Photometric determination of composition and conditional stability constants of complexes AmBn

J. inorg,nuel.Chem.,1969,Vol.31, pp. 1345to 1352. PergamonPress. Printedin Great Britain

PHOTOMETRIC DETERMINATION OF COMPOSITION AND CONDITIONAL STABILITY CONSTANTS OF COMPLEXES AmBn B. BUDESINSKY* Nuclear Research Institute, Czechoslovak Academy of Sciences, Rzhezh near Prague

(Received 15 May 1968) Abstract-The method of proportional absorbances has been generalized and applied both to the determination of absolute values of complex composition AmBn and of its conditional stability constant. The method of continuous variation or of molar ratio is suitable for determination of absolute values of the composition of the very stable complex AraBn. INTRODUCTION

THE PHOTOMETRIC analysis of solutions of complexes ABn has been worked out in detail[l,2], but the same is not true for the complexes AmBn. No convenient

photometric method exists for the determination of absolute values of m and n. Several authors[3-5] have reported that the continuous variation curves of isomolar series are concave at their ends if complex species Arab n are formed, where m and/or n i> 2. A recent paper[5] has shown this to be true from a strictly theoretical viewpoint, and experimental verification would be satisfying. Unfortunately the deviations in the measurement are so large in practice (/> 1 per cent in absorbance) that no example is known of a complexing reagent where the concave course of the curve is explicitly perceptible. Our experiences with about 500 continuous variation curves during the last 8 yr, and also the examples given elsewhere[5-7], confirm this view. An experimental study can be made of the course of the curve at its end, in the range of several hundredths of units of CA/(CA+ CB) (see below), but the measurement will be very inaccurate, particularly in the case of a colour reagent, and the evidence is not reliable if the mechanism of reaction in presence of so large excess of one of the components is the same as the mechanism in the region where the molar ratio is m/(m+ n). The correct practical application of the method of continuous variation and of the molar ratio method is described in paper[8]. This paper, along with many others, attributes the concave shape of continuous variation curves to the impure nature of the reagent and to the presence of undesirable side reactions. The formation of * Present address: Department of Chemistry, University of Waterloo, Waterloo, Ontario, Canada, 1. 2. 3. 4. 5. 6. 7. 8.

H. L. Schl~ifer, Komplexbildung in LOsung. Springer, Berlin ( 1961 ). M. M. Jones, Elementary Coordination Chemistry. Prentice-HalL Englewood Cliffs. N 3. (1964). E. Asmus,Z. analyt. Chem. 183,321 (1961). K. S. Klausen and F. J. Langmyhr, Analytica chim. Acta 28. 335 (1963). K. S. Klausen and F. J. Langmyhr, A nalytica chim. A cta 40, 167 (1968). B. Budesinsky, Z. analyt. Chem. 206,401 (1964). 1. Dahl and F. J. Langmyhr, Analytica chim. Acta 35, 24 (1966). B. Budesinsky, Colln. Czech. chem. Commun. 32, 167 (1967). 1345

1346

B. B U D E S I N S K Y

complexes where m and/or n I> 2 is often accompanied by side reactions due to stepwise complex formation. Several years ago, the method of proportional absorbances was recommended for the determination of conditional stability constants, and for classification of complexes when m = n = 1 and m = n > 1 [9]. This paper endeavours to show the general application of this method to the determination both of absolute values of m and n and of the conditional stability constant, for any useful concentration of the complexing component. In addition, the use of the continuous variation method to determine absolute values of m and n for very stable complexes is presented. EXPERIMENTAL Stock solutions of metal ions were prepared from the usual metal salts (i.e. nitrate or chloride). The pH-adjustment was performed by the use of Michaelis acetate-veronal buffer[10] or hexamine-nitric acid buffer[11], and the pH-setting was controlled potentiometrically by a "Vibron pH-meter, model 39A" (Electronic Instruments, Ltd., Richmond, Surrey). All chemicals were B.D.H. "Analar" products or "Analar" products of Lachema (Czechoslovakia). The photometric measurements were performed with a "Beckman DB Spectrophotometer" (Beckman G M B H , Munich). A "Farrand Optical Co., Catalogue No. 104244" spectrofluorometer (New York) was used for fluorimetric measurements; the spectral characteristics of this instrument were described earlier[12]. 1 cm quartz cells were used throughout. THEORETICAL

Calculation o f composition and conditional stability constants The conditional stability constant of the complex AmBn is given by the wellknown equation

(l)

Kmn = C(CA - - m c ) - m ( C B - n c ) - n ,

where c is the concentration of the complex AmBn, CA and cB are the total concentrations of the components A and B. The method of continuous variation and the molar ratio method give the values U and V for the complex AmBn where U = m/(m+ n) i.e. n / m = I / ( U - 1),

(2ab)

V = m/n.

