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Polarography of metal complexes
Polarography of metal complexes Introductory lecture
GUIDO SARTORI Istituto Chimico dell'Universitk d! Trieste - Italia
S"mm"'7: T h e p o U i b f l i t i e s of the p o l a r o g r a p h i c m e t h o d i n the study of complexes i n s o l u t i o n a r e d i s c u s s e d e i t h ~ w h e n r e d u c t i o n follows t h e d i s s o c i a t i o n m" w l m n t h e c o m p l e x il the r e d u c i b l e e n t i t y .
The study of complex ions in solution by the polarographyc method started already thirty years ago with the classical work of Heyrovsky in 1927. Since then both theory and experimental methods have developed to a closed chapter which presents to day a complete picture and enumerates beyond his achievements not only the individuation of the complex or the complexes present in solution, but also the determination of the coordination number, of the stability constant and some time the right interpretation of the reduction mechanism. If one wishes to remember the names of the scientists which have given the most theoretical contributions to this field of research, I would recall the works of Stackelberg in 1939, of Lingane in 1941, of Souchay and Foucherre in 1947, of Gierst and Juliard in 1950 and of De Ford and Hume in 1951. It is a special merit of these workers particularly and of all those which has contributed theoretically and experimentally, in this connection I will remember the italian schools of Boma and Padova, if to day we have a closed picture of the complex phenomenology. As it is already known, the half wave potential for a simple ion is expressed through the formula: (1)
(E~i)~ = E*. +
RT nF
In a . ,
RT nF
In
y a. (D~)~ (y,)~. D~
EO standard potential of amalgam, i.e.e.m.f, of the pile: Ref. E1./M ~ +/M (1-19) when C°. %/a~g. C~,. %--= 1 % pratical molar activity coefficient of the metal in the amalgam
POLABOGBAPHY
(Y,)t~
OF METAL
197
COMPLEXES
pratical molar activity coefficient of the metallic ion at ionic strenght
all0 activity coefficient of H 9 in the amalgam Da diffusion coefficient of the metal in amalgam (D,)~ diffusion coefficient of the ionic species at ionic strenght It is also known that the diagnostic prouve of the formation of a complex in solution is the horizontal shift of the half wave potential by adding to the solution eomplexing agent. From (1) it is clear that the potential is a function of the activity coefficient of the reducible ion and therefore of the ionic strength of the solution. lndeed we have: (2)
AE~ : (E½)g - - (E~t)tt_ o
_
RT
In(yo)g +
nF
RT
nF
In ( D ~ ) t ~ - °
(D~)~.
The accurate studies of Lingane, De Ford and Papoff show that this shift, on changing the ionic strength from O to I are approximately 15 mV for a bivalent ion. This behaviour leads to the necessity to work al constant iou strength or to know exactly this effect' to be able to differenziate it from the effect of the eomplexing agent. From all this, the principle of calculation of the effect of a complexing agent follows immediately when it is possible to postulate the strictly reversibility of all the phenomena. We make for the moment the assumption that the reversibility of the reducing process extends to every elementary act at the electrode, also to the dissociation of the complex in metallic ion and complexing agent. The effect of adding complexing agent is till given by (2) where ~y,) will be changed in (y,)I,, it is the activity coefficient referred to the stoichiometric concentration of the metal but function of the complexing agent (subtraction of metallic ions and electrostatic action). This new activity coefficient is defined as follows:
(~.,)~. [M'~+] = (~.,)~. [M~+1
(3) [M ~+]
stoichiometric concentration of metallic ion in solution
[/}P +]
concentration of metallic ion at equilibrium
