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A correlation between the overall stability constants of metal complexes—III
I. inorg, nucl. C h e m . , 1974, Vol. 36~ pp. 121 125. Pergamon Press. Printed in Great Britain.
A CORRELATION BETWEEN THE OVERALL STABILITY CONSTANTS OF METAL COMPLEXES--III APPLICATION TO DATA OBTAINED BY THE POLAROGRAPHIC METHOD FOR THE CADMIUM-, COPPER-, LEAD- AND ZINCFORMATE SYSTEMS M. H. MIHAILOV, V. Ts. MIHAILOVA and V. A. KHALKIN* Bulgarian Academy of Sciences, Institute of Nuclear Physics, Sofia 13, Bulgaria (Received 20 December 1972)
Abstract--The experimental data published on the formate complexes of cadmium, copper, lead, and zinc have been analyzed. As a result, the validity of the following relation connecting the overall stability constants of consecutive metal complexes has been confirmed : an
~. = A ~ where : ft, is the overall stability constant of the ~/L, complex ; n is the number of ligands in a particular complex, (I ~< n ~< N); N is the coordination number of the complex forming metal M and A and a are constants.
INTRODUCTION As A PART of a program involving the reevaluation of the data on the stability constants of metal complexes in order to check the validity of the relation connecting the overall stability constants of metal complexes derived in a previous paper of this series[l], the data on the formate complexes of cadmium, copper, lead, and zinc were analyzed. These four systems had been studied by Hershenson, Brooks and Murphy[2] using the polarographic technique as developed by DeFord and Hume [3] at a constant temperature of 25.0 +_ 0.2°C and a constant ionic strength of 2 maintained by adding an appropriate amount of N a C 1 0 , in the solution as an inert electrolyte. The formate ions came in the solution from the dissociation of Na-formate. CALCULATION OF THE STABILITY CONSTANTS As far as the mathematical model of the complex formation process in aqueous solution is presented elsewhere[l], here only the main equations will be given. Here it has to be remembered that the above mentioned mathematical model is concerned with a system of metal M m+ and univalent ligand L - in an aqueous solution. It is based on the assumptions (which are usually made for this kind of system) that : (i) the * Joint Institute for Nuclear Research, Dubna, U.S.S.R.
metal M and the ligand L form only mononuclear complexes; (ii) the complex forming metal M does not participate in side reactions with the ligand L' coming from the dissociation of the electrolyte CL' used as an inert electrolyte as well as with O H - ions; (iii) no formation of ion-pairs (outer-sphere complexes) as a result of the interaction of the negatively charged metal complexes and the C ÷ ions which come from the dissociation of the CL and CL' electrolyte of the solution occurs; (iv) the activity coefficients of all species in the solution are constant formally equalled to unity. It was found that for such system the overall stability constant ft. of the complex M L . , formed as a product of the reaction: M + n L ~ M L , (the coordinated water molecules and charges are dropped for clarity) should satisfy the following relation : an
/L = A ~
(1)
where A and a are constants. Then, the complexity function as defined by Froneus[4] will be transformed to : N n a n F = 1 + A . ~ , ~.v(L)
(2)
where: (L) is the ligand concentration and N is the coordination number of the complex forming metal M. 121
122
M. H. MIHAILOV, V. TS. MIHAILOVAand V. A. KHALKIN
If the values of F as a function of the free ligand concentration of the solution are determined from experimentally measurable quantities as directed, for example, by the DeFord and Hume's method[3], taking two data points, let us say, F' at ligand concentration (L') and F" at ligand concentration (L") with a simple rearrangement of Eqn (2) the following two equations could be written : (F'-
" a"- x(L")" 1)~ nr 1
" a"- I(L')" 1)Z n! = 0
(F" -
(3) f'-
1
(4)
~ a"(L')" n!
