THE STUDY OF COMPLEXED METAL IONS BY POLAROGRAPHIC METHODS By D. R. CROW (WEST HAM COLLEGE OF TECHNOLOGY LONDON) and J. V. WESTWOOD (SIR JOHN CASS COLLEGE LONDON) 1. Introduction THE polarographic reduction wave of a simple (aquo) metal ion is usually shifted in the direction of more-negative potential on addition of a com- plexing agent. Direct measurement of the shift in half-wave potential can serve for the determination of stability constants of complexes in solution provided that reactions at the dropping-mercury electrode occur reversibly. With suitable modifications however irreversible cases can be dealt with in many instances. Comparatively few systems are such that direct application of simpler theory serves for stability constant determination. In many instances reduction waves are both kinetically and diffusion controlled and the rates of dissociation of complex species control to a greater or lesser extent the shapes of the waves.Systems consisting of several complexes are encountered in which both reducible and non-reducible species occur in which equilibria between some species are less mobile than others. The overall kinetics of several such systems have been elucidated and the nature structure and behaviour of each species identified in addition to the calculation of their stability constants. From such applications the polarograhic method has been developed as a useful tool for the deter- mination of the structure of complexes. In the past stability constant data and little more resulted from the study of complex systems. While there is still a lack of such important data work in recent years has more constructively turned to the problem of determining the overall mechanism by which complexes of different known structures undergo reduction.Unknown structures may then be inferred from similar polarographic behaviour or confirmation may be given for structures already indicated by other techniques. 2. Stability Constant Determinations from Reversible Reductions (a) Formation of a Single Complex.-The Heyrovsky-Ilkovic equation1 expresses the half-wave potential (Ei) for the reduction of a metallic species in terms of the diffusion current (id) and the current(i) at any other point on the polarographic wave corresponding to the potential (Ed. e.) applied to the mercury drops. RT i &.e. = Ei - n~ In 1 J. Heyrovsky and D. Ilkovic Coll.Czech. Chem. Comm. 1935 7 198. 57 58 QUARTERLY REVIEWS The validity of this equation has received ample experimental verifica- t i ~ n . ~ ~ Provided that the reductions occur reversibly the E& value of a com- plexed species MXj will have a more-negative value than that of the “simple” (aquo) species M. A measure of this shift serves to determine both the co-ordination number and the stability constant of the complex. It can fairly simply be shown that the shift to a first approximation by ignoring activity coefficients may be expressed in the following relation- ship :4 in which (E& and (Ei) are the half-wave potentials of the simple and complexed species respectively j is the co-ordination number of the complex flj its stability constant and Cx the ligand concentration.Half-wave potentials of complexed metal ions shift with changing activity of the complexing ligand in accordance with Hence the number of ligands j bound in the complex is found from a plot of log, C against (E&c which in the present case should be linear. By the application of eqn. (2) Lingane4 showed the presence of the hydrogen plumbite ion in a strongly alkaline solution of lead hydroxide whose reversible reduction is represented by HPb02- + H20 + 2e + Pb(Hg) + 3 OH- (4) @) Formation of a Series of CompIexes with a Single Ligand Type.-In this category there are essentially two classes of systems viz (i) those in which each complex species exists only within a definite region of ligand concentration (i.e. two or more complexes with different co-ordination numbers are not present together) and (ii) systems which consist of a series of complexes in step-equilibrium (i.e.two or more complexes co-exist but different species predominate at particular ligand concentra- tions). (i) Occasionally a plot of logloCx against (E+) produces a segmented curve indicating the presence of a series of complexes whose stability constants and formulae may be found from the various segments. Such behaviour is shown for example in the ferro-ferri-~xalate,~~~ the zinc- ammonia,6 and cadmium-pyrazole systems. ’ It has been demonstrated a I. M. Koltlioff and J. J. Lingane Chem. Rev. 1939 24 1. J. Tomes Coll. Czech. Chem. Comm. 1937 9 150. J. J. Lingane Chem. Rev. 1941 29 1. M. von Stackelberg and H. von Freyhold Z. Elektrochem. 1940,46 120. R. Cernatescu I. Popescu A.Cracium M. Bostan and N. Iorga Studii si Cercetarj Sti. Chim. (Fil. h i ) 1958 9 1. 