GEORGESCATCHARD AND P. C. BRECKENRIDGE
596 u(z) =
La
ties of the function p(R), this calculation will not be presented here. If we examine the roots Zn of the equation
e-SR v(R)dR
i a k ( R - r)a(r)dr = f ( R ) ; 0 5 R 5 a
z2
zn =
where the path of integration is to be taken along a line parallel to the imaginary axes with the constant C lying between zero and the least positive real part of the zeros of the function z 2 - K~ cosh za. The function p(R) is then given by the expression
A. = zn2
-R(
22.
-
KZ
-2,
- K 2 a sinh z. cash
ZU .
=
VOl. 58
a
0
where the sum extends over all zeros of z2 - K~ cosh xa with positive real part. The explicit determination of the coefficients A , may be carried out after numerical solution of the fifth of cq. 29, an integral equation for p(R) on the finite interval 0 _< R 5 A . Since we desire only to discuss the general proper-
cosh za = 0 CY.
f ip,
(31)
we find that if a is set equal to zero, there is only one root Z , = K with positive real part, and the solution, eq. 27 for p(R) reduces to the DebyeHuckel first approximation e-KR. For finite a and small KU there are two positive real roots, the smaller one approximating K a t low ionic strengths, which makes the dominant contribution to p(R), so that p(R) resembles the Debye-Huckel form. As the ionic strength increases, the two real roots approach equality, become equal a t Ka = 1.03 and then move into the complex plane as complex conjugates. For values of Ka greater than 1.03, p(R) exhibits the oscillations characteristic of the potentials of mean force in the liquid state, and around each ion there develops a statistical stratificat,ion of the average space charge due to the other ions with alternating zones of excess positive and excess negative charge. It is attractive to consider the application of these ideas to an elucidation of the structure of concentrated electrolyte solutions and fused salts, although it would no doubt be necessary to go beyond the linear approximations, eq. 24 to the potentials of mean interionic forces in order to obtain more than a, qualitative description.
ISOTONIC SOLUTIONS. Ir. THE CHEMICAL POTENTIAL OF WATER IN AQUEOUS SOLUTIONS OF POTASSIUM AND SODIUM PHOSPHATES AND ARSENATES AT & S O 1 BY GEORGESCATCHARD AND R. C. BRECKENRIDGE Department of Chemistry, Massachusetts InatitUte of Technology, Cambridge, Mass. Received April 19, 1964
The chemical potential of water in aqueous solutions of potassium and sodium primary and secondary phosphates and arsenates and in equimolal mixtures of the secondary and tertiary salts at 25' was determined by isotonic com arison with sodium chloride solutions from 0.1 to 1.1 M NaCI. The osmotic coefficients are expressed as the Debye-Huctel term and LaMer-Gronwall-Greiff second term for a size of 5.35 A. plus a term Bc/( 1 Dc), in which B and D are specific parameters. This permits analytic expressions for the activity coefficients. The coefficients of the ternary salts are computed by the specific ion interaction assumption that vi3 and D are linear functions of the equivalent fraction. The activity and osmotic coefficients of the primary sodium salts are less negative than those of the potassium salts, the coefficients of the sodium secondary and tertiary salts are more negative. The coefficients of the arsenates are less negative than those of the corresponding phosphates. The basis of Bronsted's theory of specific ion interaction is reexamined, and the theory is used to determine the coefficients in mixtures and specific effects on chemical equilibria.
+
Our interest in nquems solutions of sodium and potassium phosphates was aroused by the fact that sodium phosphate buffers are much more acid than potassium buffers of the same ratio of acid to basea2 The effect may be as great as one pH unit in one molal solution. Measurements of the osmotic coefficients also permit a continuation of the comparison of the individual salts started by Scatchard and Prentissaa ( 1 ) Adapted from the Ph.D. Thesis of Robert G. Breokenridge. M.I.T.. 1942. (2) E. J. Cohn, private communication (about 1938). (3) G. Scatchard and S. 9. Prentiss, J . Am. Chern. Soe., 66, 807 (1934).