From Eqns (3) and (4) the values of the constants A and a could be obtained by the conventional methods. RESULTS A N D D I S C U S S I O N
The values of Foxp as a function of the ligand concentration were taken from the corresponding Tables of [2] for each system, where they are denoted as F o. Those functions were analysed as described above. As a result the values of the constants A and a were obtained. Inserting these values in Eqn (1) we get the whole set of stability constants for the given m e t a l ligand system. Then, the Fexp at each data point was compared with the values of the FCa~ functions, the first one obtained when using the set of stability constants reported in [2] and the second one calculated by using the set of stability constants obtained in this work. The quantity AF = 1/Fexp(Fcalc- Fexp)100 was introduced as a measure of the fit at a given data point. As a measure of the overall goodness of the fit the quantity: S = const-Smi n = { ~1E W i [ ( F e a l c ) i -
(at) 2 = const (Fexp) 2
(Fexp)~]2} const
(5) was used throughout this paper[5], where: K is the
Cadmium-formate system For this system the authors of [2] reported the following set of stability constants : fll = 3 ___ I ; f12 = 13 ___ 1 ; fi3=22+7 and f14= 1 6 + 4 . As a result of the mathematical analysis performed as described above we have found the following values of the stability constants: fll = 4.4; f12 = 10.8; f13 = 17.6 and fi4 = 21.6. They are related according to Eqn (1), where A = 0.90 and a = 4.90. Table 1 compares the experimental complexity function (Fexv) with the theoretical ones (Fcalc) obtained with the two sets of stability constants just described. The S-factors are: St21 = 8-1 × 10 -3 and STh i. . . . k = 4.1 × 10 -3. In Fig. 1 the mole fractions of the cadmium-formate complexes vs the formate concentration are plotted. The solid line presents the results obtained when the set of stability constants of this work is used and the dashed line presents the results with the set of stability constants reported in [2].
Lead-formate system For this system the following set of stability constants is reported in [2]: f l 1 = 6 1; f l 2 = 2 3 + 4 ; fla= 27 ___ 10 and f14 = 15 _ 4. We have found the following values of the constants A and a in Eqn (2): A = 2.00 and a = 4-00 respectively. Using Eqn (1) the following values of the consecutive stability constants of the lead-formate complexes are obtained: fll = 8'0; f12 = 16"0; f13 = 21'0 and f14 = 21"0. In Table 2 the comparison of the Fexp and Fcalc functions is presented. T h e S-factors for this system are: S[21 = 2.0 x 10 -2 and
Table 1. Comparison of the experimental and calculated complexity functions for the cadmium-formate system Hershenson et al.
L
Fexv
Feaze
0.09 0-19 0.49 0-79 0.89 0.99 1.79 1.98
1.40 2.28 9.53 26.6 37.0 55.2 349 525
1-39 2-21 9.10 28.6 39.5 53.4 338 475
(M)
(6)
because no estimation of Wi from any experimental consideration was possible.
1
and A - - -
number of degree of freedom of the system; (Fexp)i is the experimental value of F in the ith data point; (Fcal~)i is the calculated value o f F in the ith data point; Wi = 1/(aF)~ and (ae) ~ is the variance of F in the ith data point. Introducing the S-factor as defined in Eqn (5) we have made the assumption as others did[6] that the following relation holds:
This work
AF
Fcal¢
-0.71 - 3.07 -- 4-51 7-52 6.76 -3.26 - 3-15 -9.52
1.50 2.37 9.06 28.3 39-4 53.8 366 521
(%)
AF
(%) 7.14 3-95 - 4.93 6.39 6.49 --2-54 4.87 --0.76
Stability constants of metal complexes--III
f o r m a t e complexes in t h e s o l u t i o n calculated o n the basis of the b o t h sets of stability c o n s t a n t s are p l o t t e d as a function of the ligand c o n c e n t r a t i o n .
