THE CALCULATION OF THE STABiLITY CONSTANTS OF SOME

USING A HIGH SPEED DIGITAL COMPUTER LANTHANIDE-a-HYDROXY-CARBOXYLATE COMPLEXES

H. DEELSTRA, W. VANDERLEEN and F. VERBEEK

SUMMARY

The four successive stability constants and their standard deviations have been determined for the a-hydroxy-isobutyrate and lactate complexes of the lanthanide elements and yttrium. A least squares treatment of the experimental data obtained by the Calvin-Bjerrum potentiometric titration method using an I.B.M. 1620 high speed digital computer is described.

On complex formation between metal ions and ligands, several stepwise formed complexes ML, ML2 ... MLN may coexist in solution over a wide range of concentration. The relative abundances of the complexes are depending on the concentration of the free ligand [L]. According to Bjerrum (I), the relationship between [L] and E, the average number of ligands bound per metal ion present, is given by the formation function

i’ K1 [L] + 2K1 Kz LLI2 F ... + NKi K2 ... KN [LIN 1 + K i [L] + Ki Kz [L] 2 . . . + K i K2 ... KN [LIN

k?=-

where K1, KZ ... KN are the stepwise formation constants of the com- plexes, resp. ML, ML2 ... MLN.

P N = K1 . Kz ... KN, the over-all stability constant of the complex MLN.

When a set of numerical values of ri and [L] have been obtained by potentiometric or other measurements, equation (1) can be solved among other possibilities by Bjerrum’s half 6-method (I), Fronaeus’ extrapolation method ( 2 ) , and also by a least squares treatment of ri

and [L]. Such calculation methods using a high speed digital computer

(l) J. BJERRUM, “Metal Ammine Formation in Aqueous Solution”,

(2) S. FRONAEUS, Actu Chem. Scud., 4, 72 (1950); 5, 139 (1951); 6, 1200 P. Haase and Son, Copenhagen, 1941.

(1952).

CALCULATIONS OF STABILITY CONSTANTS 633

are described by Rydberg and co-workers ( 3 9 , Chopoorian and co- workers (7, and Sillen and Ingri (839).

In this paper a somewhat similar procedure for an IBM 1620 high speed digital computer is used. This method is applied to determine the four individual stability constants K1, K2, K3 and K4 of the lanthanides and yttrium with lactate and a-hydroxy-isobutyrate. Choppin and Chopoorian (lo) already determined K1, K2, K3 but not Kq for yttrium and nine of the lanthanides with the same ligands. The experimental arrangement of their potentiometric titration technique was a modifi- cation of the one used by Fronaeus ( z ) , while in the present investigation the Calvin-Bjerrum potentiometric titration (11) is used.

CALCULATION PROCEDURE

The expression of a regression function can often be represented by means of an mth degree polynomial

m

In general the distribution function is not known so that the regres- sion curve must be found empirically. In this way, one arrives at

m

where bmj are the guessed or estimated parameters with a “true” value B,i. I t is assumed that the recorded values of x are known exactly. In fact, it is sufficient that the errors on the x’s are small compared with

_. ~ - -~ ~

c?) J. RYDBERG and J.C. SULLIVAN, Acta Chem. Scand., 13, 186 (1959). (4) J.C. SULLIVAN, J. RYDBERG and W. MILLER, Acta Chem. Scancl., 13.

(5) J. RYDBERG and J.C. SULLIVAN, Acta Chem. Scarrd., 13, 2057 (1959). (G) J. RYDBERG, Acta Chem. Scand., 14, 157 (1960). (7) J.A. CHOPOORIAN, G.R. CHOPPIN, H. c. GRIFFITH and R. CHANDLER,

(8) L.G. SILLBN, Acta Chem. Scand., 16, 159 (1962). (’9 N. INGRI and L.G. SILLBN, Acta Chem. Scand., 16, 173 (1962). (lo) G.R. CHOPPIN and J.A. CHOPOORIAN, J. Znovg. Nucl. C h m ~ . , 21, 97

( * I ) M. CALVIN and K. W. WILSON, J. Amer. Chem. Soc., 67, 2003 (1945).