Experimental The apparatus used was that of Scatchard, Hamer and Wood,' and the only change in procedure was that two of the cups contained the reference sodium chloride solutions, and each of the other four contained a solution of a salt being studied. The four primary and secondary phosphates were studied together in one series, the corresponding four arsenates in another, and the four secondary-tertiary mixtures in a third series. The salts used for this study were prepared in this Laboratory and a t the Harvard Medical School. The sodium chloride was prepared by precipitation by hydrogen chloride gas from a saturated solution of Malliiickrodt C.P. analytical reagent sodium chloride in conductivity water. The product was (4) G. Scatchard, (1938)
W. J.
Hamer and S. E. Wood, ibid., 60, 3001
Aug. , 1954
ISOTONIC SOLUTIONS
dried at 300" for 24 hours, recrystallized once from conductivity water and again dried a t 300' for 48 hours. It was pulverized in an agate mortar, air dried at room temperature and then stored over calcium chloride. The primary and secondary phosphates were prepared by Dr. T. L. McMeekin in the Laboratory of Physical Chemistry a t the Harvard Medical School. Pure primary phosphates were made as described by SBrensens and these were converted to the secondary saks by adding the proprr amounts of pure hydroxides. For these measurements, the potassium dihydrogen phosphate was dried at 120" for 24 hours, then pulverized, air dried and stored over calcium chloride, Primary sodium phosphate crystallizes as a hydrate, so t,his salt was made up as a stock solution in conductivity water and its concentration determined by precipitating the phosphorus as magnesium ammonium phosphate and weighing as magnesium pyrophosphate.6 The secondary salts were received as so1ut)ionsand were used without further treatment, the concentrations being determined by analysis as for the primary sodium phosphate. The primary and secondary arsenates were made by adding the calculated amount of Mallinckrodt C.P. hydroxide to a sample of Mallinckrodt analytical reagent, arsenic oxide and recrystallizing the salt thus obtained. The sodium salts were recrystallized three times. Since these are obtained as hydrates, stock solutions were made up and the concentrations determined by precipitation of the arsenic as magnesium ammonium arsenate and weighing as magnesium pyroarsenate. The primary potassium arsenate was recrystsllized five times from conductivity water and dried a t 120' for 48 hours. The salt was then powdered, air dried and stored as before. The secondary potassium arsenate was prepared by adding pure hydroxide to the pure primary salt and recrystallizing twice. The Ralt was used from a stock solution similar to the sodium salts. The mixtures of secondary and tertiary salts were prepared by adding the calculated amount of pure alkali hydroxide to a solution of the corresponding secondary Ralt. Highly concentrated solutions of Malliiickrodt C.P. hydroxide were prepared in conductivity water. After several hours digestion on a steam-brtt>hthey were filtered through hardened paper from the insoluble carbonate? The filtrates were diluted with carbon dioxide free conductivity water to about 2.4 normal and stored in paraffined bottles equipped with dispensing burets and soda lime guard tubes. These solutions were standardized against rimnry-standard grade potassium acid phthalate that had gee, dried for 12 hours a t 120". Three or four stock solutions of each mixture were made by weighing the proper amount of each solution, and the ratio of total cation concentration to total anion concentration was determined for each stock solution.
Results The results of the isotonic measurements are given in Table I as the equilibrium coricentrations, and the values of M $ , in which M is the total number of moles of ions per kilogram of water, and $t the osmotic coefficient, 4 = (In ao)/0.01816M,a,, is the activity of the water. The values of the osmotic coefficient of sodium chloride were calculated from the equation which corresponds to that for the activity coefficient given by Scatchard, Batchelder and Browns q5 =
1
- ( A / a ) 2(2)/2 + 0.0126~1+ 0 . 0 1 4 1 ~(1) ~~
in which
A a ea/kTDo a(8?rNeapo/1000kTD0)'/2 z = 0(2irnizi*/2)'/2 (4) z - 1/(1 2 ) - 2 In (1 z ) 1 / ~ 2 (5)
LY
X
=
[I
+
=
+
+
(5) S. P. L. SBrensen, Compt. rend. trau. Lab. C u r l ~ b ~8, r ~1 ,(1909); Biochem. z., ai, 131 (1909); sa, 352 (1909). (6) F. P. Treadwell and W. T. Hall, Analytical Chemistnd, John Wiley and Sons, Inc., New York, N . Y.. 1935. (7) J. B. Niederl, "R4icro-Quantitative Organic Analyses b y the Method of Pregl," John Wiley and Sons, Inc., New York, N. Y.,1938. (8) G. Scatchard, A. C. Batchelder and A. Brown, J . Am. Chem. SOC.,68, 2320 (1946).