7 A. C. Andrews and J. K. Romary Znorg. Chem. 1963,2 1060. CROW AND WESTWOOD COMPLEXED METAL IONS 59 that for the segmentation of the curves to be sufficiently pronounced for the calculation of stability constant data the constants must differ by a power of ten or more. (ii) For a system of complexes in step-equilibrium the plot of log,,C against (E+)c is a continuous curve. The first serious attempt to consider the full implications of step-equilibria in the study of complex-formation by the polarographic method was made by DeFord and Hume8 who obtained the following relation between the change in half-wave potential and the free-ligand concentration [XI for a reversible metal deposition N [a = (Ed* - ( m c l Here IM and Ic are the diffusion-current constants for aquo and complex metal ions respectively the y’s are activity coefficients the P’s are stability constants (Po = 1 for the “zero” complex) and [XI = free-ligand concentration.The current-potential curve for such a deposition has been shown to be:g N RT i ~ nF nF id - i &i.e. = E0 - - In 2 &[X]i - (7) 0 The graphical method due to Ledenlo is then applied. If allmeasurements are made at the same ionic strength and the activity quotients dropped from eqn. (6) the right-hand side may be denoted by Fo[X] indicating that it is a function of free-ligand concentration and written as follows &[XI = P o + Pl[XI + P,[XI2 + * - * 3- P“X1N (8) A new function Fl[X] may then be defined by In a similar manner other functions may be derived giving finally * D.D. DeFord and D. N. Hume J. Amer. Chem. Soc. 1951,73,5321. P. Kivalo and H. A. Laitinen J. Amer. Chem. Soc. 1955 77 5205. lo I. Leden 2. phys. Chem. 1941,188 160. 60 QUARTERLY REVIEWS From eqn. (9) a plot of Fl[X] against [XI gives a curve having a limiting slope of p2 at [XI = 0 and an intercept of pl. Such plots are made for each function until all the complex species are accounted for the penulti- 1 that for the highest com- axis. Fig. 1 is a schematic mate graph being linear with positive slope and plex a straight line parallel to the concentration r 1 I I BI - FIG. 1. Schematic plots of F[X] functions for a system of three complex species in step-equilibrium. representation of a set of graphs expected for such a system with three complex species formed.In the original DeFord-Hume derivation,8 (E#) was designated (E+O) to indicate that it was the half-wave potential of the simple ion when its activity was unity. There is then little justifica- tion for assuming that the first term in eqn. (6) is unity it is in fact p0/.yM. As used above (E& refers to the value obtained in a medium of the same ionic strength as that used in the measurements on the com- plexed cation. For a particular system the ligand number i is definedll by so that a plot of log,J,[X] against loglO[x] gives a curve whose slope at any point gives the value of ii corresponding to a particular value of [XI. i then represents the average composition MX, of the species present in solution. On increasing [XI the composition will approach that of the highest complex MXN and eqn.( 5 ) may be expressed in the form Eqn. (12) is identical with eqn. (2) under these limitingconditions if it is assumed that I = Ic and [XI = Cx. The earliest practical applications of the theoretical work of DeFord and Hume were to the cadmium and l1 H. Irving “Advances in Polarography,” Pergamon London 1960 vol. 1 p. 49. CROW AND WESTWOOD COMPLEXED METAL IONS 61 zinc thiocyanate complexes. Hume et a2.l2 determined the half-wave potentials of the cadmium ion in potassium nitrate-thiocyanate mixtures over a thiocyanate concentration range 0-1-2-0 M at a constant ionic strength of 2 M. The above treatment revealed the presence of four com- plex species CdSCN+ Cd(SCN), Cd(SCN), and Cd(SCN),2- with consecutive stability constants 11 56 6 and 60 respectively.Over this thiocyanate concentration range the half-wave potential was observed to change from -0.5724 v to -0.6646 v [against standard calomel electrode (S.C.E.)] with a corresponding decrease in the diffusion current from 7.56 PA to 7-03 PA. Such small shifts in Ei values demand very careful measurement and this can only satisfactorily be done by the use of manual equipment. In the case of the zinc complexes the overall shift was from -0.9977 v to -1.079 v (against S.C.E.),13 so small that earlier investigators had reported the absence of c~mplex-formation.~~ A recent study of the cadmium-azide system15 revealed the presence of the five complex species GIN,+ Cd(N3)2 Cd(N,),- Cd(N,),,- and Cd(N3)2-. (c) Formation of Mixed-ligand Complexes.-Since the above early applications many other systems have been studied.A notable contribu- tion which seems to have aroused little interest is the extension of the DeFord-Hume approach by Schaap and McMasterP to deal with mixed- ligand systems. For a complexing reaction of the type M + iX +jY + . . . . = MXtYi.. . . (1 3) in which i j . . . . are stoicheiometry numbers and X Y . . . . are different ligands the DeFord-Hume expression for the Fo[X] function may be extended to give a new function Fo0 .... [X,Y . . . .I given by N AEh + log - M As before each term in eqn. (14a) is determinable except for y,,which is included in the resultant equilibrium constants. For a total of three bound ligands of the type X and Y factorisation of the Fo0 function leads to I;~o[XYYI = (180 P01fyI P0zI?12 1803[Y13) [XI0 (810 IBIIIYI P12[YI2IEXI 4- ( 8 2 0 + /%i[Y]>[Xl2 (15) (16) $- 1 P 3 o ) [XI3 or&,IX,Yl = A + B P I + C[X12 + D[XI3 la D.N. Hume D. D. DeFord and G. C. B. Cave J. Amer. Chem. SOC. 1951 73 5323. l3 R. E. Frank and D. N. Hume J. Amer. Chem. SOC. 1953 75 1736. P. R. Stout and J. Levy Coll. Czech. Chem. Comm. 1938 10 136. l6 P. Senise and E. F. de Almeida Neves J. Amer. Chem. Soc. 1961,83,4146. l6 W. B. Schaap and D. L. McMasters J. Amer. Chem. SOC. 1961 83,4699. 62 QUARTERLY REVIEWS where for a given w] A By C and D are constants. The original graphical solution may be applied to the Foo data if the activity of one of the ligands is held constant while that of the other is varied. For the copper- and cadmium-ethylenediamine-oxalate systems the oxalate concentration was held constant (at several fixed values) while that of ethylenediamine was varied.Thus [XI = [en] and [Y] = [ox]. For the Cd-en-ox system in which possible mixed complexes are Cd(en)(ox) Cd(en),(ox) and Cd(en)(ox),2- values of A and D were known from studies of the simple Cd-en and Cd-ox systems. B was obtained graphically from the Flo function defined by Foo - I;;o = [+] = B 4- c[en] + D[enI2 by plotting Flo against [en]. Similarly C is given by the Fzo function With a knowledge of C the mixed-complex stability constant ps1 may be calculated but in order to determine isll and isl2 B must be evaluated for at least two different oxalate concentrations. 