On substituting in (2) we have: (4) - - A E~ = (E~)ott - - (E~) t,.. o
_
RT
nF
In
o (y~)~ ~ Rn TF In (D,~)I*(D~)~
198
GUIDO SARTORI
(D~)~ -- square-root of the directly measurable apparent diffusione coefficient. If we make the assumption that the only one complex is present in solution
MX~,
(M ~ + ) ~ =- (M" +),~ + MXp
¢5)
(M~+)~ ----(M~+)"" 7' (~, +
K'(X)P)7 mx,
y mx~, activity coefficient of the complex (X) effective concentration of complex forming material K , constant of formation By comparing (5) with {3)
(5 bis)
I
1/%-
+
K, (X)' 7 mx,
7,
If we substitute (5bis) in' (4)
AE½__(E~)a.o+(E½)~a - RTnFIn(-~s-}" (6)
..}.K~, (X)"
RT
log (D~)~ ffio
/ (6) gives a relation between the shift of the half wave potential and the equilibrium constant of the complex. If one works at constant ionic strenght, with enough complexing agent, and it is permissible to make the usual simplifing assumption common in polarography, one may deduce from (6) the practical relationship
¢7)
E½
8 log (X)
_
0,058 p n
n valence of the ion M" +. (7) allows the calculation of the coordination number p from the shift of E~ as a function of the concentration of X. (8)
(E~) ~ - - (E~)tt = o ---- - - 0,058 log K , n
POLA~OGRAPHY OF METAL COMPLEXES
199
gives the stability constant of the complex from the difference between half wave potential of the complexing solution at ionic strenght tt and the potential of the simple ion at ionic strength ~----0. If the reduction procedes only to a lower valency state (7) changes to (9) (9)
8E~ _ log (X)
0,058 ( p _ p , ) n
p coordination number of the oxydized form p' coordination number of the reduced form and the value of E~ for X ~ 1 minus the redox potential of the simple ion gives (10) (10)
A E ~ --
--0,058 n
log Ko K~
K o stability constant of the complex in the oxydized state KR stability constant of the complex in the reduced state It follows from (9) that if the coordination numbers of the oxydized and reduced form are equal the half wave potential does not shift at all with the change of concentration of the complexing agent. If the form of the complex is MXIpl Xs~ one may change alternatively the concentration of X1, keeping Xs constant and one gets
(11)
E~ _ log (X1)
0,058 n
8 E% log (Xs)
0,058 n
_
Pl
Pz
Even the polynuclear complexes may be examined on the same way. If they are formed by a reaction like mM q- p X ----M m X p on let the concentration of M change and on can derive the values of m by the relation: (11 bis)
BE½ 8 log c
_
0,058_ ( 1 - n
1 ) m
It should be noticed that this relation is less accourated than the preceedings but usually it unables to state if a polynuclear complex is present or not.
200
GUIDO SARTORI
When more complexes are present at the same time in solution relation (5) will be modified as follows [M"+]tot = [ M " + ] ~ y , . [1/¥,] + (12)
+
...
K1 ( X - )
+
K, (X-) z
y mx
+
.fmz 2
K, (X-)" y MX,
( X - ) effective concentration of complex forming agent supposed to be monovalent 7 may, activity coeffmient of corresponding complex M X , K~ constant of formation of the complex, which can be written in the simplified form [M "+] tot -- [M~+] ~ Y,. ~ g~ ( X - ) ~ - [M" +]., v,. X ,:o -f rnx,
(12 bis)
By analogy with the previous proceeding (13)
A E~ = (E~)~_ o -- (E~)~ =
R T ln Z - n---F
RT nF
In
D~v- °
It is useful to adopt now the method of De Ford and define a functioa F(X) as follows (14)
nF
Fo ((X-)) -- [ M " + ] ~ "f, R!n + - - Z = I O + !/, log -
Dg=o D~v.
+
AE,t. RT2,303 1
/
F (X) results a power series whose coefficents may be calculated by subsequent derivation for the value (X) = 0. The results are
¢15)
I (0) = Ao - -
I" (o) -
1
2 K, y lllx~
I'{0) = A t =
1" (0) - -
K,
pt K p y m.T,~
The expansion of F (X) for X - = 0 gives obvious only approximative results of the constant, while when more complexes are possible, in this region of concentration of X the complexes at higher coordination numbers may not be present at all.