597
and E is the protonic charge, k is Boltzmann's constant, T is the absolute temperature, N is Avogadro's number, Doand po are the dielectric constant and density of the solvent, zi is the valence of the species i, and a is the nearest distance of approach of two ions. For water a t 25") A is 2.342 X los and TABLE I ISOTONIC RATIOS NaCl
' IZIinPO4
0,95289 1.25414 ,92773 1.21154 ,89267 1.15619 ,88989 1,15350 .79265 1,00109 ,78377 0.98764 ,96882 .77198 .72093 ,89257 ,85528 .69512 .74374 .61902 .73077 .60786 .72163 .60115 .55410 .65576 .54381 .64174 .48114 .55594 .46114 .53097 .44452 .50715 .42844 ,48849 .40561 * 45757 .38264 .42976 .37021 .41453 ,36330 .32862 .24445 .26331 .20494 ,21872 .19085 .20218 ,18440 .19548 .12992 ,13535 ,11553 .12018 NaCl
1.10091 1.05953 0.94722 .go452 ,87140 .83938 ,76296 ,71489 ,66215 .61583 ,60048 .58344 .52662 ,52469 .51910 .46 120 .42670 .42578 ,31380 ,29636 .28545 .24142 ,23250 ,21205 .15172
-
Phosphates m NaHaPO4 KaHPOi
NaaHPO4
1 16786 0.87287 1.13048 ,84496 1,08708 ,81181 1,07920 ,80950 0,94255 ,71311 ,93161 ,69869 ,92395 ,69160 .84678 ,63960 ,81201 ,61294 ,53961 .71390 .5269 1 ..... ,68806 ,52661 ,63224 ,47941 . 61672 . . . , . ,53736 ,40794 ,51426 . . . . . ,49313 ,37579 ,4739I ..... . . . . . ,34194 ,41913 ,31915 .30597 ..... ,35475 ,26985 .25964 ,19786 .21513 ,16452 ,19933 ,15183 ,14636 .19285 ,10129 , 13383 ,08947 .....
0.99143 1.7794 ,96044 1.7310 . . . . . 1.6636 .91084 1.6582 ,78756 1.4728 .77661 1.4559 . . . . . 1.4335 ,69920 1.3368 ,66951 1.2881 .57979 1.1444 .56745 1.1235 ,56186 1.1113 ,51034 1.0234 .49806 1.0029 ,42892 0.8876 .40728 .8505 ,39045 .8205 . .... .7900 .35275 .7478 .32961 .7054 ,31453 .6825 ,27747 .6060 ,20016 ,4515 .16467 .3792 ,15253 ,3534 .14731 ,3414 ,10115 .2418 ,09002 .2154
-
Arsenates, m KHaAsOi NaHsAsOi KnHAsOd NaaHAsO4
1.38515 1.33171 1.16684 1.10423 1.06030 1.01075 0.90912 ,84383 ,77330 ,71072 .I38949 .67013 ,59678 .59387 .58436 .51520 ,47451 ,47147 ,33904 ,32000 ,30649 .25765 ,24666 ,22398 .15898
1.28602 0.88646 1.23628 .85201 1.08849 .75837 1.03391 ,72368 ,69447 0.99270 ,95284 ,66878 ,60558 .85606 ,79636 .56473 ,72935 ,52417 ,67560 .48434 ,65594 ,47161 .63707 .45604 .57119 ,41130 ,56771 .40982 ,55621 .40362 ,49268 ,35522 ,45236 ,33076 ,45179 .32942 ,32708 ,32740 ,30785 ,22601 ,29819 ,21645 ,24940 ,18247 ,23956 .17347 ,21845 ,15898 .11193 .11283
1.02910 0.98517 .86336 ,81780 ,78341 ,75385 ,67167 .62295 ,56957 .52613 ,51041 ,49363 .44103 ,43950 .42755 ,37976 .34878 .34666 .24892 ,23449 .22463 .18803 .BO30 ,16358 .11481
R40
M+
2.0655 1.9841 1.7685 1.6865 1.6230 1.5471 1.4048 1.3160 1.2154 1.1277 1.1048 1.0789 0.9729 .9686 .9584 ,8506 ,7868 ,7851 .5787 .5467 .5267 .4460 .4297 .3922 .2817
