2126
R. H. STOKESAND R. A. ROBINSON
Interactions in Aqueous Nonelectrolyte Solutions.
I.
SoluteSolvent Equilibria
by R. H. Stokes’ Chemistry Departmat and Institute for E n z m Research, UniversQy of W i s w m ’ n , Madison, Wisconein
and R. A. Robinson Natwnal Bureau of 8tandards, Washington, D . C.
(Received Odober 19, 1966)
Solutes which interact with the solvent by a series of solvation equilibria to form species which mix according to the ideal solution law are considered. General expressions relating the solvent activity to the molality and the equilibrium constants are given. Sucrose solutions can be described with considerable accuracy by the assumption of a number of possible solvation sites equal to the number of oxygen atoms in the solute molecule, with a single equilibrium constant given the same value for each site. Mixed solutions of several solutes conforming to this model are shown to obey very simple equations relating the molalities at isopiestic equilibrium between solutions of the separate and mixed solutes. A similar relation between the activity coefficients is given. Examples of systems which conform to these mixture relations are given, and it is suggested that cases of large departures from the relation may be taken as evidence of specific solute-solute interaction.
Introduction While a rigorous treatment of concentrated aqueous solutions remains a distant goal of solution theory, there is great practical interest in such solutions, and some value in incomplete theoretical approaches in which certain major effects are singled out. In the present work we consider one such effect, that of interactions between solute and solvent which may be treated in terms of solvation equilibria. These are likely to be important in aqueous solutions of highly soluble nonelectrolytes such as sugars which have polar groups capable of hydrogen bonding with water molecules. We shall use the concept of the semiideal (or speciesideal) solution introduced by Scatchard2 and used earlier without this name by D~lezalek.~Such a solution is one in which all the departures from ideal behavior are attributed to chemical reactions, and the activity of each actual species in the solution is equal to its actual mole fraction when the chemical reactions have reached equilibrium. The Hildebrand-Scatchard‘ theory of nonelectrolyte The Journal of Physical Chemistry
solutions makes it clear that one may in general expect considerable departures from ideal behavior even when no specific interactions of a “chemical” kind are considered. Nevertheless, SarolBa-Mathot5 and McGlashan and RastogP have successfully treated acetonechloroform mixtures and dioxanechloroform mixtures by the assumption of “semiideal” behavior. For aqueous solutions there are at least two other effects which may be of importance. One is the wide disparity in size between the molecules of water and the sugars which are our chief objects of interest. The other is the “structured” nature of liquid water. The main purpose of the present work is to obtain predictions of the activities in solutions of several solutes from known (1) On leave from the University of New England, Armidale, N. 8. W.. Australia. (2) G. Scatchard, J. A m . Chem. Soe., 43,2387 (1921). (3) F. Dolezalek, 2.Phyaik. Chem., 64, 727 (1908). (4) J. H. Hildebrand and R. L. Scott, “The Solubility of Nonelectrolytes,” Dover Publications, New York, N. Y., 1964. (5) L. Sarolba-Mathot, Trans. Faraday Soc., 49, 8 (1953). (6) M. L. McGlashan and R. P. Rastogi, ibid., 54, 496 (1958).
INTERACTIONS IN AQUEOUS NONELECTROLYTE SOLUTIONS
data for single solutes; it may be hoped that the ignored effects will to a large extent cancel between the several-solute and single-solute systems.