Cd (CH 00) 2
°°I
Copper-formate system
75
% x
,#
50
123
~
Ct4
,
~t ~
/ / a4
25
F o r this system the following values o f the stability c o n s t a n t s are r e p o r t e d in [2]: fll = 37 _-4-8; f12 = 166 + 22; f13 = 112 + 40 a n d f14 = 283 + 23 respectively. T h e analysis o f the Fexp function gives the following values o f the c o n s t a n t s : A = 6.55 a n d a = 5.80. F o r the stability c o n s t a n t s E q n (1) gives: fit = 38; fie = 110; f13 = 213 a n d f14 = 309. T h e c o m p a r i s o n of b o t h fits is p r e s e n t e d in Table 4. T h e S-factors calculated are: S[2 ] = 7.1 x 10 - 2 a n d S T h i . . . . k = 5"5 X l 0 - 2 . In Fig. 4 the mole fractions of the c o p p e r - f o r m a t e c o m plexes calculated by using b o t h sets o f c o n s t a n t s are I00
0"2
0"4
0"6
0"8
I'0
I-2
1"4
1'6
1'8
'
Pb(CH
00) z
2"0
(L)
Fig. 1. Distribution of complex species in the cadmium formate system. SThis work = 2.7 x 10-2. In Fig. 2 the mole fractions of the lead complexes vs the f o r m a t e c o n c e n t r a t i o n in the solution are plotted. Solid line is used w h e n the set of stability c o n s t a n t s o f this work is used a n d a d a s h e d line for the set of [2].
75
~
~o -"---.
~-
-a~ - - -~....~
Zinc-formate system In [2] t h e following set o f stability c o n s t a n t s is r e p o r t e d : fll = 4 4- 1; f12 = 9 -I- 2; fla = 3 ___ 1 a n d f l 4 = 6 + - - 3 . O u r results are: fll = 3 " 1 5 ; f l 2 = 5 ' 5 ; f13 = 6"4; a n d f14 = 5'6. These c o n s t a n t s satisfy E q n (1) w h e r e A = 0.90 a n d a = 3.49. T h e c o m p a r i s o n o f the Fexp a n d Fcalo functions is p r e s e n t e d in Table 3. T h e S-factors are: S[2 ] = 5-0 × 10 - 2 a n d S T h i . . . . k = 4'0 × 10 -2. O n Fig. 3 the m o l e fractions of the z i n c -
0
0-2
0'4
0-6
0"8
I-0
I-2
[-4
1'6
1"8
2"0
(LI
Fig. 2. Distribution of complex species in the lead-formate system.
Table 2. Comparison of the experimental and calculated complexity functions for the lead-formate system Hershenson et al.
This work
L (M)
Fop
Fcalc
AF (%)
Fc=lc
AF (%)
0.04 0.09 0.19 0.29 0-49 0-79 0"89 1.29 1.59 1.79 1.98
1.30 1.32 3.18 5.48 14.5 40.5 51.7 146 256 411 570
1.28 1.75 3.17 5.44 13.5 39.2 53.0 146 273 394 543
- 1.54 32.57 -0.31 -- 0-73 - 6.90 - 3-21 2.51 0.00 6.64 -4.14 -- 4.74
1.34 1-87 3.27 5.33 12.4 35.8 48.8 141 273 403 565
3.85 41-67 2.83 - 2.73 -- 14.48 - 11.60 - 5.65 - 3.42 6.64 - 1.95 - 0.88
124
M. H. MIHAILOV, V. TS. MIHAILOVAand V. A. KHALKIN Table 3. Comparison of the experimental and calculated complexity functions for the zinc-formate system Hershenson et al.