POLAROORAPHY
OF
METAL
COMPLEXES
~01
Otherwise it is possible to calculate the coefficients of F (X) by graphical extrapolation to X - : O. De Ford suggests to define such function that (16)
lim Fo ( X - ) = Ko / ~,, X-~ 0 F1 (X-) = (F (X-) - Ko/y,) / ( X - )
The peculiarity of thise functions allows to recognize immediately the highest complex, F,,(X)-- is in this case a straight line with slope equal zero; the second, the function F . _ ~(X-)is still a straight line with a positive slope, and the other F . _ . ( X - ) are higher order curves. From the function F ( X - ) it is also easy to calculate the mean coordination number ~ for every value of X - , its maximun value and the concentration of X - at which every type of complex is present at maximum concentration. Relation (17) gives the mean coordination number (17)
P -- ~ log F ((X-))
log (X-)
RT. 2,303 2F
~ E~ 8 log ( X - )
and the following the percentage ~p of the complex with coordination number p (18) ~ lg =p -~ E~. 2,303. R T p __ 0 ~ 8 lg ( X - ) 8 Ig ( X - ) 2F I believe that it is more correct to calculate from the extrapolation of F(X) for X ~ 0 the first three stability constants and to solve for all other the complete system of linear equation of p-3 unknown values. All thin has a meaning if the complex dissociates reversibily in solution at a speed which is high compared with diffusion velocity, so that the simple ion is the entity which undergoes reduction. This is not always true. There are already many examples of polarographic reduction of complexes where the simple schema outlined before cannot be applied. I have only to remember the works of Schwarzenbach, of Koryta and Kossler, of Tanaka and Tamamushi and our .works in Rome about Mn (CN)~- and (Co (NH.~)s CI)z+ Recently Gierst and Juliard and later Delahay working on the theory of electrolysis at constant current, diffusion controlled, presented a method to choose between the three different reduction possibilities of a complex: 1) the complex is directly reducible; 2) the electrochemical reduction follows a chemical equilibrium; 3) both reduction mechanisms occur at the same time.
202
GUIDO SARTORI
The theory is rather simple: the potential of the electrode on which substance Ox is reduced is as follows, t seconds after the beginning of the electrolysis
(19)
E ~ E o 4- R T in /°D'~
nF
4- R T
/,Do~
nF
In C° - - pt~ Pt ~
where E o is the standard potential for the couple Ox-R(d, the f's are the activity coefficients of substances Ox and Red, and P is defined as (20)
2 io ~ nFDo~
P --
In equation (20), i o is the constant current density for the polarizable electrode, D ° the diffusion coefficient of substance Ox, and the other notations are conventional. The most characteristic value in this relation is called transition time and corresponds to the time after which C ° ---- p ~ , so that the potential of the electrode grows to infinity. ~ is proportional to the concentration of the reducible matter and •inversely to the current density. If the electrochemical process follows a reaction of the type Z -----Ox + X and an excess of X is present, io :~ is no more constant but changes as a function of io. If some limiting conditions imposed to the error function which appears in the relation are observed, the variation of i~~ w~th io is given by (21)
-
io -
8 io
---
-
-
x .
2
K , (kb -t- kl) ~
where K is the equilibrium constant of the chemical reaction and k~ and k~ the rate constants for the back and forth reactions. Two examples will make the problem clear: in the reduction of Cu (en)] + by changing io from 1 to 8,8 mAmp. io ~ is constant which demostrates t h a t the complex is directly reduced. By the reduction of Cd(CN)~- io • ~ slowe down on increasing io; this demonstrates that some dissociation takes place before the electrochemical reduction. In 1954 Gerischer was able to show t h a t this equilibrium is: Cd (CN)~- -~ Cd (CN)~ 4- O N -
and the reduction Cd (CN)~ d- 2 (--) -~ Cd + 3 C N -
I believe that this short report has demonstrated that the polarographic method, even with its limitation, compares favourably with the other items we are dealing off in this Congress, to give us a better inside view of the behaviour of complexes in solution.