3. Stability Constant Data from Irreversible Waves There are two essential approaches to the determination of stability constants from irreversible waves.These involve the use on the one hand of diffusion-current measurements and of half-wave-potential data on the other. In the majority of cases a process of competition for a ligand by the metal studied and an “indicator ion” is used. (a) The Use of Current Measurements.-In all the relations so far given it is usually assumed that the diffusion coefficients of the various species in solution do not differ to any great extent from one another and that the limiting current is therefore independent of ligand concentra- tion. Should there be a measurable difference between the diffusion co- efficients of a metal M and its complex MX the stability constant of the latter species may be determined by observing the change of diffusion current with ligand concentrati~nl~’~~ and applying the relation19 where DM and DMx are the diffusion coefficients of the free and com- plexed metal ions respectively and D is the observed mean coefficient in a solution containing the two species.Eqn. (19) can only be used for systems containing a complex with one bound ligand but since no poten- l7 V. KaCena and L. MatouSek Coll. Czech. Chem. Comm. 1953 18 294. l8 Z . Zabransky Coll. Czech. Chem. Comm. 1959 24 3075. lS J. Koryta “Progress in Polarography,” ed. P. Zuman Interscience New York 1962 vol. 1 p. 291. CROW AND WESTWOOD COMPLEXED METAL IONS 63 tial measurements are needed it may be applied to irreversible processes. Use of the method assumes a rapid attainment of equilibrium between the metal ion and the complex. In the case of very stable complexes for which completely non-labile equilibria may be assumed current measurements may also serve to pro- vide stability constant data by use of a method of competitive complex- formation largely developed by Schwarzenbach and his co-workers.20-22 Suppose that a metal complex MX is irreversibly reduced or is even electro- inactive at the dropping-mercury electrode.In order that the stability constant of MX may be determined it is necessary to have access to a complex NX whose stability constant is known and is of the same order as that of MX. The wave for uncomplexed N must be reversible occur at a more positive potential than that of M and be undetectable in the presence of NX when N and X are present together in equivalent concen- trations. Briefly the experimental procedure is as follows polarograms are obtained for N in both the presence and absence of X.A measured quantity of M is now added which competes for the ligand X some of which it abstracts from the previously formed NX finally setting up the equilibrium By this action ions of N are liberated and reduction waves are observed for both N and NX whose heights are directly proportional to the concentrations (Fig. 2). By observing the above conditions waves for M and X do not interfere; that of MX might well appear after that of hydrogen. The equilibrium constant for reaction (20) is given by M + N X + M X + N (20) Hence when KNX is known KMx is determinable. This technique has been used by Schwarzenbach and S a n d e ~ a ~ ~ in studies of vanadium complexes with ethyleiiediaminetetra-acetic acid (E.D.T.A.) using copper as the indicator ion N MX being NaJOY and NX Na,CuY.Many metals have been employed as the auxiliary cation in this method e.g. manganese zinc cadmium mercury copper and iron in the deter- mination of stability constants of aminopolycarboxylate complexes of many metal ions including the lanthanide~.~~-~~ 2o G. Schwarzenbach and H. Ackermann Helv. Chim. Acta. 1952,35,485. z1 G. Schwarzenbach R. Gut and G. Anderegg Helv. Chim. Acta. 1954 37 937. 22 K. Bril and P. Krumholz J. Phys. Chem. 1953 57 874. 23 G. Schwarzenbach and J. Sandera Helv. Chim. Acta. 1953 36 1089. 24 L. Holleck and G. Liebold Naturwiss 1957 22 582. a5 D. M. H. Kern J. Amer. Chem. SOC. 1959 81 1563. 26 F. H. Spedding J. E. Powell and E. J. Wheelwright J. Amer. Chem. SOC. 1956 78. 34.. - I - 27 G. Schwarzenbach and R. Gut Helv. Chim. Acta. 1956,39 1589. E. J. Wheelwright F. H. Spedding and G. Schwarzenbach J. Amer. Chern. Sac. 1953,75,4196. 64 QUARTERLY REVIEWS t FIG. 2. (a) Polarogram of metal ion N. (6) Polarogram of complex-ion NX. (c) Super- position of wave due to N upon that of NX on addition of metal ion M. (b) The Use of Potential Data.-A method due to S~brahrnanya~~ utilises a modification of the treatment of Lingane by Tamamushi and Tanake30 for an irreversible process i.e. (22) - j x 2.303RT - - A E d log c x an F By using the modified Heyrovsky-Ilkovic equation for an irreversible reduction 2RT anF i.e. Es - Ep = - . In 3 from which an was evaluated. Thus j was solved and the dissociation constant Kc deduced from the expression RT anF an F (Et) - (E& = - In Kc --lEln Cx R.S. Subrahmanya “Advances in Polarography,” Pergamon London 1960 vol. 2 p. 674. a. R. Tamamushi and N. Tanaka Bull. Chem. SOC. Japan 1949 22 227. CROW AND WESTWOOD COMPLEXED METAL IONS 65 The method has been applied to the study of the mono- di- and tri- ethanolamines of iron cadmium nickel cobalt copper lead and zinc at 30” in alkaline media. A few of the mono- and di-ethanolamine compounds are reversible but the triethanolamine