CONSECUTIVE FORMATION CONSTANTS FROM
xov., 1951
and vanadatesI4 and the respective peroxy compounds'b is also inadequate, and no quantitative ollgcm. Che% 411, 49 (1933); (14) G. Jander and IC Jahr, z. 11%1 (1933); G. Jand-, Jahr and H. Witzmann, i b j d . ~417, 65 (1934); J. Meyer and A. Pawletta. Z. angcw. Chcm., 8% 1.284 (1926); P. Souchay and G. Carpeni, Bull. $06. chim.. 160 (1946). (15) K. Jahr, Fiat Rer. Germ. Science, 111, p. 180, 1939-1946. I n g . chim.; Aiigew Chem., 64, 94 (1941); M. E. R. Nordmann, Comfit.
[CONTRIBUTION FROM THE CHEMICAL
POLAROGRAPHIC DATA
532 1
interpretation of the effect of the various factors presented is this paper can be given. rend., 414, 485 (1946); K. Jahr, Ber. Gcs. Frcund Tech. Hochschulc, Berlin, 1941, No. 1, 5 5 ; 2. Elcklrochcm., 47,810 (1941); J. Meyer and A. Pawletta, 2. ghysik. Chcm., 190, 49 (1927); M. E. Rumpf, Ann. chim., 11E.485 (1937).
MINNEAPOLIS, MINN.
RECEIVED MAY7, 1951
LABORATORY OF THE UNIVERSITY
OF KANSAS]
The Determination of Consecutive Formation Constants of Complex Ions from Polarographic Data BY DONALD D. DEFORD'AND DAVIDN. H U M E ~ A method for the mathematical analysis of the change in half-wave potential of a metal ion with changes in the concentration of the complexing agent is derived. The identification of the successive complex ions formed and the determination of their formation constants are thus made possible.
The polarograph has been extensively used in the study of complex ions which are reversibly electroreduced at the dropping mercury electrode. If a metal ion forms only one complex over a considerable range of ligand concentration, it is possible to determine from a plot of half-wave potential against the logarithm of ligand concentration, the number of ligands bound in the complex, and by the shift in half-wave potential from that of the simple ion, the dissociation constant.a Hitherto, data derived from systems involving mixtures of consecutively formed complex ions have not been interpretable. During the last ten years a great deal of interest has been developed in the study of consecutively formed complex ions largely through the pioneering work of B j e r r ~ m . ~Ledens has described an ingenious and useful method of taking and interpreting potentiometric data for the evaluation of consecutive formation constants. In this paper is described a method of mathematical analysis of the shift of half-wave potential with ligand concentration which makes possible the identification of the successive complex ions formed and the evaluation of their formation (or dissociation) constants. The sign conventions are those of reference 3. For complex ions of a metal which is soluble in mercury, the reduction to the metallic state (amalgam) a t the dropping mercury electrode may be represented by MX,f(n-ja)
+ ne JT M(ama1gam) + jX-s
(1)
where X-b is the complex-forming substance. For convenience, this reaction may be regarded as the sum of the partial reactions M x i +(n-jb))
M fn
+ jx-a
where M+n symbolizes the simple (hydrated) metal ions. If the electrode reactions are reversible, the potential of the dropping electrode is given by (4)
where C: is the concentration of the amalgam a t the electrode surface, C& is the concentration of simple metal ions at the electrode surface, and the f ' s are the corresponding activity coefficients. Since the amalgams formed at the electrode surface are very dilute, f a may be considered to be unity and will be neglected hereafter, If the formation of the complex ions is rapid and reversible, then for each individual complex CMXjfMXj = KjcYfM(cX)i(fX)i
(5)
where K, is the formation constant of the complex MXj+("-jb), CMXj is the concentration of the complex in the body of the solution, CM is the concentration of the simple metal ion in the body of the solution, CX is the concentration of the complexforming substance, and the f's are the corresponding activity coefficients. Also Gxjfaaxj
Kj c",faa(Cx)'(fx)'
(6)
where the zero superscripts refer to concentrations a t the electrode surface. It is assumed that the complex-forming substance is present in relatively large excess so that the concentration of this substance a t the electrode surface is virtually equal to the concentration in the body of the solution. Addition of the equations for the individual complexes as represented by equation (6) and rearrangement of the resulting equation shows that
(2) C G X J
and M+"
+ ne
M(ama1gam)
(7)
(3)
( 1 ) Department of Chemistry, Northwestern University, Evanston, Illinois. (2) Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts. (3) I. M. Kolthoff and J. J. Lingane. "Polarography," Interscience Publishers, Inc., New York, N. Y., 1941. (4) J. Bjerrum, "Metal Amine Formation in Aqueous Solution," P Maase and Son, Copenhagen, 1911. (5) I. Leden, 2. physik Chcm., USA, 160 (1941).