Solvation Equilibria and Average Hydration Numbers for One Solute and Several Solutes Denote the anhydrous solute by So,the monohydrate by S1, and so on. Let A be the average hydration number, i.e., the average number of molecules of bound solvent per solute molecule. This number will decrease with increasing concentration of solute. For semiideal solutions, Scatchard2has shown that ti is given by
h = - 55.51 -m
2127
55.51 - a, MB 1 -a,
+
55.51
+ A,
MC
-
a, 1 -aw
hB
The corresponding result for the mixture is
Multiplying eq 7 by eq 9 gives
mB,
eq 8 by
mc, adding, and using
- + - - =mc I MB Mc mB
a, 1-a,
Equation 10 generalizes to any number of solutes where m is molality and a,the water activity. If we suppose that a solute molecule has n sites where a water molecule may be attached, the ith stepwise hydration equilibrium Si-1
+ HzO
Sj
(i = 1, 2,
. . . n)
(2)
Equation 10 is an alternative and neater expression for the situation discussed by Robinson and Bowere where the isopiestic ratio, R, of the mixture, defined by
has the equilibrium constant
Ki: = Nt/(Ni:-law)
(3)
where N denotes mole fraction. Then by methods familiar from treatments of stepwise metal-ligand equilibria,’ we obtain
A
= u p
(4)
where
z:
= 1
+ Klaw + K1Kzaw2+ . . . + K1K2... .Knawn
M C
R= mB
+ mc
is linear in the fraction of B in the mixture, defined by YB
mB
= mB
+ mc
Where this linear relation, or eq 10, holds, the McKay-Perring’O equation for the activity coefficient of either solute in the ternary system can be shown to reduce to
(5) A-3
=
UY ~
d In a,
=
Kla,
+ 2K1KzaW2+ . . . . + nKIKz...Knawn (6)
It is important to note that according to eq 4,5, and 6,
where r B is the molal activity coefficient of B in a solution of B alone having the same water activity as the mixed solution, and YB is the activity coefficient of solute B in the mixed solution. An equation similar to (12) of course holds for solute C. It can also be shown that subject to eq 11 holding, the activity coefficient of solute L in any mixture is
depends only on the water activity. Now the study of solutions of several solutes is readily made by isopiestic vapor pressure measurements, in which by isothermal distillation B solution of the mixed solutes is brought to the same water activity as separate solutions (13) of the single solutes.* For a given water activity, a,, let be the molality of a solution of B alone, and MC (7) J. Bjerrum, “Metal Ammine Formation in Aqueous Solutions,” that of a solute C alone, while the mixture contains P. Haase and Sons, Copenhagen, 1941. solute B at molality m B and solute C at molality m ~ . (8) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” 2nd ed, Butterworth and Co. Ltd., London, 1959, Chapter 16. Then f i and ~ hc will have the same values in the mix(9) R. A. Robinson and V. E. Bower, J. Res. Natl. Bur. Std., 69A, ture as in the single solutions, since each is determined 19 (1965). by the water activity. (10) H. A. C. McKay and J. K. Perring, Trans. Faraday Soc., 49, We have, from eq 1 163 (1953). Volume YO, Number 7 July 1966
R. H. STOKESAND R. A. ROBINSON
2128
Comparisons with Experiment (a) Single Sugar Solutions. Equations 1 and 4-6 permit the calculation of the molality m corresponding to a chosen a, if the K I are known. This is of little value, since it seems likely that every oxygen atom in a sucrose molecule constitutes a possible hydration site, If we are free to assign 11 Ki values, we can be assured of a good fit to any reasonably simple experimental curve such as the osmotic coefficientof aqueous sucrose. Scatchard2 chose the rather drastic course of assuming that only one hydrate (for preference, the hexahydrate) existed in significant amounts. A more acceptable simplifying assumption is to suppose that all the s t e p wise K , are equal (though statistical objections can be raised against this assumption also). Then we obtain
- - -55.51 m
a,
U
(14)
I-%+:
where
+ Ka, + . . . + (Ku,)" u = Ka, + . . . + n ( K G ) n
(15)
2: = 1
(16)
are functions of a, and the single equilibrium constant
K . For sucrose solutions we put n = 11 and find that K = 0.994 gives a good fit to the osmotic coefficient data, as shown in Table I." It is not advisable to attempt the calculation of a, a t a given value of m for any selected value of K and n. It is better to reverse the calculation to find m a t a given value of aw. The calculation is made as follows. First, a table is constructed of values of a, ranging from 0.98 to 0.84 a t intervals of 0.02, and then of corresponding values of aw/(l - a,) and of -55.51 In a, (= mp). ~
Table I: Osmotic Coefficients of Aqueous Sucrose Solutions a t 25" m
9calod"
9erptlb
1.092 1.189 1.285 1.374 1.450 1.510
1.085 1.186 1.286 1.376 1.450 1.508
a From eq 14-16 with n = 11, K = 0.994. measurements (see ref 11) cp = (55.51/m) In a,
' From isopiestic
-
A value of K is selected and the corresponding values of 2 , U , and u / Z are calculated with the aid of tables. Knowing a / Z and a,/(l - h), we calculate m by eq 1 with h = u / Z . As m is known for each given aat, we The Journal of Physical Chembtry
can evaluate p for each value of m. Graphical interpolation is then adequate to give values of p at round values of m. The calculation is then repeated with different values of K until the best fit is found with the experimental data. Other compounds of the sugar type for which isopiestic data exist are mannitol, glucose, and glycerol. The solubility of mannitol (-1.2 m) is too limited to make a useful test of eq 14. For glucose, there is some difficulty in obtaining good isopiestic equilibrium at low concentrations, which may be connected with the mutarotation phenomenon. However, some results obtained by Dr. P. N. Henrion a t the University of New England, probably reliable within 0.5% in the osmotic coefficient, are given in Table 11. For glucose we put n = 6 and find that K = 0.786 gives a good representation of the results up to saturation. For glycerol, the isopiestic data of Scatchard, Hamer, and Wood12 are available. Osmotic coefficients of glycerol derived therefrom have been calculated to conform with the reference data for sodium chloride13 so that the osmotic coefficients of glycerol are consistent with those of sucrose and glucose. For glycerol we put n = 3, since there are three oxygen sites, and obtain a good fit to 7 m with K = 0.720. Table I1 : Osmotic Coefficients of Glucose and Glycerol Solutions a t 25"
m
1 2 3 4 5 6 7 7.5
-Glucose------CalcdD
Exptl
1.028 1.054 1.080 1.105 1.128 1.147 1.166 1.175
1.020 1.050 1.079 1.105 1.128 1.149 1.166 1.173
--GlycerolCalcdb
1.011 1.021 1.031 1.039 1.047 1.055 1.062
...