GUIDO SARTORI Istituto Chimico dell'Universitk d! Trieste - Italia
S"mm"'7: T h e p o U i b f l i t i e s of the p o l a r o g r a p h i c m e t h o d i n the study of complexes i n s o l u t i o n a r e d i s c u s s e d e i t h ~ w h e n r e d u c t i o n follows t h e d i s s o c i a t i o n m" w l m n t h e c o m p l e x il the r e d u c i b l e e n t i t y .
The study of complex ions in solution by the polarographyc method started already thirty years ago with the classical work of Heyrovsky in 1927. Since then both theory and experimental methods have developed to a closed chapter which presents to day a complete picture and enumerates beyond his achievements not only the individuation of the complex or the complexes present in solution, but also the determination of the coordination number, of the stability constant and some time the right interpretation of the reduction mechanism. If one wishes to remember the names of the scientists which have given the most theoretical contributions to this field of research, I would recall the works of Stackelberg in 1939, of Lingane in 1941, of Souchay and Foucherre in 1947, of Gierst and Juliard in 1950 and of De Ford and Hume in 1951. It is a special merit of these workers particularly and of all those which has contributed theoretically and experimentally, in this connection I will remember the italian schools of Boma and Padova, if to day we have a closed picture of the complex phenomenology. As it is already known, the half wave potential for a simple ion is expressed through the formula: (1)
(E~i)~ = E*. +
RT nF
In a . ,
RT nF
In
y a. (D~)~ (y,)~. D~
EO standard potential of amalgam, i.e.e.m.f, of the pile: Ref. E1./M ~ +/M (1-19) when C°. %/a~g. C~,. %--= 1 % pratical molar activity coefficient of the metal in the amalgam
POLABOGBAPHY
(Y,)t~
OF METAL
197
COMPLEXES
pratical molar activity coefficient of the metallic ion at ionic strenght
all0 activity coefficient of H 9 in the amalgam Da diffusion coefficient of the metal in amalgam (D,)~ diffusion coefficient of the ionic species at ionic strenght It is also known that the diagnostic prouve of the formation of a complex in solution is the horizontal shift of the half wave potential by adding to the solution eomplexing agent. From (1) it is clear that the potential is a function of the activity coefficient of the reducible ion and therefore of the ionic strength of the solution. lndeed we have: (2)
AE~ : (E½)g - - (E~t)tt_ o
_
RT
In(yo)g +
nF
RT
nF
In ( D ~ ) t ~ - °
(D~)~.
The accurate studies of Lingane, De Ford and Papoff show that this shift, on changing the ionic strength from O to I are approximately 15 mV for a bivalent ion. This behaviour leads to the necessity to work al constant iou strength or to know exactly this effect' to be able to differenziate it from the effect of the eomplexing agent. From all this, the principle of calculation of the effect of a complexing agent follows immediately when it is possible to postulate the strictly reversibility of all the phenomena. We make for the moment the assumption that the reversibility of the reducing process extends to every elementary act at the electrode, also to the dissociation of the complex in metallic ion and complexing agent. The effect of adding complexing agent is till given by (2) where ~y,) will be changed in (y,)I,, it is the activity coefficient referred to the stoichiometric concentration of the metal but function of the complexing agent (subtraction of metallic ions and electrostatic action). This new activity coefficient is defined as follows:
(~.,)~. [M'~+] = (~.,)~. [M~+1
(3) [M ~+]
stoichiometric concentration of metallic ion in solution
[/}P +]
concentration of metallic ion at equilibrium
On substituting in (2) we have: (4) - - A E~ = (E~)ott - - (E~) t,.. o
_
RT
nF
In
o (y~)~ ~ Rn TF In (D,~)I*(D~)~