(2c)

and

Introduction of the expressions c A n / m = c~,

(3ab)

C/CA = Zmn

into (1) gives -1 n CA 1-(rn+n) = Zmln(1 __ mZmn)rn+n Kmn(m/n)

=

y,

9. B. Budesinsky,Z. analyt. Chem. 209,379 (1965). 10. A. M. James, Practical Physical Chemistry, p. 329. Churchill, London ( 1961 ). 11. B. Budesinsky and K. Haas, Z. analyt. Chem. 210,263 (1965). 12. G . F . Kirkbright, T. S. West and C. Woodward, Talanta 12, 517 (1965).

(4)

C o m p o s i t i o n and conditional stability c o n s t a n t s of c o m p l e x e s AmBn

1347

where the ratio m/n is determined from the continuous variation or molar ratio method (see (2b) or (2c)). I f c Ais the concentration of A in a second solution, then ' :CA/a, cA

(Sa)

t

(5b)

and Zmn

=

c'/c A,

where a is an optional constant (0 < a + 1), cf. Equation (3b). Therefore, from Equation (4) -1 n t 1--(m+n) Kmn(m/n) CA =

(6)

Zm n'-I ( l - - mZ'nan)m+n.

By substituting for CAfrom (5a) rearranging the following equation is obtained

K-lmn(m/n)n CAl-(m+n) ---- a l-(m+n) Zmn -1! ( 1 __ mZmn ) , m+n .

(7)

Combination of this equation with (4) gives z~(1-

m Z m n ) m+n = a l - ' m + n ) Z m n l ( l __ m Z m) ,n r e + n =

y.

(8)

If A and A' are the measured absorbances of the complex AmBn for concentra! tions CA and c A, i.e. A = kc, A' = kc', where k is the proportionality factor and is represented by the molar absorptivities of individual absorbing species*, then Zmn/Z 'mn - - c / ( a c ' ) =

A/(aA')=

Xmn.

(9)

The functions Y(Zmn) and Y(Z~n) are calculated from (4) and (8) and are given graphically, in semilogarithmic form for a = 2, in Fig. I. The limiting values for y, Zmnand Z'm., reqmred" for experimental" measurements, can be obtained from (4) and (8) limy = ~ , zm°,z~ ~ 0

limy = 0 .

(10ab)

zm.,<,,#~ I/m

The function y(Xmn ) (see (4), (8 and (9)) may be calculated from Fig. 1 for any composition of AmBn, as one pair of values of Zmn, Zm, corresponds to the same y. The result of such a calculation, for complexes AB, AB2, AB~ and A2B2, giving the corresponding values Xla, x12, x13 and x22, is summarized in Table 1 ; a = 2 is used throughout. The limiting values of Y(Xmn), which are important for experimental measurements, may be estimated from (4), (8 and (9): lim Xmn = a m+n-l,

lim Xmn = 1.

Y~

y~0

( 1 lab)

T h e y may also be obtained by extrapolation from Table 1. * I f the basic c o m p o n e n t s A and B do not a b s o r b , m e a s u r e m e n t against a s o l v e n t blank is quite sufficient. If either of the c o m p o n e n t s A and B a b s o r b , m e a s u r e m e n t against a solvent blank of the s a m e c o n c e n t r a t i o n of the light-absorbing c o m p o n e n t is n e c e s s a r y .