2023 (1959).

J . Inorg. Nucl. Chem., 21, 25 (1961).

(1 961).

634 H. DEELSTRA, W. VANDERLEEN AND F. VERBEEK

those on the y’s. Equation (3) is linear in bmj in this case and represents a linear fundamental equation.

When one disposes of a number n of observed data yt of the physical quantities Yt, the problem consists in determining the most probable values of the bmj coefficients. These experimental data regarded as empirical values are often normally distributed around the unknown values Yr with dispersions ot. Generally the oi’s differ from one another, which means that the accuracies of the various yi-values are different. The relative accuracy is known, and consequently also the weights pi

where 02 is a proportionality factor, representing the dispersion of an observation with unit weight.

To the principle of the “least squares” underlies now the following criterion : the values bmj must be chosen in such a way that

where ri is the distance in the y-direction between the observed point and the estimated regression curve. This condition gives in + 1 linear equations in bmj or the normal system.

Because of the normal distribution of the experimental data, the best estimate for 02 can be obtained by

n

where (m + 1) is the number of parameters to be estimated,

are calculated. Therefore the inverse matrix x = m To investigate the possible statistical fluctuations, the dispersions

is computed; m

it is obvious from (5) that @jk = 2 pr .Y$ xik. i=l . -

When E (y/x) = ’ y f ( x ’ y ) d y = U (x) and the uOs are mutually Jf (x, Y ) dy uncorrelated, it is easy to point out that

T E (bm - Bm)(bm - B:) = 5’ x

CALCULATIONS OF STABILITY CONSTANTS 635

T Bm is the (rn + 1) x 1 matrix of the “true” values Bmj; b: and Bm resp. are the transposed matrices of bm and Bm.

Accordingly covariant matrices are obtained

COV (bmj, b m k ) = x j k cr2 (7)

and in particular

c3 b,j = ~ j j 02 (8)

If the parameters b,j are non-linear in the regression function, it is possible to reduce this equation to the preceding linear case by expand- ing the non linear function

(9) Y = f (x, bmj)

in a Taylor series around the good approximations bomj of bmj

The fundamental equation is now linear in Abomj and can be solved by the least squares method. The values of the estimated parameters are now

bmj = bomj + A bomj (1 I )

These newly obtained estimates bmj however are approximations because (10) is only valid for sufficient small hbo,j values, the higher terms of Abomj being neglected in the development of the series.

If these values are too large, the same procedure can be repeated with the obtained values bmj of (1 1) as new estimates.

Calculation of the stability constants

The formation function (1) can for N = 4 be written in the form

f i = f ( K i , Kz, K3, K4, [L]) (12)

where the stability constants KI, Kz, K3, K4 must be estimated from observations of f i r in function of [L].

The transformation of equation (12) into a linear form is possible by substituting these values by an estimated value KO with a deviation AK; from the “true” value Kj (thus Kj = KY + AKO).

3

I

636 H . DEELSTRA, W. VANDERLEEN AND F. VERREEK

From this the normal system follows

wherefis written forf'(Ky, K!, KE, Ki, [Lz]) and I = I , 2, 3. 4. The A K: values are obtained by solution of the system (14). Addition

of these AKY values to the first estimates K; gives new estimates for the Kj values. This procedure is repeated until the 11.K:. values are sufficient small.

Equation (14) is programmed in FORTRAN and computed with an IBM 1620 high speed digital computer. The cyclus is repeated until

!%< O 1 - KY 1000'

The standard deviation on P n is calculated by

where K1, Kz.. . , Kn are the individual stability constants, with standard deviations resp. ul, a', ... an given by the computer.

RESULTS

Experimental conditions and more complete information on the determination of f i and [L] as well as the calculation of the stability constants according to the graphical methods, mentioned in Table 11 are given elsewhere. (Iz)

(12) F. VERBEEK and H. DEELSTRA, Med. Vluamse Chem. Verenig., 24, 167 (1962).

