July, 1959
'
1041
STABILITY CONSTANTS FROM CRYOSCOPIC MEASUREMENTS
rium from one molecule to another. It would have to be due to some eo-action of the acid sites and the platinum sites. Acknowledgment.-The authors are grateful to members of the Analytical Section of the Research and Development Laboratory of the Socony
Mobil Oil Company, Inc., for devising the scheme for analysis of mixtures of deuteriobutanes, and to R. W. Baker, G. W. Munns, Jr., and R. L. Smith for determination of catalyst activities toward cyclohexane dehydrogenation of p-xylene isomerization.
CALCULATION O F STABILITY CONSTANTS FROM CRYOSCOPIC MEASUREMENTS BY THE PROJECTION STRIP METHOD BY F. J. C. ROSSOTTI AND HAZEL ROSSOTTI Contributionfrom the Department of Chemistry, The University of Edinburgh]Edinburgh]Scotland Received Seplember 8 , 1068
It is shown that the formation curve, %(log a), for a system of mononuclear complexes may be calculated readily from cryoscopic measurements in a constant ionic medium. If N < 2, stability constants are best obtained from the formation curve by the projection strip method, which is illustrated by recalculation of Kenttamaa's cryoscopic data for the sulfato complexes of magnesium, nickel(II), copper( 11) and zinc. Methods are indicated for calculating stability constants for systems in which N > 2, and the limitations of the cryoscopic method are discussed.
Introduction Although acid dissociation constants of aldehydes were calculated from freezing point measurements over fifty years cryoscopy has been little used for quantitative studies of equilibria in solution. Data for solutions of weak electrolytes in pure water must be expressed in terms of both equilibrium constants and activity coefficient^.^-^ Even if only one complex is formed, the calculated value of the equilibrium constant then depends upon an arbitrary choice of activity c~efficient,~ and interpretation is even more difficult if two or more complexes coexist. Calculation is simpler if all activity coefficients can be taken to be unity, as in studies of association of neutral molecules in organic solvents6-9 and of ionic equilibria in concentrated sulfuric acid.1° Ionic equilibria in aqueous solution are best investigated in the presence of a large excess of inert electrolyte, which keeps the activity coefficients effectively constant"; changes in the number of solute species may then be followed by measuring changes in a eutectic, or transition point.12 A number of worker^'^-^' have calculated one, or, a t the most, two equilibrium constants for data of this type, but the relationship between the equations
used, and the contemporary treatment of mononuclear step-equilibria, is not immediately obvious. It is shown below that the formation function, and hence the stability constants, of a mononuclear system may be obtained readily from cryoscopic measurements in a constant ionic medium. The method is illustrated by recalculation of datal5 for some sulfato complexes of bivalent metal ions. Theory 1. Calculation of the Degree of Formation of Mononuclear Systems.-The formation of a mononuclear complex BA, from a central group B and n ligands A in a constant ionic medium may be represented by the equilibria B
+ n A -7 BAn
Pn
(1)
BAn Kn
(2)
and BAn-1
+A
where charges are omitted. The equilibrium constants p,&and K , are the over-all and stepwise stoichiometric stability constants of BA,, valid for the particular ionic medium and temperature. The degree of formation of the system is given by18 n- = - A - a B
(1) F. Auerbach, Ber., 88, 2833 (1905). (2) H. von Euler, ibid., 89, 344 (1906). (3) M. Randall and C. Allen, J. Am. Chem. Soc., 62, 1814 (1930). (4) J. S. Johnson and K. A. Kraus, ibid., 79,4436 (1952). (5) P. G. M. Brown and J. E. Prue, Proc. R o y . SOC.(London),283A, 320 (1955). ( 6 ) B. C. Barton and C. A. Kraus, J . Am. Chem. Soc., 78, 4561
(3)
N
-
n[BAnl 0
B
(4)
where B and A are the total concentrations of central group and ligand, a is the free ligand concen(7) M. Davies and D. h4. L. Griffiths. 2.phgsik. Chem., (Frankfurt), tration, and N is the maximum value of n. The 3, 353 (1954); 6, 143 (1956). sum, S, of the concentrations of solute species is (8) J. A. Davison. J . Am. Chem. Soc., 67, 228 (1945). related to the observed lowering, 0, of the eutectic (9) N. E. White and M . Kilpatriok, THIB JOURNAL, 69, 1044 (1955). or transition point, and to the appropriate cryo(10) R. J. Gillespie, E. D. Hughes and C. K. Ingold, J . Chem. Soc., scopic constant, A, by the expression 2473 (1950). (1951).