198
GUIDO SARTORI
(D~)~ -- square-root of the directly measurable apparent diffusione coefficient. If we make the assumption that the only one complex is present in solution
MX~,
(M ~ + ) ~ =- (M" +),~ + MXp
¢5)
(M~+)~ ----(M~+)"" 7' (~, +
K'(X)P)7 mx,
y mx~, activity coefficient of the complex (X) effective concentration of complex forming material K , constant of formation By comparing (5) with {3)
(5 bis)
I
1/%-
+
K, (X)' 7 mx,
7,
If we substitute (5bis) in' (4)
AE½__(E~)a.o+(E½)~a - RTnFIn(-~s-}" (6)
..}.K~, (X)"
RT
log (D~)~ ffio
/ (6) gives a relation between the shift of the half wave potential and the equilibrium constant of the complex. If one works at constant ionic strenght, with enough complexing agent, and it is permissible to make the usual simplifing assumption common in polarography, one may deduce from (6) the practical relationship
¢7)
E½
8 log (X)
_
0,058 p n
n valence of the ion M" +. (7) allows the calculation of the coordination number p from the shift of E~ as a function of the concentration of X. (8)
(E~) ~ - - (E~)tt = o ---- - - 0,058 log K , n
POLA~OGRAPHY OF METAL COMPLEXES
199
gives the stability constant of the complex from the difference between half wave potential of the complexing solution at ionic strenght tt and the potential of the simple ion at ionic strength ~----0. If the reduction procedes only to a lower valency state (7) changes to (9) (9)
8E~ _ log (X)
0,058 ( p _ p , ) n
p coordination number of the oxydized form p' coordination number of the reduced form and the value of E~ for X ~ 1 minus the redox potential of the simple ion gives (10) (10)
A E ~ --
--0,058 n
log Ko K~
K o stability constant of the complex in the oxydized state KR stability constant of the complex in the reduced state It follows from (9) that if the coordination numbers of the oxydized and reduced form are equal the half wave potential does not shift at all with the change of concentration of the complexing agent. If the form of the complex is MXIpl Xs~ one may change alternatively the concentration of X1, keeping Xs constant and one gets
(11)
E~ _ log (X1)
0,058 n
8 E% log (Xs)
0,058 n
_
Pl
Pz
Even the polynuclear complexes may be examined on the same way. If they are formed by a reaction like mM q- p X ----M m X p on let the concentration of M change and on can derive the values of m by the relation: (11 bis)
BE½ 8 log c
_
0,058_ ( 1 - n
1 ) m
It should be noticed that this relation is less accourated than the preceedings but usually it unables to state if a polynuclear complex is present or not.
200
GUIDO SARTORI
When more complexes are present at the same time in solution relation (5) will be modified as follows [M"+]tot = [ M " + ] ~ y , . [1/¥,] + (12)
+
...
K1 ( X - )
+
K, (X-) z
y mx
+
.fmz 2
K, (X-)" y MX,
( X - ) effective concentration of complex forming agent supposed to be monovalent 7 may, activity coeffmient of corresponding complex M X , K~ constant of formation of the complex, which can be written in the simplified form [M "+] tot -- [M~+] ~ Y,. ~ g~ ( X - ) ~ - [M" +]., v,. X ,:o -f rnx,
(12 bis)
By analogy with the previous proceeding (13)
A E~ = (E~)~_ o -- (E~)~ =
R T ln Z - n---F
RT nF
In
D~v- °
It is useful to adopt now the method of De Ford and define a functioa F(X) as follows (14)
nF
Fo ((X-)) -- [ M " + ] ~ "f, R!n + - - Z = I O + !/, log -
Dg=o D~v.
+
AE,t. RT2,303 1
/
F (X) results a power series whose coefficents may be calculated by subsequent derivation for the value (X) = 0. The results are
¢15)
I (0) = Ao - -
I" (o) -
1
2 K, y lllx~
I'{0) = A t =
1" (0) - -
K,
pt K p y m.T,~
The expansion of F (X) for X - = 0 gives obvious only approximative results of the constant, while when more complexes are possible, in this region of concentration of X the complexes at higher coordination numbers may not be present at all.