TA

BL

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(10

).

638 H. DEELSTRA, W. VANDERLEEN AND F. VERBEEK

3.02 ,

Theoretically, it should be possible to solve equation (14) for the desired four constants of the lanthanide-a-hydroxy-carboxylate systems with only 4 pairs of ri and [L] data. However if the constants are to be accurate, it is necessary to use considerably more than 4 pairs of ri and [L] over a sufficient large range of [L]. In this investigation 20 to 24 of such pairs were used for the computation.

The reliability of the computer method was previously checked. Experimental ri and [L] data (12 pairs) from Choppin and Chopoorian (lo)

for the systems cerium (111)-glycolate, holmium (111)-lactate and ytter- bium (111)-lactate were directly introduced into equation (14) which was then solved for the successive constants by the proposed method. As can be seen from Table I the results show good agreement.Also the fourth constant K4 of the complex ML, not given by Choppin because of random computional errors is calculated with an acceptable deviation. The standard deviations on the constants determined by Choppin (column 2) are estimated from the graph.

3.01 & 0.06

TABLE I1

Comparison of log Pn values of some lathanide-a-hydroxy-isobutyrate complex sys fems, determined by three graphical methods and by computer calculation

from the same sets of ii and [L] data. (*)

Poulsen

Lu (111)

Yb (111)

Tm (111)

Er (111)

Fronaeu: Constants

~

3jerrurn

3.38 6.18 8.39

10.01

3.32 6.05 8.20 9.75

3.25 5.90 7.94 9.37

3.20 5.77 7.71 9.03

3.21 6.01 8.41 9.78

3.12 5.80 8.00 -

3.08 5.76 8.08 -

3.02 5.63 7.86 9.24

3.21 5.92 8.05 9.99

3.12 5.82 7.85 9.70

3.08 5.80 7.72 9.47

Cornpu ter

3.18 & 0.055 6.04 & 0.08 8.07 & 0.09 9.98 & 0.1 1

3.13 + 0.10 5.87 5 0.1 5 7.94 & 0.19 9.72 i 0.20

3.10 & 0.08 5.79 & 0.12 7.71 & 0.16 9.33 & 0.19

(*) temperature : 25.OoCc; NaC104 = 0.2 M.

CALCULATIONS OF STABILITY CONSTANTS 639

3.05 + 0.03

3.03 & 0.025

2.92 & 0.03

2.86 & 0.035

2.73 fO.04

2.59 0.045

2.65*0.045

2.57 i O . 0 5

2 .55 t0 .05

2.56 &0.055

2.80 iO.06

~~

-

Another check on the reliability is given in Table 11. The values of log pn. calculated from our experiments by this least squares technique and by the graphical methods of Bjerrum (I), Fronaeus (2) and Poulseii and co-workers (13), are compared for some lanthanide-a-hydroxy-iso- butyrate systems. The approximate constants obtained by Bjerrum's half %method (column 3) were taken as estimates and introduced into equation (14) to compute the values in column 6. It was usually only necessary to perform three successive iterations.

The lanthanide-lactate and the lanthanide-a-hydroxy-isobutyrate stability constants are given in Table 111, resp. Table IV. Considering the difference in the methods used and in the ionic strength, and the experimental errors, there is a general good agreement between the reported values and those obtained by Choppin and Chopoorian (10).

2.51 hO.03 1.41 rtO.03 0.86 *0.04 7.83 !-0.07 -

2.42 & 0.025 1.34 f 0.03 0.69 + 0.035 7.48 + 0.06

2.34 1 0 . 0 3 1.34 -I 0.035 0.65 f 0.048 7.25 3- 0.07

2.33 & 0.03 1.33 0.035 0.66 f 0.047 7.18 t 0.075

2.30 kO.05 1.27 0.06 0.67 4 0.075 6.97 CO.10

2.23 5 0.045 1.09 & 0.065 0.63 & 0.09 6.54 C 0.15

2.17+0.05 1.07f0.07 0.44+0.!0 6.33 t0 .16

2.15 i-0.05 1.02 iO.08 0.47 &0.15 6.21 f0 .17

2.12-fO.055 0.88+0.10 0.51 +0.18 6.06f0.20

2.04 *0.065 0.90 &0.12 0.41 +0.25 5.91 f 0.30

2.14 i.O.065 1.22 I-tO.10 0.55 & 0.14 6.71 -C 0.20

______

___-_______

______ _-

-~

-

-

TABLE 111

Stability constants of' various lathanide and yttrium-lactate complex systems. (*)