( 1 1 ) G. Biederinann and L. G. Sill&, Arlciv Kemi, 6, 425 (1953). (12) E . Darrnois, BUZZ.aoc. chim. France, 1 (1950). (13) P. Hagenmilller, Ann. ehim., l2:6, 1 (14) J . ICenttBmaa, Suomen kemislilehti, 39B, 69 (19513). (15) J. Kenttimaa, Acta Chem. Scand., 12, 1323 (1958). (16) R. Schaal, J . chim. phys., 6 3 , 719 (17) P. Souohay and R. Scliaal, Bull. soc. chim. France. 819 (1950).
(1951).
(19.55).
S
= Oh-1 =
A
+B -
N
n[BA,,] = A
+ B(l - 6)
0
(5) (18) J. Bjerrurn, "hletal Ammine Formation in Aqueous Solution," P. Haase and Sons, Copenhagen, 1941.
I?. J. c. ROSSOTTI AND HAZEL RossoTTr
1042 0.4
1
0.3
-
0.2
-
0.1
-
VOl. 63
described for computing stability constants of mononuclear complexes, the most satisfactory are those in which plots of experimental functions are compared with normalized curves.2o Systems in which N 6 2.-If it is found that FZ 6 2, it is possible that N 6 2 and that no complexes higher than BA2 are present under the conditions used. If only the first two complexes are formed, equation 8 may be transformed to
/
9 where
IS
a = pz'/Za LOP
a
p =
-1.0 -0.8 -0.6 -0.4 log a. Fig. 1.-Calculation of stability constants of nickel sulfate complexes in 1.151 molal aqueous potassium nitrate a t -3.06 & 0.19': (a) projection strip of experimental points, ii, log a, calculated from Kenttlimaa's measurements,16 and theoretical formation curve calculated for log pi = 0.70 and log p2 = 1.50; (b) projection strip superimposed on normalized curves log p(1og a)6 in position corresponding t o log p = -0.50 and log 02 = 0.75. -1.4
-1.2
Thus rt may be calculated readily from 0 provided that A and B are known from the analytical composition of the solution, and that X has previously been determined (e.g., by means of a completely dissociated electrolyte). If the compound B,A, is the sole source of both central group and ligand, then A = rq-lB, and equation 5 reduces to ox-' = B(7-g-1 + 1 - 5) (6) Once fi has been determined, the corresponding free ligand concentration may be calculated from equation 3, except in the case of such stable complexes that FZ A / B , and a is indistinguisha.ble from zero. Equations 5 and 6 are only valid in the absence of reactions other than (1) and (2). They cannot be used if polynuclear complexes are formed in solution, or if,A or B take part in competitive equilibria involving protons or hydroxyl ions. 2. Computation of Stability Constants from Data a,a.-Since
-
N
B =
[BAJ
(7)
0
equation 4 may be written
c N
(n - 7i)pnan = 0
(10)
Thus fi is a function of the normalized variable a and of the parameter
(8)
0
and the N stability constants 01, pz, . . . . . , P N may in principle be calculated, provided that at least N pairs of values fi,a have been determined. Of the many methods (e.g., ref. 19-20) which have been (19) F. J. C. Rossotti and H. S. Rossotti, Acta Chsm. Scand.. 9 , 1166 (1955). (20) L. G. Sill&, ibid., 10, 186 (1956).
=
01@2-'/2
(Ki/K2)l/2
(11)
which fixes the shape of the formation curve rt(1og a). Systems described by two parameters and two variables, one of which can be normalized, are best treated by the projection strip method.21 Since this device has not yet been used to calculate stability constants of mononuclear complexes, its application to systems in which N 6 2 will be described in some detail. The family of theoretical curves log p (log a); are calculated from equation 9, using a number of coiivenient values of a. Values in the range 0.1 6 rt 6 1.0 are shown in Table I, and those in the range 1.0 6 f i 2.0 readily may be obtained using the symmetry relationship