Combination of equations (4)and (7) gives
When an excess of supporting electrolyte is pres-
X3'72
n0NALD
n.DEFORD.4ND DAVIDN. HUME
ent to eliminate the migration current, the current a t any point on the wave is given by I
2
I(C~MXI -
iJ z k
= 1
G&j)
('))
I
where k is the capillary constant, t i z 1 / ? f ' / 6 , and 1, is the diffusion current constant, G07nD'/2of the complex MX,. From equations ( 3 ) , (6) and (I)), it follows that
VOl. 73
it is apparent that the value of F,(X) a t the intercept is equal to &/fMx. Likewise, the value of K z / ~ M X is %given by the value of &(XI at the intercept when the function Fz(X), defined bythe relationship F2(X) = [FdX)
- (KI/jMX)l/CxfX
(19)
is plotted against Cxfx and is extrapolated to CX = 0. The formation constants of higher complexes may be determined in a similar manner. It is apparent from equation (17) that the first i = kIo ( C X X ~- C i X J (10) derivative of Fo(X) with respect to Cxfx is equal to Fl(X), when Cx = 0. If Fo(X), including the where I, is the apparent (measurable) diffusion value F o ( X ) = 1a t CX = 0, is plotted as a function current constant, which is related to the individual of Cxfx, the slope of this curve a t CX = 0 will be I,values by equal to the value of F,(X) a t Cx = 0. Similarly, a t Cx = 0 the slope of the curve obtained by plotting Fl(X) as a function of Cxfx gives a value for F2(X) a t CX = 0, and so on for the higher complexes. Measurement of the limiting slope of the previous F ( X ) curves serves as a useful check on Similarly, the diffusion current is given by the value of Kj obtained by extrapolation. The mathematical form of the F-functions has id = k IC CMX~ (12) several interesting consequences. It will be obI t is also known that the concentration of the arnal- served that a plot of the F ( X ) versus Cxfx for the gam a t the electrode surface is related to the current Zast complex will be a straight line parallel to the concentration axis. A similar plot for the immehy diately preceding complex will be a straight line i = kIMc (13) with a positive slope, and all previous F-function where I M is the diffusion current constant of the plots will show curvature. These characteristics metal atoms in mercury. Combination of equations aid in establishing the number of complex ions (8), (lo), (12) and (13) shows that the half-wave po- formed in a given system and also provide a qualitential of the reducible ion in the presence of the tative check on the validity of the data. At the time of the development of the above relacomplex-forming substance is given by tions, the only precise data available to the authors for the testing of the theory were some measureInents on the half-wave potential of the cadmium Similar considerations show that the half-wave po- ion as a function of thiocyanate concentration in the range of 0.1 to 6.0 These measurements tential of the simple metal ion is given by were, unfortunately, not made a t constant ionic strength. Application of the method, ignoring all activity coefficients, gave values of K1 = 16, KZ = The half-wave potential of the simple metal ion 63 and K3 = -1 with indications of one or more higher complexes of relatively low stability. It when its activity coefficient is unity is given by was felt that although the data did not justify quantitative interpretation beyond the first two or three formation constants, the essential validity of Combination of equations (lis) and (16) gives, on the method had been demonstrated. More recently we have studied the cadmiumrearrangement thiocyanate system a t constant ionic strength in another laboratory and the results of these experiments, which show clearly the application of \ nF I, ! method, are described in the following paper.; antilog 10.435 RQZ [ ( . E o 1 / 2 ) , - ( E l / ? ) o ] l o g x , (17) Subsequently, with the aid of other collaborators where the symbol F o ( X ) is introduced for conven- we have applied the method successfully to the ience to represent the experimentally measurable study of the lead-nitrate, lead-thiocyanate, thallous-thiocyanate, leadacetate and cadmium-acequantity on the right-hand side of the equation. We now introduce the function F1(X)defined by tate systems. These investigations will be dcscribed in later communications. the relationship EVANSTON, ILLINOIS 4 ( X ) = [Fo(X) - (ko/fs)l/C~f~ (18) CAMBRIDGE, MASSACHUSETTS RECEIVED APRIL23, 1931 where KO is the formation constant of the zero (6) D. D . DeFord, M A. Thesis, University of Kansas, 1947. complex, which is, of course, unity. If F l ( X ) is (7) D N. Hume, D . D. DeFord and G C B Cave, T R I JOURNAI ~ x is extrapolated to CX = 0, 73,532.1 (1951) plotted against C ~ f and J
+