POLAROORAPHY
OF
METAL
COMPLEXES
~01
Otherwise it is possible to calculate the coefficients of F (X) by graphical extrapolation to X - : O. De Ford suggests to define such function that (16)
lim Fo ( X - ) = Ko / ~,, X-~ 0 F1 (X-) = (F (X-) - Ko/y,) / ( X - )
The peculiarity of thise functions allows to recognize immediately the highest complex, F,,(X)-- is in this case a straight line with slope equal zero; the second, the function F . _ ~(X-)is still a straight line with a positive slope, and the other F . _ . ( X - ) are higher order curves. From the function F ( X - ) it is also easy to calculate the mean coordination number ~ for every value of X - , its maximun value and the concentration of X - at which every type of complex is present at maximum concentration. Relation (17) gives the mean coordination number (17)
P -- ~ log F ((X-))
log (X-)
RT. 2,303 2F
~ E~ 8 log ( X - )
and the following the percentage ~p of the complex with coordination number p (18) ~ lg =p -~ E~. 2,303. R T p __ 0 ~ 8 lg ( X - ) 8 Ig ( X - ) 2F I believe that it is more correct to calculate from the extrapolation of F(X) for X ~ 0 the first three stability constants and to solve for all other the complete system of linear equation of p-3 unknown values. All thin has a meaning if the complex dissociates reversibily in solution at a speed which is high compared with diffusion velocity, so that the simple ion is the entity which undergoes reduction. This is not always true. There are already many examples of polarographic reduction of complexes where the simple schema outlined before cannot be applied. I have only to remember the works of Schwarzenbach, of Koryta and Kossler, of Tanaka and Tamamushi and our .works in Rome about Mn (CN)~- and (Co (NH.~)s CI)z+ Recently Gierst and Juliard and later Delahay working on the theory of electrolysis at constant current, diffusion controlled, presented a method to choose between the three different reduction possibilities of a complex: 1) the complex is directly reducible; 2) the electrochemical reduction follows a chemical equilibrium; 3) both reduction mechanisms occur at the same time.
202
GUIDO SARTORI
The theory is rather simple: the potential of the electrode on which substance Ox is reduced is as follows, t seconds after the beginning of the electrolysis
(19)
E ~ E o 4- R T in /°D'~
nF
4- R T
/,Do~
nF
In C° - - pt~ Pt ~
where E o is the standard potential for the couple Ox-R(d, the f's are the activity coefficients of substances Ox and Red, and P is defined as (20)
2 io ~ nFDo~
P --
In equation (20), i o is the constant current density for the polarizable electrode, D ° the diffusion coefficient of substance Ox, and the other notations are conventional. The most characteristic value in this relation is called transition time and corresponds to the time after which C ° ---- p ~ , so that the potential of the electrode grows to infinity. ~ is proportional to the concentration of the reducible matter and •inversely to the current density. If the electrochemical process follows a reaction of the type Z -----Ox + X and an excess of X is present, io :~ is no more constant but changes as a function of io. If some limiting conditions imposed to the error function which appears in the relation are observed, the variation of i~~ w~th io is given by (21)
-
io -
8 io
---
-
-
x .
2
K , (kb -t- kl) ~
where K is the equilibrium constant of the chemical reaction and k~ and k~ the rate constants for the back and forth reactions. Two examples will make the problem clear: in the reduction of Cu (en)] + by changing io from 1 to 8,8 mAmp. io ~ is constant which demostrates t h a t the complex is directly reduced. By the reduction of Cd(CN)~- io • ~ slowe down on increasing io; this demonstrates that some dissociation takes place before the electrochemical reduction. In 1954 Gerischer was able to show t h a t this equilibrium is: Cd (CN)~- -~ Cd (CN)~ 4- O N -
and the reduction Cd (CN)~ d- 2 (--) -~ Cd + 3 C N -
I believe that this short report has demonstrated that the polarographic method, even with its limitation, compares favourably with the other items we are dealing off in this Congress, to give us a better inside view of the behaviour of complexes in solution.