GEORGE SCATCHARD AND P. C. BRECKENRIDGE
598
A = 2.4915 B = 2.4917 C = 2.4976
TABLEI (Continued) Secondary-tertiary mixtures, D = 2.5311 G = 2.5063 E = 2.5198 H = 2.5710 F = 2.5043 I = 2.5020 J = 2.4820
Vol. 58
K = 2.4558 L = 2.5355
M
=
2.4000
+NanKPOa NarPO4
NaUl
M6
0.68282 ,65152 .60347
A A A
0,85427 ,79448 .72586
D D D
,56599 ,54340
A A
,67221 .63981
D D
.48890 ,48697 .43590 .42412
A
D
A C
,56679 ,57167 ,49551 ,48517
F
.32087
C
.35365
F
,27952 ,24878 .24097 .2 1807
c
R C
c C
.17328 ,15608
C
,09997 .09732 ,07762 ,07752
B
B C
c C
F
E F F F
,18125 ,16032
F E
.lo138 ,09976 .07894 .07894
E F F F
& = 1 - (A/~~)Z(~)(Zimizi')/2(Zimi) ( A/cY)'[Xn(2)/2 - 2 Y ~ x ()Z]i r n i ~ i a ) 2Zimi / ( )( Zimizi)' BcAl Dc) (6)
+
+
.-15787
H
,44975
J
with c the equivalent concentration, to determine the specific parameters B and D. The second term in this expression is that of LaMer, Gronwall and Greiff,lO who tabulate values of [X2(z)/2 - 2Yz(z)] up to z = 0.4. We used values up to z = 3.2.
K K K
,61054 .58785
K K
,39194 ,35098 .29912 ,28980 .25729
J H J H H
.23050
J
.20569 .15609 .15883
J H J
.093837
H
.08968 .07117 .06475
J J
K
.52108 .51547 .45770 .44355
D
.30027 .26710 ,25825 .234ll
0.76524 ,71218 .05819
2.0870 1.9313 1.7885 1.6902 1.6433 1.5486 1,4412 1.4249 1.4178 1 3564 1.3081 1.2270 1.1807 1.1796 1.0570 1,0299 0.9584 * 7953 7868 .7850 .6934 ,6230 .6053 ,5512 ,4459 .4455 .4048 .2817 .2685 .2629 ,2134 ,2047
I G
.53357 .53371
a l a = 0.3281 X 10'. For sodium chloride we used a = 4.72, a = 1.55. To obtain an analytical expression for the phosphates and arsenates we used the method of Scatchard and Epstein9 to determine a size a = 5.33, Q = 1.75 for all the salts. Then we used the more general expression
-
I I G I
0.77710 ,72088 .65683 .63352
1.11186 1.03158 0.94722 .90449 ,88199 ,83248 ,77602 .76745 ,76296 .73128 ,70575 .66215 .63814 .63755 .57209 ,65756 ,51910 .42884 .42670 ,42578 .37610 ,33784 .32829 .298!31 .24142 ,24114 ,21892 .I5172 ,14437 ,14143 .11444 .lo975
Af K
nf M
.32966
I
.28286 .25187 ,24421 .22056
L M
,17352 .15563
M L
,09860 ,09722 .07664
L M M
M M
H
mined by graphical integration. Table I1 contains, for steps in z of 0.2 from 0.4 to 3.2, the values of -102 [Xn(z)/2
- Y 2 ( 2 )and ] -102
[X,(z)/2
- 2Y,(r)l
which are the combinations most used in In y and in 9. For a single salt, with cation 3 and anion 4 (2i?7ZjZi2)/(ZiWLi)
and pz =
=
-
(2imizi3)/(Zimizi2)
2124, 3
$3
+
24