1348

B. BUDESINSKY

iog~2 ~

T

,

f

,

,

.4

.6

.B

1,0

-2

-4

2

r.mn(Z~n)

Fig. 1. Dependence of log y on z~, and z'n (curves with comma) for various complexes: 1,I' AB; 2,2' ABe; 3,3' AB.~;4,4' AzB~. Equation (1 la) may be used to determine absolute values of m and n. It is not necessary in practice to achieve the limiting conditions given by eqn. (11 a). It may be seen from Table 1 that, for a = 2 , the interval 0.0 < log y < 1"0 gives a reasonable characterization of individual complex species, and for the estimation of absolute values of m and n if the ratio m/n is known from the continuous variation or molar ratio methods. This interval corresponds to suitable concentrations of starting components CA and Ca (see (3a)) and to a suitable value of the conditional stability constant (see (4)), which may be achieved by a convenient adjustment of the acidity of the solution. T h e choice of a --- 2 simplifies the calculation in this case, and by using a suitable series of solutions, a difference in log y results in a range of 1.5-2.0 for all practical measurements. If the values of m and n are known, the conditional stability constant Km. can be calculated very easily from (4).

Composition and conditional stability constants of complexes AmBn

1349

Table I. Dependence of log y on Xmnvalues log y 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1-4 1.2 1.0 0.8 0.6 0.4 0-2 0-0 -0.2 -0.4 -0.6 -0.8 1.0 1-2 - 1.4 1.6 - 1.8 - 2-0 -2.2 2.4 -2.6 -2.8 -3.0 -3-2 - 3.4 - 3.6 3-8 - 4.0 4.2 4.4 - 4.6 4.8 - 5-0 -

-

-

-

Xll

1.992 1.981 1.958 1.922 1.882 1.838 1-788 1.738 1.687 1.631 1.557 1-482 1.410 1.346 1.286 1-227 1.169 1.127 1.098 1.079 1.063 1.052 1.042 1.034 1.028 1.023 1-018 1.014 1.011

a

11 23 36 40 44 50 50 51 56 74 75 72 64 60 59 58 42 29 19 16 11 10 8 6 5 5 4 3

-

Xl2

A

Xl3

a

x22

a

3.980 3.975 3-970 3.964 3-950 3.910 3.837 3.713 3.510 3.295 3-075 2.847 2.632 2.421 2.212 2.022 1.853 1.718 1-587 1.473 1.382 1-317 1.257 1-214 1.177 1-148 1.120 1-102 1.086 1.074 1.063 1.053 1.045 1.038 1-033 1.030

5 5 6 14 40 73 124 203 215 220 228 215 211 209 190 169 135 131 114 91 65 60 43 37 29 28 18 16 12 11 10 8 7 5 3

7.960 7-940 7-915 7.871 7.803 7-692 7.510 7-290 7.058 6.610 6-062 5-411 4.722 4.061 3-570 3.085 2.670 2.363 2.112 1.883 1,723 1-596 1.507 1.423 1.360 1-300 1.250 1-211 1.181 1.156 1.137 1.120 1.105 1.091 1.088 1.076 1.068 1.058 1.052 1.048 1-045

20 25 44 68 111 182 220 232 448 548 651 689 661 491 485 415 304 251 229 160 127 89 74 63 60 50 39 30 25 19 17 15 14 13 12 10 8 6 4 3

7.930 7.916 7-890 7-858 7.800 7.692 7.440 7.085 6.640 6.110 5-517 4-914 4.309 3.701 3.258 2.617 2.223 1.956 1.738 1-607 1.494 1.403 1.326 1.283 1.253 1.227 1.203 1.180 1-160 1.141 1.123 1-107 1.092 1.072 1-063 1-057 1-052 1.049 1.047 1.045 1.043

14 26 32 58 108 252 355 445 530 593 603 605 608 543 541 494 267 218 131 113 91 77 43 30 26 24 23 20 19 18 16 15 10 9 6 5 3 2 2 2

-

-

-

Determination o f m and n for very stable complexes If the complexes

formed

a r e s o s t a b l e t h a t t h e c o n d i t i o n 0 . 0 < l o g y < 1.0

c a n n o t b e a c h i e v e d b y a d j u s t m e n t o f t h e a c i d i t y o r b y d i l u t i o n , o r if a h i g h d e g r e e of inaccuracy

in a b s o r b a n c e

measurements

occurs as a result of such an adjust-

m e n t ( A , A ' < 0-01), t h e n a d i f f e r e n t m e t h o d f o r t h e d e t e r m i n a t i o n must be adopted.