This work
L (M)
Fexp
Feaze
AF (%)
F~,,]¢
AF (%)
0.04 0-09 0.19 0-29 0.49 0.69 0.89 1.09 1.29 1.49 1.69 1.84 1.98
1.34 1.8 1.8 2-8 5.4 12-8 16.0 23.3 37.9 68.8 88.7 145 259
1.17 1.4 2.1 3.0 5-8 10.4 17.6 28.4 44.2 66-4 96.9 126 160
- 12.69 - 22.22 16.67 7.14 7.41 - 18.75 10.00 21.89 16.62 - 3.49 9.24 - 13.10 - 38.22
1.13 1.3 1.8 2.6 4.9 9.2 16.2 27.2 43.5 66.7 98.6 130 164
- 15.67 - 27.78 0-00 - 7-14 - 9.26 -28.13 1-25 16.74 14.78 - 3.05 11-16 - 10.34 - 36.68
plotted as a function of the ligand concentration in the solution, The result of the analysis strongly confirm the validity of Eqns (1) and (2) and the mathematical model developed in [1] for the four formate systems studied. These results suggested a preference for this model since by two criteria this model better interpreted the experimental data than the classical one. These criteria are : first, that this model explains the complex formation process with less n u m b e r of parameters than the classical one a n d from the m a t h e m a t i c a l point of view it is the preferable one a n d second t h a t the inspection
Cu(CHO0)2
4 foe
F
x 50
///Q4
Zn(CHO0) 2 25
°°I
0
t;l3
0'2 0'4
0"6 0 " 8
I'0
1"2 1"4 1"6
I'e 2"0
(L> / / "~
50
t
I~
Fig. 4. Distribution of complex species in the copperformate system.
t
,/ / G4
' ~ (
~/11, ~ ~,
,I.
,/ II
~
~ ~ t21
t~ 0 Ct I
0"2 O'4 0"6 O'O I'O
1'2
1"4 1"6
1"8 2"0
(t.) Fig. 3. Distribution of complex species in the zinc-formate system.
of S-factor values indicated better fits in the cases when the set of stability constants found in the present work are used. For the lead-formate system the both fits could be practically considered as equal, t h a t one of [2] being better by a factor of a b o u t 1-4. But this difference is too small to need any explanation or discussion a n d some r a n d o m error introduced in the course of the measurements could cause such effect. Obviously, for systems for which the mathematical model developed in [1] is valid, the ft, values could be determined directly from Eqn (1) after A and a constants are obtained from the experimental complexity function. This is less time consuming a n d m u c h more
Stability constants of metal complexes--III
125
Table 4. Comparison of the experimental and calculated complexity functions for the copper-formate system Hershenson et al. L
F~xp
Foal,
(M) 0.04 0.19 0.29 0.49 0.69 0.89 1-09 1.29 1.49 1.69 1-84 1.98
AF
This work Fca]c
(%) 2.96 10-3 34 87 226 421 799 2197 1807 4550 4630 7725
2.76 15.2 30 88 206 422 783 1349 2190 3387 4573 5944
a c c u r a t e p r o c e d u r e t h a n t h e c o n v e n t i o n a l m e t h o d of the successive graphical extrapolation[7] usually a p p l i e d for t h e d e t e r m i n a t i o n o f t h e p o l y n o m i a l coefficients ft,.
REFERENCES 1. M. H. Mihailov, J. inorg, nucl. Chem. 36, 107 (1974).
-6.75 47.57 - 11.67 - 1.15 -8.85 0.24 -2.00 - 38.60 21.19 -25.56 -1.23 -23-05
AF
(%) 2.72 14.1 29 89 220 466 884 1546 2529 3928 5312 6910
-8.11 36.89 - 14-70 2.30 -2.65 10.69 10-64 - 29.63 39.95 - 13.67 14.73 - 10-55
2. H. M. Hershenson, R. Th. Brooks and M. E. Murphy, J. Am. chem. Soc. 79, 2046 (1957). 3. D. DeFord and D. N. Hume, J. Am. chem. Soc. 73, 5321 (1951). 4. S. Froneus, Thesis, Lund, 1948. 5. J. Rydberg and J. C. Sullivan, Acta them scand. 1 , 2057 (1959). 6. A. Sandell, Acta chem. scand. 23, 478 (1969). 7. I. Leden, Thesis, Lund, 1943.