Equations 14-16 with n = 6, K = 0.786. 16 with n = 3, K = 0.720.
Exptl
1.012 1.023 1.033 1.043 1.050 1.055 1.060 ...
Equations 14-
The data for glucose show the same characteristic as those for sucrose, a slightly low osmotic coefficient in the region below 2 m but an excellent fit at the higher concentrations. In the case of sucrose, we have tried other values of the number of sites, n, and find that (11) R. A. Robinson and R. H. Stokes, J. Phys. Chem., 6 5 , 1954
(1961). [Note that in Table V of this paper the headings MB and Mc in the first two be interchanged.1 (12) G. Scatchard, W. J. Hamer, and S. E. Wood, J. Am. Chem. Soc., 60, 3061 (1938). (13) See ref 8 , p 476.
INTERACTIONS I N AQUEOUS NONELECTROLYTE SOLUTIONS
after choosing the best value of K the fit to the experimental data is definitely inferior to that in Table I. It is obviously preferable to fix n as equal to the number of oxygen atoms in the molecule, for then eq 14 becomes in effect a single-parameter equation. It may well be asked why the equilibrium constant, K , should be noticeably lower for glucose and glycerol than for sucrose. A fairer question, in view of the extreme oversimplification of the argument, would be why the treatment works as well as it does. We have certainly neglected a great many effects which could be compensated by small variations in K . (b) Solutions of Several Solutes. Equations 10-13 hold for solutions conforming to the present model irrespective of whether the equilibrium constants, K,, are all the same or all different, known or unknown. Data have been published" for mixtures of sucrose and mannitol. Unfortunately, the limited solubility of mannitol means that direct tests of eq 10 must be confined to solutions of water activity greater than that of saturated mannitol solution. For more concentrated solutions, however, we can make the test by calculating from the data for mixtures and for sucrose alone, by eq 10, the value of ZCin a supersaturated solution of mannitol of the same water activity. The constancy of MC obtained from several mixtures of the same water activity, shown in Table 111,confirms relation 10.
2129
I.OE
LO€
1.07
1.06
1.05
E"lr"
+
fflg 1.04 1.03
1.02
LO1
1,oo
I
Table 111: Sucrose (B)-Mannitol (C) Mixtures" h / M d
2.8576 2,8576 2.8576 1.9123 1.9123 1.9123 5.497 5.497 5.497
+
(a) Direct tests of eq 10 0,6227 0.1604 0.8197 0,4597 0.3332 0.8197 0,3139 0.4880 0.8197 0,1555 0.6559 0.8197 0.7561 0.1947 1.0046 0.5594 0,4057 1.0046 0,3828 0.5953 1,0046 0.1900 0.8014 1.0046
(b) Indirect tests of 2.5321 2.2820 1,9707 1.5970 1.4767 1.1259 7.14 5.2913 7.08 5.1174 7.13 4.9550
3.401 3.390 3.381 2.180 2.182 2.184
eq 10 0.3874 0.5756 1.9707 0.3594 0.4971 0.8979 0.2669 0.4889 0.7051
0.9990 0.9996 1.0003 1.0008 0.9987 0.9993 1.0005 1 .oooo
(1.000) (1 .000) (1 ,000) (1.000) (1.000) (1.000) (1,000) (1,000) (1.000)
Figures calculated from the equation are shown in boldface type.