of m and n

A n e a r l i e r m e t h o d [8] f o r c a l c u l a t i o n o f c o n d i t i o n a l s t a b i l i t y c o n s t a n t s i n v o l v e s drawing tangents to the continuous variation curve from both end-points, so that

1350

B. BUDESINSKY

the curve representing the hypothetical infinitely stable complex is established. If a 1 per cent deviation or less is obtained for the absorbance, then the shape of the practical curve may be compared with that of the curve for an infinitely stable complex, in the interval CB/(CA+ CB) = 0" 1. The minimum stability of complexes, for which this method is applicable, is given by equation

(12)

Kmn ~-- Ioan-I(CA -]- CB)1-(m+n),

where m ~< n. The same condition may also be deduced from the molar ratio method. If the continuous variation method is used, then the conditional stability constant can be calculated from the equation / m + n \ m+n-1 A{1 Kmn = m - r a n - n | - - }

A ] -(re+n)

\cA +

(13)

'

and in the molar ratio method from the equation A' / A I \ -(rn+n) Kmn = m n-1 n - n c ~ 1-(m+n)-i'7| 1 -- -i-7/ Ai \ Ai ,/

(14)

where A, A' are the measured absorbances and Ai, A~ are the absorbanees of the infinitely stable complex. These are obtained by extrapolation (see [8]). The absolute values of m and n may be calculated from either (13) or (14) for two different concentrations, i.e. CA(l), CB,) and CA(2),%(2> or c~(~) and c~(zr The constant Kin, can then be eliminated and substitution into (13) and (14) gives Equations (15) and (16) respectively. o

m+ n = l og

(cA(~)+ c~(~))A(~)Ai(z) g

+

-

log m+ n =

~

(CA(2)"~- CB(2))Ai(1)A(2) -

CA(1V t Al(1) P A i(2) t !

l

t

CA(z)Ai(1)A(2)

t t log CA(I""'--~) (1 -- A(I--"~)~ / (1 -- A~2--~)~ c~,(2)\

(15)

A~(1)//\

(16)

A~(2)/

RESULTS AND DISCUSSION

Several examples of the application of the method of proportional absorbances to determine absolute values of m and n and conditional stability constants Kmn are presented in Table 2. Complexes of different composition have been chosen, and a good agreement between theory and practice may be seen. Examples of the determination of m and n for extremely stable complexes are given in Table 3. This particular method requires very accurate measurements of absorbance, the deviation ___0.30 being the acceptable maximum for determination of the value of m + n.

C o m p o s i t i o n and conditional stability c o n s t a n t s of c o m p l e x e s AmBn

1351

Table 2. D e t e r m i n a t i o n of c o m p l e x composition and conditional stability c o n s t a n t by the proportional a b s o r b a n c e m e t h o d Specification of c o m p l e x formation

- log CA

Xmn

log y

log Kn~n

B a - d i m e t h y l s u l p h o n a z o III [13], p H 6 . 9 9 (Michaelis), habs. = 655 mtz, m / n = 1, m+ n = 2

4.40 4.70 5.00 5.30 5.60 5.90 6,20

1-071 1.166 1.200 1.267 1.378 1-490

- 1.50 -0-81 -0'69 -0.40 -0"10 0.22

5.90 5.51 5.69 5-70 5.70 5.68

A 1 - 3-hydroxy-2-naphthoic acid [ 14], p H 6-20 (Michaelis), hexlt. = 370 m/z, hnuores. = 460 m/z, m / n = 1, m + n = 2

4"70 5"00 5"30 5"60 5-90 6"20

1"835 1 '924 1"981 1"992 1 '999

1"39 1'81 2'20 2"40 2'60

3'31 3" 19 3-10 3"20 3'30

Z r - A r s e n a z o I I 1 [6] 3-7 M perchloric acid, habs. = 665 m/z, m/n = 0"5, m + n = 3

4.70 5-00 5"30 5-60 5'90 6"20

1.653 1-870 2-410 2.860 3.580

--0.70 --0.18 0.39 0.81 1.47

9"50 9"58 9.61 9.79 9.73

A I - 4-di(carboxymethyl)aminomethyi-3-hydroxy-2-naphthoic acid, p H 6.75 (Michaelis), hexit. = 370 m/z, hfluores. = 460 m/z, m / n = 0.5, m + n = 3