Lu

Yb

Tm

Er

Ho

DY

Tb

Gd

Eu

Sm

Y

(*) temperature : 25.OoC; NaC104 : 0.2 M.

(13) K.G. POULSEN, J. BJERRUM, J. POULSEN, Acta Chem. Scand., 8, 921 (1954).

640 H . DEELSTRA, W. VANDERLEEN AND F. VERBEEK

- ~ - - ~ ~ ~~

Ele- I log K I log Kz ment I ____. - - - - _ _ _ ~

Lu 3.18 t0.055 2.863Z0.06

Yb 3.13&0.10 2.74+0.12 _-

The time necessary for computing the four constants was 10 to 15 minutes. The use of such computer programmes also greatly simplifies the calculations of these investigations and represents a considerable saving of time.

~

log F 4 I ~~

~ ~~

1 logK3 log K4

~- - -____-- ~- - -

2.03i0.055 1.92st0.045 9.9910.11

2.07&0.115 1.78+0.09 9.72I-tO.20 __c___I_ -

TABLE IV

2.79 1 0.035

Sm 2.75i0.015

Nd I 2.74 50.04

___ -____I_

Eu i I------ _--

Stability constants of the carious lanthanide and yttrium-a-hydroxy-isobutyrate coniplrx systenu. (a)

2.07 -f 0.065 1.48 0.15 1.25 & 0.21 7.59 + 0.29

2.0210.03 1.40&0.07 1.21 *0.11 7.38d~0.14

1.68 &0.075 , 1.563~0.10 0.60 & 0.19 6.58 +0.25

I___

i--- __ __

3.10 k 0.08 2.69 h 0.095 1.92 ZIZ 0.08 1.62 0.09 9.33 h 0.19 _______I_-

Er T m ! 3.01 & 0.06 I 2.69 i 0.075 I 1.87 & 0.065 I 1.45 * 0.055 I 9.02 & 0.1 -- 1

Ho 12 .98&0.08 1 2 . 5 6 5 0 . 0 9 I 1 . 9 0 i 0 . 1 1 1 1.30f0.13 18.74-kO.22

Dy ! -2.9410.07 i 2.50f0.09 1 1.843Z0.10 1 1 . 2 1 ~ 0 . 1 4 1 8.49+0.20

Tb I -2.92 1 0 . 0 3 2.32 50.02 1 1.62 50 .03 1.23 j ~ 0 . 0 3 ___I______

._______-

Pr 2.59 iO.015 I 1.78 h0.025 j 1.23 h0.05 1 0.78 h0.07 I 6.38 t 0 . 0 9

Ce 1 2.55 10.035 5.49 i 0.18 (b)

4.04 i: 0.065 (c) - ~~

Y ~ 2.92+0.13 1 2.70i0.14 1 1.72.+0.17 1 1.44+0.18 1 8.78i0.31

(a) temperature = 25.0°C; NaC104 : 0.2 M. (b) log P3. (c) log p..

CALCULATIONS OF STABILITY CONSTANTS 64 I

ACKNOWLEDGEMENTS

The authors wish to express their thanks to Prof. Dr. J. Hoste and

This investigation is in part sponsored by the Interuniversity Institute Prof. Dr. C . Grosjean for kind interest in this work.

for Nuclear Sciences (1. I. K. W.).

Laboratorium t100r Annlytische Chemie Rekenlaboratorium

RIJKSUNIVERSITEIT TE GENT

Medegedeeld aan de Vlaamse Chemische Vereniging op I I juli 1963.