which is zero for symmetrical salts. TABLE I1 ELECTROSTATIC IXTERACTION TERMS - 102 - 102 - 102 - IO? [ X * / Z - Y Z ] [ X 2 / 2 - 2Yzl x [.y2/2 - Y?lfX2/2 - 2Yzi X 0,4324 0.0982 2.0 0.2159 -0.1062 0.4 ,0548 2 . 2 ,1913 - ,1103 ,4575 .6 ,4370 ,0140 2.4 ,1702 - ,1120 .8 1.0 ,0240 2 . 6 ,1520 - ,1122 ,3975 1.2 ,3547 ,0528 2 . 8 ,1360 - ,1110 1.4 - ,0741 3 . 0 ,1223 - ,1100 ,3138 ,2769 - ,0891 3 . 2 ,1103 ,1079 1.6 ,2442 - ,0996 1.8
-
X z was calculated from elementary functions and tables of the exponential integral. Y2 was deter( 9 ) G. Scatchard and L. F. Epstein, Chern. Reus., 30, 211 (1942). (10) K. La Mer, T.H.Gronwall and L. J. Grieff, THIRJOURNAL,
V.
86, 2245 (1931).
-
The parameters B and D are listed in Table 111, which includes values for the ternary phosphates and arsenates extrapolated on the assumptions that
,
ISOTONIC SOLUTIONS
Aug., 1954
VB and D are linear functions of the equivalent fraction. These assumptions will be discussed later. TABLE I11 SPECIFICCOEFFICIENTS Salt
--B
KHgPO4 NaH2P04 K~HPOI NarHP04 IC3POI Na3P04
D
-B
Salt
D
0.303 0,639 KtHAs04 0.084 0.436 ,352 ,256 ,767 NatHAs04 .154 .2GO ,966 KHtAsOi ,262 ,762 ,139 ,335 ,256 ,503 NaH2ARO4 ,162 ,889 K ~ A s O ~ ,042 ,776 ,263 ,465 X\'ctaA~Oi ,234 ,632
Equation 6 represents the results within the scatter of the measurements for the primary and secondary salts, and for the mixtures of secondary and tertiary salts above 0.1-0.15 m. In dilute solutions of these mixtures the measured osmotic coefficients are consistently high and appear to extrapolate to a limit greater than unity. For both of the sodium salts and for the potassium phosphates the maximum discrepancy is 2%. For potassium arsenate, however, it reached 8% and the measurements in more concentrated solutions scatter abnormally. There seems to have been an impurity
which contributed about 0.003 to M (0.012 for potassium arsenate) for any value of M . This type of behavior would result from the hydrolysis or from the absorption of carbon dioxide from the atmosphere, but the amount seems very large. We have ignored the measurements in very dilute solutions. The results for the mixtures are therefore uncertain to about O.O03/M (0.012/M for potassium arsenate). The extrapolated values for the tertiary salt have about twice this uncertainty in addition to any uncertainty in the assumptions. Table IV contains the osmotic coefficients, and Table V the activity coefficients, as 1 log y, calculated from these parameters. The equation for the activity coefficients which corresponds to equation 6 is
+
-
KHnPO4
NaHaPO4
0.1 .2 .3 .4 .5 .6 .7
0.937 .928 ,925 ,923 ,922 ,921 ,921 ,920 .921 ,921 ,921 ,921 ,922
0.909 ,875 ,848 ,826 ,807 .790 ,774 ,760 ,748 ,736 ,725 ,715 ,706