complexes are invariably irrevers- ible. In addition at higher pH values OH- NH, and CO2- tend to enter the complex and by variation of the concentration of these species with constant amount of complexing agent in the base solution some of these additional species have been identified and their dissociation con- stants calculated.However the success of the method relies upon the reversibility of the simple ion in eqn. (24) and conditions giving a constant value of a so that it was inapplicable to cobalt and nickel although the formulae of the complexes including the mixed-ligand complexes were deduced. A further technique based on theoretical suggestions of Ringbom and E r i k ~ s o n ~ ~ ~ ~ also uses an indicator ion to compete with the ion studied for the complexing ligand. The method can be used in principle at least for studying systems containing a single complex or a series of complexes in mobile equilibrium which are electro-inactive or which give irreversible waves. If the indicator metal ion N is present in solution with an excess of ligand addition of ions M will decrease the free-ligand con- centration and give a consequent shift of the wave for the N-X system to more-positive values.It is necessary that M should react very rapidly with the ligand and that the complex equilibrium is established almost in- stantaneously. That the latter may not occur is a weak feature of the method A preliminary experiment is performed in which the shift in half-wave potential for the system NX is determined with increasing free-ligand concentration. The position of E+ for the system NX when M has been added then gives directly the free-ligand concentration from the preliminary calibration. The ligand number for the system MX is given by at the half-wave p~tential,,~ where Z‘ is the ligand number of the system NX [XI the free-ligand concentration Cx the total ligand concentra- tion CN the concentration of N and CM the concentration of M.If the activity coefficients are neglected the DeFord-Hume Fl function is expressed by N Fl[X] = KI 4- &D(] -l- &[XI2 -k . . . = z&[x]’-’ (27) 1 A. Ringbom and L. Eriksson Acta. Chem. Scand. 1953,7 1105. 8a L. Eriksson. Acra. Chem. Scand. 1953 7 1146. s3 F. J. Rossotti and H. Rossotti “The Determination of Stability Constants,” McGraw-Hill New York 1961 ch. 8 p. 185. 66 QUARTERLY REVIEWS It was shown by Fronaeou~~~ that N Then the graph of Fl[X] against [XI gives Kl as intercept and K2 as limiting slope. The process is continued up to FN. In this way Zabranskyl* determined the stability constants of the sodium and lithium E.D.T.A. complexes by using thallium as the indicator metal. Plumbic lead has also been used as the auxiliary cation for the determination of the stability constants of the chloro-complexes of nickel and zinc.35 The tedious nature of the technique when a series of complexes is formed and the slow attainment of equilibria in many cases may ac- count for the somewhat limited application and exploitation.4. Limitations of the Graphical Method Graphical solutions of the F[X] functions involve cumulative errors and this is reflected in the increased scattering of points in the graphs for higher complexes. This becomes even more apparent in the case of mixed-ligand complex-formation. Some attempt has been made for example by Irving,36 to use algebraic solutions for the various p’s. If more simultaneous equations of type (8) than there are unknowns can be set up from the polarographic data a least-squares treatment can be employed to find the best set of values.The ultimate equations in the DeFord-Hume and Eriksson techniques possess properties which allow of their being treated by the relaxation method. Examples of the use of the method with data obtained by using the above techniques are given in an excellent Paper by Watkins and Jones.37 Although the preceding refinements are of great importance it should be borne in mind that much improved data can be obtained by correct control of experimental conditions. Approximations valid for some sets of conditions do not hold in others. For example in the original treatment of the Cd-SCN system the total ligand concentration was used as an approximation for that of the free ligand (in terms of which the Leden functions are expressed).In this case the assumption was valid since the thiocyanate concentration was so large compared with those of the com- plexes that in the most dilute solutions with respect to the ligand the 84 S. Fronaeous Acta. Chem. Scand. 1950 4 72. a6 B. Kivalo and R. Luoto Suomen Kern. 1957 30B 163. 36 H. Irving ref. 11 p. 52. 87 K. 0. Watkins and M. M. Jones J. Znorg. Nuclear Chem. 1961,16,187. CROW AND WESTWOOD COMPLEXED METAL IONS 67 greatest error introduced was no more than 0.5 per cent. In cases where this assumption is no longer valid the ligand number n must first be found assuming at this stage that [XI = Cx. Then true (or truer) values of [XI may be calculated by substitution into the relation The fraction IM/Ic [see eqn. ( 5 ) ] should be included in the calculations since although small its neglect produces a significant positive error in the final consecutive stability constants.Many workers have ignored this fraction in their studies. Above all temperature and ionic strength must be maintained strictly constant over the entire ligand-concentration range. Any effect causing a potential shift which can be superimposed on that due to complex-forma- tion must be rigorously excluded. 