of the solubility of either of the pure solutes in the ternary solution, plus certain othcr ineasurements on the appropriate binary solution, although not without an extra-thermodynamic assumption.
McKay* has recently shown that many useful thermodynamic relations in ternary and higher order systems may be obtained by the application of the cross-differentiation relations. I n this paper McKay's results are applied to ternary systems which obey a particular activity coefficient law, which applies to such widely different ternary systems as sodium chloride-benzene-water at 25' and carbon-sulfur-iron a t 1500". The relations between various versions of this law are derived, and (1) This research was supported by the U. S. Atomic Energy Coininission. (2) H. A. C. hlcKsy, Trans. Faraday Soc., 49, 237 (1953).
the procedure for the investigation of solute-solute interaction in these systems by solubility memurements is discussed. Preliminaries.-The application of the crossdifferentiation theorem is simplest if concentrations are expressed in mole ratios or molalities. ml and m2will represent solute molalities in a ternary phase and p 1 and pz the chemiml potentia,ls3of the solutes. y1 and y2 are activit,y coe%.cient,sdefined relative to hypothetica.1 1 molal standard states of the respective solutes in the pure solvent. v l and v 2 are the (81
pi =
encrgy.
Pi in G . N. Lewis notat,ion, t h e imrtial molal Gibbs frce
HAROLD L. FRIEDMAN
162
numbers of particles produced by the dissociation of a molecule of the respective solutes a t high dilution. Then for either solute pi
= pio
+ viRT In Yimi
(&.damz)ml= ( W b m d m Z
(2)
u d d In ~ 1 l d r n 2 = ) ~ vdb ~ In wlbml)mz
(3)
and therefore The usual relations between the partial derivatives of several variables, two of which are independent, are also required. I n the dilute ternary solutions discussed here, the solutes will be designated as components 1 and 2, and are both supposed to be at concentrations of the order of one molal or less. The Limiting Interaction Coefficient.-The salting out of a non-electrolyte (component 1) from aqueous solution by the addition of an electrolyte (component 2) is usually found to be accurately described by the Setschenow equation4 (4)
where mls and mfls are the solubilities of the nonelectrolyte in the pure solvent and in solution containing electrolyte a t molality m2.6 Similar solubility relations are shown by many ternary systems of importance in m e t a l l ~ r g y , ~although J in these cases the subscripts of eq. 4 merely designate the respective components, without necessarily implying electrolytic behavior for either. It seems then, that eq. 4 expresses a law which is valid for a variety of ternary systems. I n order to express this law in a more general form, it is convenient t o first rephrase eq. 4. We define a solubility interaction coeficient kl
E
-(a
In ml/bmz)pL
(5)
This coefficient corresponds t o the constant of eq. 4 if pl is the chemical potential of pure phase 1, for then ml = mls’. However, the coefficient may be evaluated a t any other p~1value which it is experimentally feasible to hold constant. The conditions for ICl and the constant of eq. 4 to be exactly the same are discussed in a later section. It is apparent that in any dilute ternary system there is a second solubility interaction coefficient k z = - ( b In mz/dml)p,
(5’)
I n some cases both constants can be evaluated experimentally, and then the relation between them is of interest, as discussed below. By change of variables, eq. 5 becomes or
Thus to the extent that eq. 4 is valid and ml(blnyl/ ?mi),, is small (or independent of m2), p1 depends linearly upon m2, or
(1)
The cross-differentiation relation leads to2
In (mlR/m’ls)= m2 X const.
VOl. 59
kl = ( b p ~ / ~ m ~ ) ~ , / m l ( b p ~ / d m t ) m ,(6)
(bpJarnz),,/vlRT = ( d In ~ l l b m z = )~~ kl[l ml(a In ril~mi)mzl (7)
+
(4) F. A. Long and W. F. McDevit, Chem. Reus., 61, 119 (1952). (5) I n this discussion of solubility,. and those which follow in this paper, it is assumed that the solubility of a solid or a liquid is being measured. If one of the solutes of the ternary solution is a gas when pure, then its solubility at constant partial pressure (constant p) can be obtained and handled b y the equations employed here. However, a more powerful way to proceed would be to measure the solubility of the gas (component 1) over a range of partial pressures as well as a range of mp, to allow the direct calculation from the data of the derivative of (8) or (10). (6) See, for example, L. S. Darken and R. W. Gurry, “Physical Chemistry of Metals,” McGraw-Hill Book Co., Inc., New York, N. Y., 1953, pp. 134, 135. (7) J. Chipman, Disc. Furuday Soo.. 4, 23 (1948).