4.70 5-00 5"30 5"60 5.90 6'20 6"50 6"80

1.543 1.915 2'020 2-341 3.402 3.910 3-973

--0"68 --0.13 0.00 0-31 1-30 2.00 2.74

9'48 9"53 10-00 10-29 9"90 9-80 9"66

T i ( I V ) - chromotropic acid [ 15] p H 5.32 (Michaelis), habs. = 420 m/z, m / n = 0"33, m + n = 4

3-72 4-02 4-32 4"62 4.92 5"22

1.367 2"040 2'530 3.815 7.104

-- 1'77 --0.66 --0-29 0'30 1-44

11-50 I1"76 11"99 12-13 11 '87

Se( 1V) - 2-mercaptobenzthiazol [16] 7.5 M hydrochloric acid, habs. = 370 rn~, m / n = 0.25, m+n=5

4"40 4.70 5-00 5"30 5'60

1-927 2-042 2-301 4.012

F e ( I I l ) - c h r o m e azurol S [4] p H 3.40 (hexamine), habs. = 570 m/z, m / n = 1, m + n = 4

3'80 4'10 4.40 4.70 5.00 5.30 5.60

1-058 1.236 1"411 1-850 4.612 6.840

--3"10

14"49

-- 1.93

14'23 14-38 14.60 14'30 14'41

1"18 -- 0"50 0"70 --

1.49

1352

B. B U D E S I N S K Y Table 2 (Contd.) Specification of complex formation Cu(I I 1) - 4,4'-diaminostilbene- N,N,N ',N '-tetraacetic acid [ 17] pH 6.2 (haxamine), kexlt" = 385 m/z, hauores. = 440 m/z, m/n = 0.5, m + n = 3*

- log cA 4.80 5.10 5.40 5.70 6.00 6.30

Xmn

log y

1 . 3 8 2 -1"00 1-637 -0-52 1.911 -0.13 2.610 0.58 3.300 1.21

log Kmn 10"05 10" 12 10.32 10"22 10.19

*Ccu = cB, CL= CAin this case. Table 3. Determination of complex composition by the continuous variation method Specification of complex formation

- log (cA + c8)

A/AI

m+ n

Ba - dimethylsulphonazo 111 [ 13] pH 6.99 (hexamine), habs. = 655 m/z, m/n = 1-00

4.40 4"10

0.697 0.601

1-95

L a - dicarboxyarsenazo 111 [ 11] pH 3-50 (hexamine), hab~. = 645 m/z, m/n = 0"50

4.10 3.80

0-970 0.952

3"20

U(V1)-chlorophoshonazo II I [ 18] pH 1-20 (nitric acid), haas. = 665 m/z, m/n = 0.50

4-10 3.80

0-980 0.969

2-82

Self-association of reagent particles occurs frequently for many organic dyestuffs, rendering the investigation of metal complex composition very difficult, and in such a case the methods described above cannot be used. Arsenazo III, 3,6-di(o-arsonophenylazo)-4,5-dihydroxy-2,7-naphthalenedisulphonic acid, exhibits this property in the concentration range of 10-3-10 -4 M and pH 3-7. In this case self-association can be avoided by using a more dilute solution (! 0 -5 M) or by changing the pH. Acknowledgement-The author is grateful to Dr. G. F. Kirkbright, Department of Chemistry, Imperial College London, for his kindness in supplying 4,4'-diaminostilbene-N,N,N',N'-tetraacetic acid. 13. 14. 15. ! 6. 17. 18.

B. Budesinsky, D. Vrzalova and B. Bezdekova, A cta Chim. A cad. Sci. Hung. 52, 37 (1967). G . F . Kirkbright, T. S. West and C. Woodward, A nalyt. Chem. 37, 137 (1965). L. Sommer, Z.Anal. Chem. 164, 299 (1958). B. C. Bera and M. M. Chakrabartty,Analyst 93, 50 (1968). G. F. Kirkbright, D. 1. Rees and W. I. Stephen, A nalytica chim. A cta 27, 558 (1962). B. Budesinsky, K. Haas and A. Bezdekova, Coll. Czech. Chem. Commun. 32, 1528 (1967).