5. Structural and Kinetic Factors affecting Reduction The overall redox process occurring at the electrode and in its immediate vicinity may be represented diagrammatically (Fig. 3) division into the stages being quite Elect surfa I Structural changes I I influence giving I under electrode I form capable of I ' Electron direct reaction transfer I with electrode 7- I - I+- I - I Structural changes I in primary products a c Edge Dde x dou b '3 Structural changes before entering double layer ___.I Chemical 1 changes of products >f electrode 3 layer Depolarizing particles diffuse in from bulk solution t -+ Products diffuse out t o bulk solution Electrode process -+ I I I+- I I=+ overall redox process - I (a) Structural Factors.-Both the mechanism and rate of the overall electrode process depend on the energy and localization of the lowest unoccupied (or singly occupied) orbital on the reducible species. It is into such orbitals that the electrons provided at the electrode in a reduc- A. A. Vlcek Progr. Org. Chem. 1963,5,216. 68 QUARTERLY REVIEWS tion process are re~eived.~*,~~ Should the depolarizing particle have a high electron affinity direct transfer of electrons to these vacant orbitals may take place.If such a condition does not hold the electronic energy of the complex may change in such a manner that direct reaction with the electrode is made possible.41 The energy necessary to produce this change constitutes an important and sometimes the major part of the activation energy of the electrode process. Configurational changes occurring im- mediately after the electron transfer cannot be directly determined and have to be inferred from the structures of secondary products derived from them.41,42 In attempting to correlate the structure of complexes with polaro- graphic behaviour recent studies have considered cases that reluctantly undergo substitution reactions so that the structure in solution may be assumed to be little different from that in the solid state.The most useful complexes in this respect have proved to be those of trivalent cobalt and chromium. Changes such as aquation which may occur in solution can in fact be of more help than hindrance since they usually take place to a sufficient extent to allow the change in morphology of the waves with time to be followed. Recent studies by V 1 ~ e k ~ ~ s ~ ~ have shown important correlations be- tween polarographic and spectroscopic behaviour. The energy of the “reactive” or “transition” state (capable of taking part in the electron transferences) depends on the energy differences between the ground and excited states of the central metal ion. Thus the energy depends on the ligand-field strength of the complex and if this is too large the complex may well be non-reducible.Any effect which decreases the energy difference leads to easier reduction. Vlcek’s studies involved substitution in octa- hedral cobalt(II1) complexes of the type COX to form CoX,Y. In many cases a Cox,-type complex is non-reducible whereas a substituted form may be reducible owing to the splitting of the e and tzg levels caused by the substituent Y. Should both forms of the complex be reducible the substituted form has the more-positive half-wave potential. The larger the separation of the ligands X and Y in the Spectrochemical Series the larger the potential shift. Similar trends are found for the complexes of chromium(II1) and rhodium(II1). (b) Distinction between Isomeric Species.-A study of cobaltammine isomers by Willis Friend and Mel10r~~ showed that polarographic half- wave potentials may sometimes be used to distinguish between the two structural types in solution.Since this work several studies of both A. A. Vlcek Coll. Czech. Chem. Comm. 1955,20,894. 40 A. A. Vlcek Nature 1956 177 1043. 41 A. A Vlcek Coll. Czech. Chem. Comm. 1957 22,948. 4 2 A. A. Vlcek Coll. Czech. Chem. Comm. 1957,22 1736. d3 A. A. Vlcek Discuss. Faraday SOC. 1958,26 164. 4 4 A. A. Vlcek Coll. Czech. Chem. Comm. 1959 24 181. 46 J. B. Willis J. A. Friend and D. P. Mellor. J. Amer. Chem. SOC. 1945 67 1680, CROW AND WESTWOOD COMPLEXED METAL IONS 69 octahedral and planar isomeric species have been carried out. Holtzclaw and sheet^,^,,^ found that in six-co-ordinated complexes of cobalt(m) containing two negative groups the trans-isomer was reduced at a more- negative potential than was the cis-form.For octahedral cobalt complexes the reductions proceed irreversibly in two main stages viz. cobalt(m)+cobalt(iI) and cobalt(II)-+cobalt. However the first stage cobalt(In)+cobalt(II) is in many cases repre- sented by a doublet wave and in others by a single wave. For example in the case of the isomers of [Co(NH,>,(NO2),]+ a single wave is obtained for the cobalt(rI1) to cobalt(I1) step with Ei values of -0-05 and -0.21 v (against S.C.E.) for the cis- and trans-forms respectively. The same relationship between Ei values and structure occurs with the first waves of the doublets as shown in Table 1. TABLE 1. Comparison of E+ vaIues for cis and trans cobalt(n1) isomers containing two negative groups.47 Complex Reduction cobalt(m)-+cobalt(rr) First wave Second wave -(Ei)i (v) (w.S.C.E.) - ( I 3 2 (v) (w.S.C.E.) [Co(en)2(NO,) I' cis 0.24 cis 0.41 trans 0.27 trans 0.40 CCo(en),(NCS)(No& I+ cis 0.04 cis 0.38 trans 0.12 trans 0.36 The second wave which in each case appears at about -0.4 v may correspond to the reduction of an aquated form of the parent complex the reduction of the parent complex itself being represented by the first wave of the doublet.Slight discrepancies between half-wave-potential values for the second waves in Table 1 would tend to suggest that these correspond to the reduction of different intermediate forms. The aquated species may be a hydroxo-complex of some A study of similar complexes with less than two negative groups was also carried out by the above workers and Ei values are given in Table 2 for two representative pairs.TABLE 2. Comparison of Ei values for cis and trans cobalt(1n) isomers with one negative group.Q7 Complex Reduction cobalt (111)-+co balt (11) First wave Second wave -(Ei)1 v (v3.S.C.E.) -(E+)2 v (v3.S.C.E.) [Co(en),NO2NH3I2+ cis 0.21 cis 0-40 trans 0.20 trans 0.40 [ CO(~~),NH,NCS]~+ cis 0.13 cis 0.39 trans 0.10 trans 0.39 p* H. F. Holtzclaw jun. J. Amer. Chem. SOC. 1951 73 1821. 