I
YB
(mc/Mc)
MB
0.7751 0.7751 0.7751 0,7751 0.9393 0.9393 0.9393 0.9393
0.5
+
Figure 1. Plots of (mB/MB) (mc/Mc) us. Y B = mB/(mB mc), where B = sucrose and C = a second solute: 1, sucrose-sorbitol; 2, sucrose-glucose; 3, sucrose-arabinose; 4, sucrose-glycerol ; 5, sucrosetris(hydroxymeth1l)aminomethane; 6, sucrose-urea.
+
Some further isopiestic experiments on mixed solutes have been made with sucrose (B) as one solute and one of the following : sorbitol, glucose, arabinose, glycerol, tris(hydroxymethyl)aminomethane, urea, as the second solute (C). The results are given in Table IV and plotted in Figure 1. We do not claim that eq 10 will be valid for any pair of solutes. For it to hold, it is necessary that there be no interaction between the solutes themselves, although interaction between solute and solvent is permitted. Thus eq 10 describes a type of semiideal behavior to which some pairs of solutes may conform; a deviation does not mean a breakdown of eq 10 but does imply a departure from the semiideality symbolized by this VoEume 70, Number 7 Julu 1966
R. H.STOKES AND R. A. ROBINSON
2130
Table IV: Mixtures of Sucrose (B) and Other Substances (C)
‘WB
lWC
Sucrose-sorbitol 2.5881 3.0584
2,5974
3.0664
Sucrose-glucose 3,2588 3,8757
3.2916
3.9233
Sucrose-arabinose 2.5299 3.0535
2.5846
3.1370
Sucrose-gly cero1 3.0975 3.8427
3,1193
3.8766
(mB/MB) (mc/Mc)
+
mB
mc
2.3125 1.6930 0.8984 2.0678 1.2683 0.4604
0.3241 1.0593 2.0015 0.6222 1.5709 2.5249
0.9995 1.0005 1.0015 0.9990 1.OW5 1,0008
2,8039 2.3504 1.2166
0.5542 1.0996 2.4489
1.0034 1.0049 1,0052
1.7391 0.8286 0.3648
1.8737 2.5903 3.4947
1.0060 1.0038 1.0016
2.0592 1.6592 1,1853
0.5885 1.0813 1,6606
1.0066 1.0099 1.0123
1,9049 1.3632 0.5191
0.8524 1 ,5200 2,5289
1,0087 1.0119 1,0070
2,0405 1,0701 0,3278
1.3615 2.5640 3.4537
1.0131 1.0127 1,0046
2.4640 1.5022 0,6493
0.8524 2.0650 3,1014
1,0098 1.0143 1.0082
equation. An inspection of Figure 1 shows that sucrose and sorbitol conform to eq 10 as do sucrose and mannitol. But for mixtures of sucrose with glucose, arabinose, and glycerol there are small but significant departures from eq 10 amounting to a maximum of 1.5% in the case of sucrose-glycerol, With sucrose-urea, however, deviations amounting to almost 10% are found; this is not surprising, for urea itself gives a solution which is by no means ideal, since there are significant amounts of dimer formed.14 More surprising is the deviation shown in the sucrose-tris(hydroxymethy1)aminomethane system, for the latter has been shown15to be almost ideal in its own solution. We suspect that the nearly ideal behavior of this solute is an accidental consequence of the near cancellation of two effects, a solvation which tends to increase the activity coefficient and a solute-solute association which tends to decrease the activity coefficient, Le., to a combination of the effects which characterize sucrose and urea in their separate solutions. The Journal of Physical Chemistry
Sucrose-tris( hydroxymethy1)aminomethane 2.4772 3.1036
(mB/MB) (mc/Mc)
+
mB
mc
2.0264 0.5808
0.6385 2.4761
1.0260 1.0329
MC
MB
2.4990
3.1095
1.6532 0.8438 0.2963
1.1837 2.1918 2,8023
1.0418 1.0426 1.0198
2.7286
3.4372
2.5095 1.9921 1 ,4088
0.3243 1.0591 1.8323
1.0141 1.0382 1,0494
0.4488
2.4058
1.0463
1,8948 1.3823 0.6283 1.6317 1.2148 0.5438
0.4851 1.2714 2.2345 0.9731 1.5880 2.4114
1.0400 1 ,0788 1,0684 1.0662 1.0859 1.0616
Sucrose-urea 2.1740 2.8645 2.1762
2.8659
2.2143
2.9553
2.2596
3.0184
2.1232 1,4008 0.9072
0.2402 1,3967 2.0582
1.0192 1,0826 1.0834
2.2669
3.0317
1.8178 1.1582 0.3220
0.7704 1,7477 2.7258
1.0560 1.0874 1.0441