A CORRELATION BETWEEN THE OVERALL STABILITY CONSTANTS OF METAL COMPLEXES--III APPLICATION TO DATA OBTAINED BY THE POLAROGRAPHIC METHOD FOR THE CADMIUM-, COPPER-, LEAD- AND ZINCFORMATE SYSTEMS M. H. MIHAILOV, V. Ts. MIHAILOVA and V. A. KHALKIN* Bulgarian Academy of Sciences, Institute of Nuclear Physics, Sofia 13, Bulgaria (Received 20 December 1972)
Abstract--The experimental data published on the formate complexes of cadmium, copper, lead, and zinc have been analyzed. As a result, the validity of the following relation connecting the overall stability constants of consecutive metal complexes has been confirmed : an
~. = A ~ where : ft, is the overall stability constant of the ~/L, complex ; n is the number of ligands in a particular complex, (I ~< n ~< N); N is the coordination number of the complex forming metal M and A and a are constants.
INTRODUCTION As A PART of a program involving the reevaluation of the data on the stability constants of metal complexes in order to check the validity of the relation connecting the overall stability constants of metal complexes derived in a previous paper of this series[l], the data on the formate complexes of cadmium, copper, lead, and zinc were analyzed. These four systems had been studied by Hershenson, Brooks and Murphy[2] using the polarographic technique as developed by DeFord and Hume [3] at a constant temperature of 25.0 +_ 0.2°C and a constant ionic strength of 2 maintained by adding an appropriate amount of N a C 1 0 , in the solution as an inert electrolyte. The formate ions came in the solution from the dissociation of Na-formate. CALCULATION OF THE STABILITY CONSTANTS As far as the mathematical model of the complex formation process in aqueous solution is presented elsewhere[l], here only the main equations will be given. Here it has to be remembered that the above mentioned mathematical model is concerned with a system of metal M m+ and univalent ligand L - in an aqueous solution. It is based on the assumptions (which are usually made for this kind of system) that : (i) the * Joint Institute for Nuclear Research, Dubna, U.S.S.R.
metal M and the ligand L form only mononuclear complexes; (ii) the complex forming metal M does not participate in side reactions with the ligand L' coming from the dissociation of the electrolyte CL' used as an inert electrolyte as well as with O H - ions; (iii) no formation of ion-pairs (outer-sphere complexes) as a result of the interaction of the negatively charged metal complexes and the C ÷ ions which come from the dissociation of the CL and CL' electrolyte of the solution occurs; (iv) the activity coefficients of all species in the solution are constant formally equalled to unity. It was found that for such system the overall stability constant ft. of the complex M L . , formed as a product of the reaction: M + n L ~ M L , (the coordinated water molecules and charges are dropped for clarity) should satisfy the following relation : an
/L = A ~
(1)
where A and a are constants. Then, the complexity function as defined by Froneus[4] will be transformed to : N n a n F = 1 + A . ~ , ~.v(L)
(2)
where: (L) is the ligand concentration and N is the coordination number of the complex forming metal M. 121
122
M. H. MIHAILOV, V. TS. MIHAILOVAand V. A. KHALKIN
If the values of F as a function of the free ligand concentration of the solution are determined from experimentally measurable quantities as directed, for example, by the DeFord and Hume's method[3], taking two data points, let us say, F' at ligand concentration (L') and F" at ligand concentration (L") with a simple rearrangement of Eqn (2) the following two equations could be written : (F'-
" a"- x(L")" 1)~ nr 1
" a"- I(L')" 1)Z n! = 0
(F" -
(3) f'-
1
(4)
~ a"(L')" n!