47 H. F. Holtzclaw jun. and D. P. Sheetz J . Amer. Chem. SOC. 1953,75,3053. 70 QUARTERLY REVIEWS In their original publication the authors stated that in octahedral com- plexes containing only one or no negative groups there was no apparent difference between the polarographic behaviour of isomeric species.However as has been the data tend to suggest that slightly easier reduction of the trans-isomer occurs in such cases. of cis- and tvans-[Rh(en),Cl,]+ have shown that the half-wave potential of the cis-isomer is considerably more-negative than that of the trans-form. This is in contradistinction to the case of cobalt(m) complexes of similar type. (See Figs. 4 and 5 for polarograms of the two Recent FIG. 4. Polavograms of trans (n and cis (11) [Co(en),(NO&]+ (ca. 2.5 X M) in 0.1 M-NaClO,. Both waves start at 0.0 v. FIG. 5. Polavograms oftrans (I) and cis (TI) [Rh(en),Cl,]+ (ca. 2-3 x M) in 0 . 1 ~ - cases.) In our view it seems reasonable to explain the effect in terms of reduction occurring on different sides of the electrocapillary maximum for the cobalt and rhodium cases.In view of the greater internal dipole which exists within the cis-isomer of an octahedral complex containing two negative groups Holtzclawso suggested that in the absence of a supporting electrolyte this isomer on account of its correspondingly greater orientation in the unsymmetrical field around the dropping-mercury electrode should migrate to this electrode at a rate different from that of the trans-form. Investigations of NaC10,. Wave I starts at 0.3 v; wave I1 at 0.5 v. 48 J. R. Hall and R. A. Plowman Austral. J. Chem. 1956,9 14. 49 D. R. Crow and J. V. Westwood to be published. 6o H. F. Holtzclaw jun. J. Phys. Chem. 1955 59 300. CROW AND WESTWOOD COMPLEXED METAL IONS 71 various pairs of isomers in solution under these conditions showed that the different migration effects were not reliable as a means of distinguishing between the two structures.Half-wave-potential data on the other hand have found use in the identification of the different ~ p e c i e s . ~ ~ ~ ~ Similar studies have been carried out on the four-co-ordinated complexes of platinum and p a l l a d i ~ m . ~ ~ ~ ~ Platinum(r1)-tetra-amine ions in which the ligands are ammonia methylamine dimethylamine ethylenediamine pyridine aniline or combinations of these show a polarographic distinc- tion between cis- and trans-isomers. Trans-forms undergo easier reduction than do the cis-forms suggesting the greter thermodynamic stability of the latter. Chakravarty and Bane~jea~~ observed that for platinum(I1) and palladium(@ complexes containing two negative groups there is a signific- ant difference (of the order of 60 mv or more) in the E+ values of the isomeric forms the cis- now being reduced at a more-positive potential than is the trans-form.The general behaviour of such complexes is very similar to that of six-co-ordinated cobalt(m) complexes. For diaquo- and chloroaquo-complexes Ei values are almost identical. A striking difference between the planar and octahedrally co-ordinated structures is that for the former a definite difference between the behaviour of cis- and trans- forms is observed even when all groups are neutral. (c) Kinetic Factors.-Very often the rate of dissociation of a complex as well as the rate of diffusion controls the limiting current. It has proved possible to determine stability constants from the half-wave potentials and kinetic limiting currents obtained from dissociation-rate-controlled waves.The half-wave potential of the kinetic wave is less negative than that of the hypothetical diffusion-controlled wave and the difference can be used to compute consecutive stability constants in the normal way provided that the dissociation reaction is followed by a reversible dis- charge process. The relation for a diffusion-controlled deposition [see eqn. (7)] has been modified by K ~ r y t a ~ * - ~ ~ and Matsuda and Ayabe5’ to 0 Here i k replaces id of eqn. (7) and a new term is added. This in effect “corrects” the half-wave potential of the kinetic wave for the slowness of dissociation relative to the diffusion rate of a particular complex. Then 61 Z. E. Gol’braikh Zhur. neorg. Khim. 1956,1 1739. s3 B. Chakravarty and D.Banerjea J. Inorg. NucZear Chem. 1961,16 288. 64 J. Koryta Coll. Czech. Chem. Comm. 1958 23 1408. 65 J. Koryta Coll. Czech. Chem. Comm. 1959 24 2903. s6 J. Koryta Electrochim. Acta 1959 1 26. 67 H. Matsuda and Y. Ayabe Bull. Chem. SOC. Japan 1956,29 134. E. A. Maksimyuk and G. S. Ginzburg Doklady Akad. Nauk S.S.S.R. 1959 124 1069. 