(8)
(dp1/brn2)~~ = constant
This sort of interaction, with ml fixed either a t ml,, the solubility of 1 in pure solvent, or a t another constant value, has been observed in the dilute region in several ternary liquid metallic solutions at high temperatures8 as well as in ternary amalgams (e.g., Na-K-Hg) at room temperature.9 It can be deduced from the equations proposed to describe the behavior of ternary mixtures of liquid organic compounds1° and ternary regular solutions” that eq. 8 applies to the dilute regions of these systems too. Therefore 4 and 8 are alternative expressions for a limiting law which is applicable t o a wide range of ternary systems. A third expression for this law is given byeq. 94912,13 In
YI = a m 1
+ a12m2 + bllm12 + b ~ r n 2+~c l m m + . . .
(9)
for if m2 is sufficiently small, ( b In yl/bmz)ml is independent of mz, as required by eq. 8. Although 9 is an unnecessarily restrictive statement of the general law, it makes it clear that the simplest interaction coefficient is lini (dplldmz)ml
(10)
mi=O mz=0
which is designated the limiting interaction coeficient, and is equal to vlRT a12if eq. 9 is valid. This limiting interaction coefficient is symmetrical in the components, as may be seen by comparison with eq. 2. Therefore, if eq. 9 is valid, and if an analogous relation holds for the activity coeffcient of the other component In YZ =
a2m1
+
~221722
+ ...
(11)
then nu12
(12)
= vzaz1
Equation 9 also leads to an insight into a class of dilute ternary solutions to which (8) does not apply. If component 1 is an electrolyte and the solvent is water, we may write In
71
=
+
F( d1)
allrnl
+
u12rn2
+ ...
(13)
Here I is the ionic strength. If component 2 is not an electrolyte, this solution will obey the general law, but if component 2 is also an electrolyte, then I is a function of m2 as well as ml, and eq. 13 is no longer consistent with (S), even in the limit of low ml and m2. Ternary solutions in which both solutes are electrolytes exhibit a more restricted sort of regular behavior.I4 An appropriate generalization would seem to be that the law of eq. 8 is valid if components 1 and 2 are solutes a t low concentration in a multicompo(8)J. A. Kitchener, J. 0. Bockris and D. A. Spratt, Trans. Faraday SOC.,48, 608
(1952).
(9) K. Schug and H . L. Friedman, unpublished observations. (10) R. R. White, Trans. Am. Inst. Chem. Engrs., 41, 555 (1945). (11) J. H . Hildebrand and R. Scott, “Solubility of Non-electrolytea.” Reinhold Publ. Corp.. New York, N. Y . , 1952, p. 202. (12) Ref. 6, eq. 15-9. (13) G. Scatchard in “Proteins, Amino Acids and Peptides,” E. J . Cohn and J. T.Edsall, Reinhold Publ. Corp., New York, N . Y . , 1943. (14) H . S. Harned and B. B . Owen, “The Physical Chemistry of Electrolytic Solutions,” Reinhold Publ. Corp.. New York, N, Y., 1950, Chap. 14.