List of Symbols Ki: Stepwise equilibrium constant for formation of S t a, Water activity 2: = 1 Kla, K1KzaW2 . . . K1K2.. . Knawn u = Kta, 2K1K2aW2 . . . nK1K2... Knawn rp = - (55.51/m) In a, (osmotic coefficient) 5 Average hydration number = u/2: -,JStoichiometric molal activity coefficient of solute M B , Mc Molalities of solutions of solutes, B or C, having the same water activity as the mixed solution me, mc Molalities of solutes, B and C, in mixed solution r B Stoichiometric activity coefficient (molal) of solute B alone a t molality M B YB = mB/(mB mc)
+
+
+
+
+
+
+
+
Acknowledgments. Thanks are due to the National Science Foundation and to the University of Wisconsin
(14) R. H.Stokes, J. Phys. Chem., 59, 4012 (1965). (15) R. A. Robinson and V. E. Bower, J. Chem. Eng. Data, 10, 294 (1965).
DISSOLUTION OF SOLIDOXIDES IN OXIDEMELTS
for a Senior Foreign Scientist Fellowship award to R. H. S., during the tenure of which this work was done.
2131
The Office of Saline Water also is thanked for a grant which assisted the experimental work.
Dissolution of Solid Oxides in Oxide Melts. The Rate of Dissolution of Solid Silica in Na,O-SiO, and K,O-SiO, Melts
by Klaus Schwerdtfeger Edgar C. Bain Laboratory for Fundamatal Research, United States Steel Corporation Research Center, Monroeville, Pennsylvania (Received October 36,1966)
The rate of dissolution of solid silica in static sodium and potassium silicate melts was studied in the temperature ranges 1000-1400 and 1000-1200", respectively, and in sodium silicate melt, stirred by a rotating silica disk, at 1400". Concentration profiles obtained in the static melts were measured for selected slag composition and temperature. It is concluded from the experimental data that the dissolution process is controlled by mass transfer in the liquid. Interdiflusivities as calculated from the dissolution rates under static and stirred conditions and as determined from the concentration profiles are consistent with each other.
Introduction The rates of dissolution of solid oxides in oxide melts are of considerable interest for numerous metallurgical and ceramic processes. I n the present work, the rate of dissolution of solid silica in static sodium or potassium silicate melts was studied in the temperature ranges 1000-1400 and 1000-1200", respectively, and in sodium silicate melts, stirred by a rotating silica disk, a t 1400". Previous work on the dissolution of solid silica in liquid sodium metasilicate, as done by Shurygin, Barmin, and Esin' by using the rotating-disk method, revealed that the dissolution process was controlled by mass transport in the liquid. The authors, however, state that their data are only exact to the order of magnitude because of the experimental difficulties encountered in determining the corrosion rate accurately. Experimental Section Methods. Three different experimental methods were
employed for the dissolution of silica into static melts. (a) I n the first method, silica was dissolved from silica plugs placed at the bottom of cylindrically shaped platinum crucibles. With this plug position, convection by density differences is assumed to be avoided since, according to data in the l i t e r a t ~ r e ,the ~ , ~density is lowest in the silica-poorer slag a t the top of the crucible. As an additional precaution to prevent convection currents, the crucible was suspended in a vertical tube furnace at the lower part of the hot zone so that the temperature at the top of the melt was about 3" higher than that at the bottom. After the experiment, the sample was rapidly cooled to room temperature. The platinum crucible was then sliced and the change in thickness of the plug was measured using a cathetometer. In order to have a reference point at the bottom of the plug for the length ~~
~
(1) P. M. Shurygin, L. N. Barmin, and 0.A. Esin, Izv. V ~ s s h i k h . Uchebn. Zavedenii Chernaya Met., 5 , 5 (1962). (2) J. O'M. Bockris, J. W. Tomlinson, and J. L. White, Trans. Faraday SOC.,5 2 , 299 (1956). (3) G . Heidtkamp and K. Endell, Glastech. Ber., 14, 89 (1936).
Volume YO, Number 7 July 1966