72 QUARTERLY REVIEWS and RT id + - I n nF i k (33) The method involves an estimated id value based on a reasonable value for D and a knowledge of n. These equations have found important uses for the determination of stability constant data for several systems. After “correcting” for the half-wave potential the procedure follows the DeFord-Hume graphical method.55p58p59 It is rare especially if faced with several step-wise equilibria to be able to know precisely the nature of the species present in solution.Not only are there often many species but added complications may arise due to binuclear complex-formation and also the low mobility of equilibrium between some species in the step complexes. Further not all the com- plexes may be electro-active at the mercury electrode. In some cases it has proved possible to determine both the chemical reaction controlling the overall rate and its rate constant. KorytaS0 studied the cadmium- cyanide system ([CN] = 0-005-0-1 M) and obtained a limiting kinetic current ik related to id the limiting (diffusion) current for rapid dis- sociation by use of the expression. = constant x (p - N - 8)ln[X] (34) where p and N represent the composition of the complex whose dissocia- tion is rate-determining and that of the highest complex present respect- ively.Experimental determination of the quotient d{ In [ik/(id - ik)]}/ d{ln[X]} yields a value for p when N is known. In the system quoted N = 4 and the quotient was found to have the value -3/2. Hence p = 3 and the reaction is rate-controlling. Should there be in a mobile system of complexes a species whose reduction (electrode reaction proper) controls the overall rate the com- position of this species may be found by a similar treatment. If p again represents the composition of the unique species and r p the rate constant of its reduction process the current relationship is Cd(CN),- = Cd(CN) + CN- (35) i id - i - . - 0.886 rD t ) D-5 ( k ~ . . . . kp+l.[X]N-p)-l (36) J. Koryta Coll. Czech. Chem. Comm. 1959 24,3057.69 D. Konrad and A. A. Vlcek Cull. Czech. Chem. Comm. 1963,245,808. 6o J. Koryta 2. Efektrochem. 1957,61,423. CROW AND WESTWOOD COMPLEXED METAL IONS 73 where t and D have the usual significance and the k’s are stability con- stants. Again p is found from the change in current with ligand concen- tration. In cases of more-complex behaviour both equations above have to be combined to take account of the two effects. 6. Solvent Variables In studies of complex-formation which involve measurements of half- wave-potential shifts it is necessary that care be taken to ensure that the observed shifts are due to complex-formation-especially that involving the required ligand-without interference arising from other ligands. It has already been shown that the nature of the complexes present in solution can differ widely with changing concentrations of the complexing ligand.If indifferent electrolytes with good co-ordinating ability are used these can have a pronounced effect on both half-wave potentials and diffusion currents. For example HoltzclaW46 investigated the behaviour of cis- and trans-dinitrotetramminecobalt(1ir) chloride with increasing concentrations of chloride tartrate and citrate and Laitinen et carried out similar studies on hexamminecobaltic chloride. Half-wave potentials were shifted in the negative-potential direction and diffusion-current constants were at the same time reduced. Such behaviour is attributed to “supercomplex” formation due to clustering of base electrolyte anions by ion-dipole and electrostatic attraction about the central complex.The formation of supercomplexes is closely allied with the phenomenon of ion-pairing the nature of which has been largely clarified by the work of BjerrumGa and Fuoss and K r a u ~ . ~ ~ Tur’yan and Bondarenkoe4 studied the effect of non-aqueous solvents on the polarographic behaviour of many complexes. The main feature of this work was the use of various concentrations of methanol or ethanol in water. The dielectric constant was found to have a pronounced effect on the stability of cadmium thiocyanate complexes; the stability con- stants varied inversely as the dielectric constant. In such mixed-solvent systems lower members of the Cd-SCN series of complexes are formed preferentially with a lower concentration of higher complexes and a general increase in stability with increasing coiicentration of non-aqueous solvent.Solutions of ethanol 2-methoxymethanol and dioxan in water have been used as polarographic solvents for copper chelates of 1,3-dike- tonesG5 and for copper and cadmium complexes with thiourea and its homologues. 66 Nightingale and HoltzclawG5 derived the expression 61 H. A. Laitinen J. C. Bailar H. F. Holtzclaw jun. and J. V. Quagliano J. Amer. Chem. SOC. 1948 70 2999. 62 J. Bjerrum Kgl. danske Videnskab. Selskab. Mat.-fys. Medd. 1926 7 No. 9. 83 R. M. Fuoss and C. A. Kraus J. Amer. Chem. SOC. 1933,55,1019,2387. 64 Ya. 1. Tur’yan and N. I. Bondarenko Zhur. neorg. Khim. 1959,4,1070. 66 E. R. Nightingale and H. F. Holtzclaw jun. J. Amer. Chem. Soc. 1959,81 3523. 66 T. J. Lane J. W. Thompson and J. A. Ryan J. Amer. Chem. SOC. 1959,81,3569. 74 QUARTERLY REVIEWS dE+ = constant x A k) (37) relating the half-wave-potential change and the corresponding change in dielectric constant E .Experimentally the value of the constant was found to be -3.5 in fair agreement with the calculated value of -4.09. The same workers investigated the effect of viscosity changes on the reduction of the ketone chelates and found that the Stokes-Einstein relation was in general obeyed. Values of diffusion coefficients needed for computation of the Stokes-Einstein product DT were obtained from the Lingane- Loveridge m~dification~~ of the Ilkovic equation which had not been rigorously tested for non-aqueous-solvent systems until this study. In a more recent study the effects of solvent isotopes have been ex- amined. Light and heavy water were used as solvents for the polarographic determination of stability constants of cadmium and copper oxalate complexes.6s The constants were found to be greater in heavy water than in the more-strongly-solvating light water.Increase in the ionic strength of an electrolytic