0.913 ,884 ,862 ,844 ,829 ,816 ,804 .794 ,784 .776 ,768 ,761 ,755
.8
.9 1.0 1.1 1.2 1.3
KHrAsOi NaHnAsOi
0.913 ,883 ,861 ,842 ,827 ,813 ,801 ,790 .781 ,772 ,674 ,757 ,750
0.924 ,902 .887 .874 .862 .852 .842 ,833 .825 ,817 ,809 .802. ,796
Secondary Salts m
Debye
KzHPO4
NazHPO4
&HA804
NarHAsOc
0.1 .2 .3 .4 .5 .6 .7
0.849 ,839 ,839 .839 .840 ,843 ,845 .847 ,849 .851 ,853
0.808 .764 .739 ,722 ,708 ,698 ,690 .684 ,679 ,674 .671
0.802 .754 .720 ,693 .670 .651 .634 .620 ,608 .596 .586
0.833 .811 .799 .790 .784 .779 .775 .771 .769 .766 ,764
0.820 .785 .761 .742 .726 .712 .700 .689 .679 .670 .663
.8
.9 1.0 1.1
+
+
Tertiary Salts m
Debye
KsPOa
NasPO4
K3As04
NanAsOi
0.1 .2 .3 .4 .9 .6 .7
0.748 ,742 ,746 ,752 ,759 ,766 .772
0.709 ,678 ,665 ,658 ,655 ,654 ,653
0.678 .618 ,579 ,550 .527 .508 .492
0.738 .724 .724 .726 .730 .734 ,738
0.689 .640 .612 .593 ,579 ,569 ,561
+ Z-)~(A/OI)* + BcAl + Dc) Id)
Since the experimental work of this paper was completed, Stokes" has published the results of isotonic (isopiestic) measurements on sodium and potassium primary phosphates. Her results for the TABLE V ACTIVITY COEFFICIENTS AS 1 Primary Salts
OSMOTIC COEFFICIFNTS Primary Salts Debye
+ +
In y & = z + z - ( A / ~ ~ ) z / S ( l 2) z+z-(z+ [X2(2)/2 Y&c)l (B/D) In (1 Dc)
TABLE IV
m
599
m
Debye
KHnPO4
NaHzPO4
0.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3
0.897 .873 .858 ,848 ,839 ,833 .828 .823 .819 ,815 .812 ,809 .807
0.871 .824 .789 .759 ,732 .709 .688 ,668 .650 .633 .618 .603 .589
0.876 .833 .801 .775 .752 .732 ,715 ,698 .684 .670 .657 .645 .634
m
Debye
K I H P O ~ NanHPO4
0.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1
0.722 .671 .642 .622 .607 .596 .586 .578 .571 .565 .560
.0683 .600 .546 ,504 .471 -444 .420 ,398 .380 .363 .348
m
Debye
KtPOi
0.1
0.529 ,449 .406 .377 .356 .340 .328
0.494 .388 ,324 ,279 .244 .216 ,193
+ log
y
KHnAsO4 NaHzAsOc
0.875 .832 .799 .773 .750 .730 ,712 .695 ,680 .666 ,653 .641 .630
0.885 .850 .824 .804 .786 .770 .755 .742 .730 .718 ,707 .697 .687
Secondary Salts .OM1 .593 .532 ,484 .443 .408 .377 .349 .234 .301 .279
K&L4sOc NanHAsOc
.0709 .644 ,606 .576 .553 .533 .517 502 ,489 .477 .467
0.697 .622 .572 .533 .500 .472 ,447 424 .404 .385 .367
NalPOc
KrAsOc
Na&Oc
0.467 .335 .247 .180 $126 .079 .039
0.520 .432 ,383 .350 .326 .306 .290
0.475 ,353 ,275 ,217 ,171 ,133
Ternary Salts .2
.3 .4 .5 .6 .7
(11) J. M. Stokes. Trans. Faraday Soc., 41, 685 (1945).
.loo
600
GEORGESCATCHARD AND P. C. BRECKENRIDGE
sodium salt average 0.4% higher than ours, for the potassium salt hers are 0.1% lower. Only in the dilute solutions is the difference from this average greater than 0.2yo.