From Eqns (3) and (4) the values of the constants A and a could be obtained by the conventional methods. RESULTS A N D D I S C U S S I O N
The values of Foxp as a function of the ligand concentration were taken from the corresponding Tables of [2] for each system, where they are denoted as F o. Those functions were analysed as described above. As a result the values of the constants A and a were obtained. Inserting these values in Eqn (1) we get the whole set of stability constants for the given m e t a l ligand system. Then, the Fexp at each data point was compared with the values of the FCa~ functions, the first one obtained when using the set of stability constants reported in [2] and the second one calculated by using the set of stability constants obtained in this work. The quantity AF = 1/Fexp(Fcalc- Fexp)100 was introduced as a measure of the fit at a given data point. As a measure of the overall goodness of the fit the quantity: S = const-Smi n = { ~1E W i [ ( F e a l c ) i -
(at) 2 = const (Fexp) 2
(Fexp)~]2} const
(5) was used throughout this paper[5], where: K is the
Cadmium-formate system For this system the authors of [2] reported the following set of stability constants : fll = 3 ___ I ; f12 = 13 ___ 1 ; fi3=22+7 and f14= 1 6 + 4 . As a result of the mathematical analysis performed as described above we have found the following values of the stability constants: fll = 4.4; f12 = 10.8; f13 = 17.6 and fi4 = 21.6. They are related according to Eqn (1), where A = 0.90 and a = 4.90. Table 1 compares the experimental complexity function (Fexv) with the theoretical ones (Fcalc) obtained with the two sets of stability constants just described. The S-factors are: St21 = 8-1 × 10 -3 and STh i. . . . k = 4.1 × 10 -3. In Fig. 1 the mole fractions of the cadmium-formate complexes vs the formate concentration are plotted. The solid line presents the results obtained when the set of stability constants of this work is used and the dashed line presents the results with the set of stability constants reported in [2].
Lead-formate system For this system the following set of stability constants is reported in [2]: f l 1 = 6 1; f l 2 = 2 3 + 4 ; fla= 27 ___ 10 and f14 = 15 _ 4. We have found the following values of the constants A and a in Eqn (2): A = 2.00 and a = 4-00 respectively. Using Eqn (1) the following values of the consecutive stability constants of the lead-formate complexes are obtained: fll = 8'0; f12 = 16"0; f13 = 21'0 and f14 = 21"0. In Table 2 the comparison of the Fexp and Fcalc functions is presented. T h e S-factors for this system are: S[21 = 2.0 x 10 -2 and
Table 1. Comparison of the experimental and calculated complexity functions for the cadmium-formate system Hershenson et al.
L
Fexv
Feaze
0.09 0-19 0.49 0-79 0.89 0.99 1.79 1.98
1.40 2.28 9.53 26.6 37.0 55.2 349 525
1-39 2-21 9.10 28.6 39.5 53.4 338 475
(M)
(6)
because no estimation of Wi from any experimental consideration was possible.
1
and A - - -
number of degree of freedom of the system; (Fexp)i is the experimental value of F in the ith data point; (Fcal~)i is the calculated value o f F in the ith data point; Wi = 1/(aF)~ and (ae) ~ is the variance of F in the ith data point. Introducing the S-factor as defined in Eqn (5) we have made the assumption as others did[6] that the following relation holds:
This work
AF
Fcal¢
-0.71 - 3.07 -- 4-51 7-52 6.76 -3.26 - 3-15 -9.52
1.50 2.37 9.06 28.3 39-4 53.8 366 521
(%)
AF
(%) 7.14 3-95 - 4.93 6.39 6.49 --2-54 4.87 --0.76
Stability constants of metal complexes--III
f o r m a t e complexes in t h e s o l u t i o n calculated o n the basis of the b o t h sets of stability c o n s t a n t s are p l o t t e d as a function of the ligand c o n c e n t r a t i o n .