INTERACTION BETWEEN SOLUTES IN DILUTE TERNARY SOLUTIONS
Feb., 1955
163
nent system in which long range interactions are Often it will suffice to prove that the correction not important in determining the activities of term in (15) is negligible compared to kl, and in such both of these components. In comparing eq. 8 with cases the ideal-dilute solution approximation (ie., Henry's law, it should be noted that Henry's law activity coefficients equal to unity) will be adequate deals with the first-order effects of concentration for the estimation of the correction term from measupon activity, neglecting the second-order effects ured concentrations. I n more difficult cases it implied in the use of activity coefficients. On the would be appropriate to formulate the solute activother hand, eq. 8 describes a general behavior of ity coefficients as in eq. 9 and 13, using estimated or the activity coefficients, or essentially second-order independently measured values of the coefficients. effects, arising from solute-solute interaction. Hen- For the ideal-dilute solution approximation ,we have ry's law fails for solutes which dissociate in the ob(bpl/dln mlb)mnb = vlbRT (16) servable low concentration region, while eq. 8 fails dpJdmzb = (&i/dnzza)mBB Xdmzn/dmzb under the more subtly defined condition discussed ( d ~ ~ l / ~ m 3 & ) ~ ~ a d m (17) 3~/dm~~ above. (dpi/dmzB)ma& = -RTMiVZa (18) Aside from the problem of formulating the observed regularity in ternary solutions, there is the (dp1/dm3a)~p= -RTMIvP (19) matter of choosing the solute-solute interaction co- drnp/dmzb = -(vlb X dmlb/dmzb vZb)M3m3a/v38 (20) efficient which is most suitable for comparison from system to system, for the calculation of the entropy M1 and 1cfz are the kilogram-molecular weights of and enthalpy of the interaction from temperature the respective components. Equation 20 is obcoefficient data, or for interpretation in terms of tained from the condition of equilibrium for transsolution structure and molecular interactions. It fer of component 3 from phase a to phase b. Then seems clear that of the constant of eq. 4, kl, this approximation yields (b 1n71/dm2)m,., and the limiting interaction co- kl = -(dln mIb/dmzb) ( v p / v l b ) M 1 X dmza/dmzb efficient, only the last is independent of mb, -MIMamas(dmlb/dm$ - v2b/vlb) (21) a quantity which enters not as an intrinsic property of the system, but as a result of the use of the Evidently the first and second correction terms in solubility method for the investigation of the free (21) arise from the solubilities of components 2 and energy relationships. In the past, careful workers, 3, respectively, in phase a. (2) The limiting value of kl as m2 = 0 may be employing the solubility method, have chosen combined with (b In yl/bml),, according to eq. 7 systems in which all of these coefficients are equal. The solubility method is so often the most con- to yield ( b In yl/dm2)m,s. The subscript is written venient method for the investigation of solute- as ml, in this case to emphasize that the solubility solute interaction that it seems of use to discuss its procedure described determines ( b In yl/dm2)mrfor application t o systems less ideally suited to it. this particular value of ml, i.e., for the saturation Experimental Aspects.-A determination of concentration of component 1in the absence of comthe limiting interaction coefficient in dilute ternary ponent 2. Since the kl employed is valid for m2 = solutions may proceed by measurements of the 0, ( b In yl/bml),, may be evaluated from data for solubility of either of the solutes, depending on the binary 1-3 system. I n general, ( b In yl/ which is most convenient. Assuming that the bml),,=o is a function of ml, so the value a t solubility of 1 is measured, the determination may ml, should be used. If this derivative is not a function of ml, then it reduces to In yl/ml and the corconsist of the following steps. (1) In mils is plotted as a function of m2. If the rection involved in this step (for the variation of data are suitable, and eq. 8 applies to the system, ml in the measurement of kl) reduces to the selfthen the slope (approx. -kl) will be constant from interaction correction of Long and M ~ D e v i t . ~ (3) The limiting interaction coefficient may be m2 = 0 well up into the experimentally accessible range. Omitting this step, as is sometimes done8~l5 determined from (b In yl/dm2)m,sonly if the desacrifices an opportunity to judge the consistency pendence of this quantity upon concentration (ml) of the data. However, the slope of this graph is is known. There is no way t o obtain this dependnot generally exactly equal to - kl, because what is ence if only solubility measurements are available measured in general in a solubility experiment is for the ternary system, so in this step some sort of not the solubility of pure component 1 in the ter- assumption must be made. For one thing, it may nary solution, but the distribution of component 1 be hoped that if ml, is small enough so that the In y1 between two phases, one consisting of nearly pure in the binary 1-3 system may be expressed without component 1 (phase a) and the other consisting the use of terms in eq. 9 higher than the first power, then the higher order interaction terms will not chiefly of component 3 (phase b). By expressing In mlb as a function of mzband pl be important either. It seems unlikely that this assumption will always be valid, but it may be d In mib = (d I n mlb/dmzb)p, dmzb + tested in a particular case by proceeding both from (a In mlb/ddm2bd,ul (14) measurements of kl and k2. In the case of saltwe can obtain a relation between the slope of the non-electrolyte-water mixtures, the limiting interaction coefficient with a given lion-electrolyte graph and kl ought to be an additive property of the ions of the [d In m ~ ~ / d m ~=b ] electrolytes investigated. -ki [ d ~ i / d m z ~ ] / [ d ~1n i / mib]m2b d (15) Relation between the Solubility Interaction Coefficients.-McKay concluded,2 in the present ter(15) A. P. Bltsliuller and H. E. Everuon, J . Am. Chem. Soc., 75, 4823 (1953). minology, that vl1c~= u21c2, but his application of the
+
+
+
+
164
GEORGEJ. LEVINSICAS AND WILLIAM F. NEUMAN
thermodynamic relations was not exact in this instance. The exact relation between kl and k, may be obtained by calculating the limiting interaction
Vol. 69
coeficient from each of them and equating the resulting expressions. This reduces to McKay's relation as ml. and m2. both become very small.
THE SOLUBILITY OF BONE MINERAL. I. SOLUBILITY STUDIES OF SYNTHETIC HYDROXYLAPATITE1 BY GEORGE J. LEVINSKAS~ AND W . F . NEUMAN School of Medicine and Dentistry, University of Rochester, Rochester, N . Y . Received August 6 , 1064
By means of high speed centrifugation, which achieves an efficient separation of the basic calcium phosphates from aqueous solutions, the problem of the solubility of the bone mineral has been reinvestigated using a relatively pure sample of crystalline hydroxylapatite. The steady state was approached sat,isfactorily from undersaturation and supersaturation. A number of variables: time, solidjsolution ratio, pH, Ca/P ratio, etc., were studied. The solubility (in watcr containing traces of chloroform) of hydroxylapatite at 1 g./l., 24", p = 0.165 (NaCl), pH M for Ca and 7.0 averaged 1.23 and 0.711 X P, respectively. This crystalline solid exhibited incongruent solubility, Le., the solutions gave a higher Ca/P ratio than that present in the solid phase in most instances.
Numerous conflicting studies of the solubility of bone mineral have been reported.a-z2 Not one of these studies has offered conclusive evidence that the system under study was a t equilibrium or even reversible. By means of high speed centrifugation, which achieves an efficient separation of the basic calcium phosphates from aqueous solutions, the problem has been reinvestigated. Evidence is presented that the system, under the conditions of the present study, is reproducible and reversible. If not a t final equilibrium, the system a t least attains the same end-point whether approached from supersaturation or undersaturation. Experimental Materials.-Because bone mineral itself is variable in composition and requires drastic treatment for its isolation, a relatively pure sample of commercial hydroxylapatite, (1) Abstracted from the Ph.D. Thesis of George J. Levinskas, June, 1953, University of Rochester. (2) Atomic Energy Commission Predoctoral Fellow, 1949-1952. (3) N . Bjerrum, Paper read a t the Nordic Scientist Meeting in Helsingfors, August 11, 1936. Translated from the report of the meeting, 344. (4) C. Blarez, Compt. rend., 103, 264 (1886). (5) A. A. Browman, Doctoral Thesis, Univ. of Chicago. 1953. (6) F. K . Cameron and L. A. Hurst, J . A m . Chem. Soc., 2 6 , 885 (1904). (7) H. Danneel and K. W. Frohlioh, Z . anorg. aElgem. Chom., 186, 14 (19hO). (8) Y. Eriosson, Acta Ondont. Scandinauica, 8, Supplement 3 (1949). (9) I. Greenwald, J. Eio!. Chem., 143, 703 (1942). (10) E. Hayek, F. Mtlllner and I