Discussion The comparison of the osmotic coefficients may be made without involving any theory except that necessary for interpolation, and it makes but little difference which concentration is chosen for the comparison. The osmotic coefficients are all smaller than those calculated from electrostatic theory. The primary phosphates have the smallest coefficients of any salt of the same cation and a univalent anion listed by Robinson and Stokes.I2 Primary sodium arsenate also lies below any other sodium salt, but potassium primary arsenate has a larger coefficient than the nitrate or bromate. The coefficient of every arsenate is larger than that of the corresponding phosphate, and the difference is such that the smaller arsenate coefficient is nearly the same as the larger phosphate coefficient. For the primary salts the coefficient of the potassium salt is smaller than that of the sodium salt, for the secondary salts the coefficient of the potassium salt is larger. The difference between the primary arsenates is somewhat less than that for the nitrate, chlorate and perchlorate ions, and that for the phosphates is very much less. Scatchard and Prentiss3 divided the alkali salts of univalent anions into three classes: the first includes the halides except the fluorides and their osmotic coefficients decrease with increasing size of the cation; the second includes the hydroxides, fluorides and salts of carboxylic acids and their osmotic coefficients increase with increasing size of the cation; the third consists of salts of oxygenated anions such as the nitrates, chlorates and perchlorates, and the order is the same as in the first, but the salts of larger cations have very much smaller osmotic coefficients. The difference between the third and first class is less marked a t 25" than a t 0", but is still significant. The primary phosphates and arsenates appear t o combine the characteristics of the second and third classes. The potassium salts behave like the nitrates and perchlorates, the sodium salts have an additional factor reducing their osmotic coefficients, which is greater for the phosphate than for the arsenate. The primary phosphate ion is the weaker base, so the basic strength cannot explain quantitatively the behavior of class two. Perhaps the smaller size of the phosphate ion is more important in its interaction with sodium ion than in that with a proton. This fact is brought out even more clearly, however, by a comparison of the hydroxide and fluoride ions. Our knowledge of the alkali salts of bivalent and trivalent anions is much more limited than that of the salts of univalent anions, but the same factors are obviously operative. For the potassium salts 4'" decreases continuously as the valence of the anion changes, and for the sodium salts it increases, though to a smaller extent. The phosphates always have larger values than the arsenates. (12) R. A. Robinson and R. H. Stokes. Trans. Faraday Soc., 45, 642 (1949).
Vol. 58
Salt Mixtures T o relate our results to the pH measurements, or to any study of chemical equilibria, it is necessary to consider the activity coefficients of salts in salt mixtures. We use essentially the "specific ion interaction" theory of B r 0 n ~ t e d . l ~If we expand the term B c / ( l Dc) in equation 6 for the osmotic coefficient, it becomes
+
+----
BC - B D c ~
This corresponds to a virial expansion, so for mixtures the first term must be a t most a quadratic function of the composition and the second a cubic, Then our parameters must have the forms B = Zicibi/c Zijcicjbij/2c2 (10)
+
D = Zicidi/c
(11)
in which c, is the equivalent concentration of species i, c = Zi+q = Zj-c, is the total equivalent concentration and YC = Zicivi = Zimi = ZiNi/V&o, if Zi+ is the sum over all cations and 2,- the sum over all anions. The parameter bi corresponds to the interaction of the ion i with the solvent, and may be interpreted as a chemical solvation or as a change in the association of the solvent. The parameter bij corresponds to the interaction of a n i ion with a j ion. Bronsted called bici the salting out term, and bijCicj/c the interaction term. The specific ion interaction theory is merely the statement that bij is zero if the product of the valences Zizj is positive. It is powerful enough, however, to determine the osmotic coefficient and the mean activity coefficient of any salt in any mixture of salts of the same valence type from the osmotic coefficients or the mean activity coefficients of the single salts. For mixtures of different valence types a further assumption is necessary. Since ternary collisions between ions, and therefore the third virial coefficient, must involve a t least two ions of the same sign, Bronsted's theory is obviously limited to the second virial coefficient. However, since D for a mixture can also be derived from those of the component salts solutions, there is no loss of power in the extended equation, although there is considerable loss in convenience. To save space we will derive immediately the expression for the general case, without Bronsted's theory, and derive the simpler cases from the general expression. We will use the superscript ' to denote the Debye-Huckel approximation, " to denote the Gronwall-La Mer second term, and "' to denote the short range term in solutions so dilute that Dc