Cd (CH 00) 2
°°I
Copper-formate system
75
% x
,#
50
123
~
Ct4
,
~t ~
/ / a4
25
F o r this system the following values o f the stability c o n s t a n t s are r e p o r t e d in [2]: fll = 37 _-4-8; f12 = 166 + 22; f13 = 112 + 40 a n d f14 = 283 + 23 respectively. T h e analysis o f the Fexp function gives the following values o f the c o n s t a n t s : A = 6.55 a n d a = 5.80. F o r the stability c o n s t a n t s E q n (1) gives: fit = 38; fie = 110; f13 = 213 a n d f14 = 309. T h e c o m p a r i s o n of b o t h fits is p r e s e n t e d in Table 4. T h e S-factors calculated are: S[2 ] = 7.1 x 10 - 2 a n d S T h i . . . . k = 5"5 X l 0 - 2 . In Fig. 4 the mole fractions of the c o p p e r - f o r m a t e c o m plexes calculated by using b o t h sets o f c o n s t a n t s are I00
0"2
0"4
0"6
0"8
I'0
I-2
1"4
1'6
1'8
'
Pb(CH
00) z
2"0
(L)
Fig. 1. Distribution of complex species in the cadmium formate system. SThis work = 2.7 x 10-2. In Fig. 2 the mole fractions of the lead complexes vs the f o r m a t e c o n c e n t r a t i o n in the solution are plotted. Solid line is used w h e n the set of stability c o n s t a n t s o f this work is used a n d a d a s h e d line for the set of [2].
75
~
~o -"---.
~-
-a~ - - -~....~
Zinc-formate system In [2] t h e following set o f stability c o n s t a n t s is r e p o r t e d : fll = 4 4- 1; f12 = 9 -I- 2; fla = 3 ___ 1 a n d f l 4 = 6 + - - 3 . O u r results are: fll = 3 " 1 5 ; f l 2 = 5 ' 5 ; f13 = 6"4; a n d f14 = 5'6. These c o n s t a n t s satisfy E q n (1) w h e r e A = 0.90 a n d a = 3.49. T h e c o m p a r i s o n o f the Fexp a n d Fcalo functions is p r e s e n t e d in Table 3. T h e S-factors are: S[2 ] = 5-0 × 10 - 2 a n d S T h i . . . . k = 4'0 × 10 -2. O n Fig. 3 the m o l e fractions of the z i n c -
0
0-2
0'4
0-6
0"8
I-0
I-2
[-4
1'6
1"8
2"0
(LI
Fig. 2. Distribution of complex species in the lead-formate system.
Table 2. Comparison of the experimental and calculated complexity functions for the lead-formate system Hershenson et al.
This work
L (M)
Fop
Fcalc
AF (%)
Fc=lc
AF (%)
0.04 0.09 0.19 0.29 0-49 0-79 0"89 1.29 1.59 1.79 1.98
1.30 1.32 3.18 5.48 14.5 40.5 51.7 146 256 411 570
1.28 1.75 3.17 5.44 13.5 39.2 53.0 146 273 394 543
- 1.54 32.57 -0.31 -- 0-73 - 6.90 - 3-21 2.51 0.00 6.64 -4.14 -- 4.74
1.34 1-87 3.27 5.33 12.4 35.8 48.8 141 273 403 565
3.85 41-67 2.83 - 2.73 -- 14.48 - 11.60 - 5.65 - 3.42 6.64 - 1.95 - 0.88
124
M. H. MIHAILOV, V. TS. MIHAILOVAand V. A. KHALKIN Table 3. Comparison of the experimental and calculated complexity functions for the zinc-formate system Hershenson et al.