solution shifts the half- wave potentials to more-negative values. In using the DeFord-Hume method activity coefficients are assumed to remain constant and inde- pendent of the concentration of the complexing agent by holding the ionic strength at a constant value. This assumption is not always justifiable in cases where concentrated solutions of mixed electrolytes are used. The polarographic method indeed has been applied to the determination of formation constants in LiN03-KN03 melts at 180°.69 In this way the chloro-complexes of nickel cadmium and lead were studied by the DeFord-Hume method.A similar application involved the formation of the species AgCI AgCI,- and Ag,C1+ in molten KN03.70 7. Thermodynamic Quantities Evaluation of AGO AHo and ASo is in principle possible from measurements of the formation constants at various temperatures al- though relatively few of such studies have been made polarographically. Earlier work on the complexes of nickel and copper with glycine and valine over the range 25-40' gave results in reasonable agreement with those obtained from potentiometric studies. 71 The AHo values varied from -14.0 to -21-0 kcal. but the ASo values varied considerably with both positive and negative results. A more recent study of cadmium-pyrazole complexes7 confirmed the presence of 1 1 1 :2 and 1 3 meta1:complex 67 J.J. Lingane and B. A. Loveridge J. Amer. Chem. SOC. 1950 72 438. 68 D. L. McMasters J. C. Raimondo L. H. Jones R. P. Lindley and E. W. Zelt- 6 9 J. H. Christie and R. A. Osteryoung J. Amer. Chem. SOC. 1960 82 1841. 'O J. Braunstein M. Blander and R. M. Lindgren J. Amer. Chem. SOC. 1962 84 mann J. Phys. Chem. 1962 66 249. 1529. N C. Li J. M. White and R. L. Yoest J. Ainer. Chem. SOC. 1956 78 5218. CROW AND WESTWOOD COMPLEXED METAL IONS 75 ratios with mean values of AHo of -3.94 -8.10 and -12.14 kcal. respectively for the overall reactions over the temperature range 0-45’. A regular increase of AHo thus occurs with each ligand attachment with a corresponding negative entropy increase. The variability in entropy values for complex-formation is well shown by the values obtained for the mercury(Ir) cadmium and zinc complexes with ethylenediamine with values of -5 +3.4 and +10.7 e.u.re~pectively,~~ and more recently by the cadmium complexes of histamine and related compounds 73 where ASo varied from -19 e.u. for antistine to +4 e.u. for benadryl. The loss of translational degrees of freedom by ligand attachment would suggest a greater degree of order and hence a negative entropy value but this may be offset by a loss of hydration of the metal ion. The latter may in fact be sufficient to swing the value over to the positive side. The field force due to the central ion may still be sufficient to maintain some degree of order- ing of the solvent outside the sphere of the ligand (the “iceberg” concept of Frank and Wen74).With a multidentate ligand more water tends to be displaced from the metal ion and the entropy changes are greater. The entropy values give some indications as to the nature of the complex and show whether inner-sphere or outer-sphere complexes are formed. FIG. 6. Plots of loglo kj against (j - 1) for several systems of complexes. 0 Cu-F I3 (ref. 5). Cd-OX*- (ref. 16). a Cd-N8+ (ref. 15). 0 Cd-thiourea (T. J. Lane “Advances in Polarography,” Pergamon London. 1960 vol. 2 p. 797). 0 Pb- thiourea (ref. as for Cd-thiourea). 72 D. K. Roe D. B. Masson and C. J. Nyman Analyt. Chem. 1961,33 1464. 73 A. C. Andrews and J. Kirk Romary J. 1964,405. 74 H. S . Frank and W. Y . Wen Discuss. Fmaaby Soc. 1957 24 133. 76 QUARTERLY REVIEWS The values of the stability constants for step-wise complexes show an interesting correlation.Van Panthaleon van E c ~ ' ~ suggested a relationship of the form where X is an empirical parameter for each system. This however appears to have been little exploited. Accordingly values for some of the systems which have been reported have been calculated from the overall step constants and are shown in the form of a graph of log, Kj against ( j - 1) (see Fig. 6). These systems give reasonably linear graphs and this suggests an equal change in free energy at each increase in ligand. How- ever in some systems e.g. cadmium thiocyanate,12 the results are not so encouraging. 8. Conclusion Few methods are available for studying step-wise complex-formation in solution and of these the polarographic method has some advantage over spectral methods since the identification and the determination of the properties of more species is possible simultaneously.However it must be admitted that since the measurement of very small differences in half-wave potentials is often involved the accuracy in the determination of the higher stability constants so far obtained is often not as great as is desired. This is unfortunate since the potentialities for obtaining thermo- dynamic data are considerable. There are however indications that recent workers are realising these advantages and greater care is being observed over solution factors. More information is required for families of compounds of similar composition to relate general electrode behaviour with structure. So far only cobalt compounds have been studied in considerable detail particularly by Vlcek but other elements e.g. chromium rhodium and iridium would be interesting. In particular there is wide scope for studying the electrode mechanisms of complex species. This is not an easy field being somewhat bedevilled by the complications arising from the electrical double layer but is a fascinating and rewarding one. 75 C. L. van Panthaleon van Eck Rec. Trav. chim. 1953 72 529.