This work
L (M)
Fexp
Feaze
AF (%)
F~,,]¢
AF (%)
0.04 0-09 0.19 0-29 0.49 0.69 0.89 1.09 1.29 1.49 1.69 1.84 1.98
1.34 1.8 1.8 2-8 5.4 12-8 16.0 23.3 37.9 68.8 88.7 145 259
1.17 1.4 2.1 3.0 5-8 10.4 17.6 28.4 44.2 66-4 96.9 126 160
- 12.69 - 22.22 16.67 7.14 7.41 - 18.75 10.00 21.89 16.62 - 3.49 9.24 - 13.10 - 38.22
1.13 1.3 1.8 2.6 4.9 9.2 16.2 27.2 43.5 66.7 98.6 130 164
- 15.67 - 27.78 0-00 - 7-14 - 9.26 -28.13 1-25 16.74 14.78 - 3.05 11-16 - 10.34 - 36.68
plotted as a function of the ligand concentration in the solution, The result of the analysis strongly confirm the validity of Eqns (1) and (2) and the mathematical model developed in [1] for the four formate systems studied. These results suggested a preference for this model since by two criteria this model better interpreted the experimental data than the classical one. These criteria are : first, that this model explains the complex formation process with less n u m b e r of parameters than the classical one a n d from the m a t h e m a t i c a l point of view it is the preferable one a n d second t h a t the inspection
Cu(CHO0)2
4 foe
F
x 50
///Q4
Zn(CHO0) 2 25
°°I
0
t;l3
0'2 0'4
0"6 0 " 8
I'0
1"2 1"4 1"6
I'e 2"0
(L> / / "~
50
t
I~
Fig. 4. Distribution of complex species in the copperformate system.
t
,/ / G4
' ~ (
~/11, ~ ~,
,I.
,/ II
~
~ ~ t21
t~ 0 Ct I
0"2 O'4 0"6 O'O I'O
1'2
1"4 1"6
1"8 2"0
(t.) Fig. 3. Distribution of complex species in the zinc-formate system.
of S-factor values indicated better fits in the cases when the set of stability constants found in the present work are used. For the lead-formate system the both fits could be practically considered as equal, t h a t one of [2] being better by a factor of a b o u t 1-4. But this difference is too small to need any explanation or discussion a n d some r a n d o m error introduced in the course of the measurements could cause such effect. Obviously, for systems for which the mathematical model developed in [1] is valid, the ft, values could be determined directly from Eqn (1) after A and a constants are obtained from the experimental complexity function. This is less time consuming a n d m u c h more
Stability constants of metal complexes--III
125
Table 4. Comparison of the experimental and calculated complexity functions for the copper-formate system Hershenson et al. L
F~xp
Foal,
(M) 0.04 0.19 0.29 0.49 0.69 0.89 1-09 1.29 1.49 1.69 1-84 1.98
AF
This work Fca]c
(%) 2.96 10-3 34 87 226 421 799 2197 1807 4550 4630 7725
2.76 15.2 30 88 206 422 783 1349 2190 3387 4573 5944
a c c u r a t e p r o c e d u r e t h a n t h e c o n v e n t i o n a l m e t h o d of the successive graphical extrapolation[7] usually a p p l i e d for t h e d e t e r m i n a t i o n o f t h e p o l y n o m i a l coefficients ft,.
REFERENCES 1. M. H. Mihailov, J. inorg, nucl. Chem. 36, 107 (1974).
-6.75 47.57 - 11.67 - 1.15 -8.85 0.24 -2.00 - 38.60 21.19 -25.56 -1.23 -23-05
AF
(%) 2.72 14.1 29 89 220 466 884 1546 2529 3928 5312 6910
-8.11 36.89 - 14-70 2.30 -2.65 10.69 10-64 - 29.63 39.95 - 13.67 14.73 - 10-55
2. H. M. Hershenson, R. Th. Brooks and M. E. Murphy, J. Am. chem. Soc. 79, 2046 (1957). 3. D. DeFord and D. N. Hume, J. Am. chem. Soc. 73, 5321 (1951). 4. S. Froneus, Thesis, Lund, 1948. 5. J. Rydberg and J. C. Sullivan, Acta them scand. 1 , 2057 (1959). 6. A. Sandell, Acta chem. scand. 23, 478 (1969). 7. I. Leden, Thesis, Lund, 1943.