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A critical study of the linear least-squares method applied to the spectrophotometric determination of protonation constants of diprotic acids
Spectrochimico Acta. Vol.
0584-8539/82/020247-06503.00/0 @ 1982 Pergamon Press Ltd
38A. No. 2, pp. 247-252, 1982
Printed in Great Britain.
A critical study of the linear least-squares method applied to the spectrophotometric determination of protonation constants of diprotic acids
Department
C. MONGAY, G. RAMIS and M. C. GARCIA of Analytical Chemistry, Faculty of Chemistry, University of Valencia, Valencia, Spain (Received 25 July 1981)
Abstract-The method of multiple linear regression applied to the spectrophotometric determination of the protonation constants and the molar absortivity coefficient of the monoprotonated species of diprotic acids, is studied critically. The best way of fitting the data, according to the shape of the absorbance vs pa-curve, is established. The conclusions are applied to real systems.
INTRODUCTION Spectrophotometry is, after potentiometry, the most widely-used technique in the determination of protonation constants of acid-base systems. Although spectrophotometry is more time-consuming and is considered as less precise, it is an ideal method for the study of sparingly soluble substances or those substances, which constants are particularly low or high. Many diverse methods of calculation have been proposed for the spectrophotometric determination of the two protonation constants of a diprotic acid, when they lie far enough, since in this case the same treatment used for the monoprotic acids may be applied [14]. However, when the two protonation constants are too close, the problem becomes more complex because not only the two protonation constants are unknown, but also the molar absortivity coefficient of the monoprotonated species, which cannot be directly determined experimentally. The methods proposed in this case[S-81 use to be very laborious, often restrictive and they are usually based upon iterative procedures, including occasionally weighting factors. These methods require the use of high speed computers and very sophisticated programs. In this work, the application of the linear leastsquares method in the determination of the protonation constants of diprotic acids with overlapping equilibria, is studied critically. This method has the advantage over other methods of requiring only a simple program, of being very rapid, noniterative and of being overcome by using programmable pocket calculators. In order to be able to discuss not only the precision, but also the accuracy of the method, hypothetical acids of prefixed protonation constants and molar absorptivity coefficients are assumed. Since the different ways of fitting the data are not equivalent, the conditions of applicability of each one are studied.
Finally, the conclusions are applied to the determination of the protonation constants and molar absorptivity coefficient of the monoprotonated species of three real systems, and the results are compared with others from the literature.
SIMULATION OF THE EXPERIMENTAL CURVES The absorbance of a solution of a diprotic acidbase system measured in a cell of l.OOOcm of optical pathlength, at a determined wavelength, is given by
eo+ e&h + e2P2h2 A=
c
l+P,h+&h*
where h is [H’], C the analytical concentration of the acid-base system and eO, c, and l2 the molar absorptivity coefficients of the nonprotonated, monoprotonated and diprotonated species, respectively. The cumulative protonation constants are PI and p2. In order to generate the ‘experimental’ curves, a diprotic system with determined values of its protonation constants and molar absorptivity coefficient of its different forms is assumed. About 40 points of the theoretical curve A, = f(pH,) given by equation (1) are calculated for a certain value of C, and from this ‘exact’ curve, 15 simulated ‘experimental’ curves are obtained, transforming the different theoretical points (A,, pH,) into experimental points (A,, pH,), assigning to the former gaussian errors by means of the expressions A. = A, +
uAz
(2) PI-L = pf-5 +
CpHZ
being (T,, and gpR the standard deviations associated to the absorbance and pH, which values are 241
248
C. MONGAY et
also assumed, and z an aleatory number, which follows the normal distribution law. The different values of z have been generated making use of the central limit theorem, which postulates that the distribution of probability of the sum of K independent and identically probable values of a certain aleatory variable approximates to a normal distribution as K increases [9]. Therefore a computer has been used, which provides uniformly distributed aleatory numbers, ri, between null and the unity, and the equation
al.
Y and 2 to slight experimental errors in the absorbance in the rejected zones. For this reason, the number of useful points, N, is reduced to about 30 in each determination. Each one of the four equations in Table 1 is fitted by the least-squares procedure by solving the system of normal equations CZ=a,N+a,
xX+a2x
Y
C,Xz=ao~x+~*~XZ+a2~xY
has been applied. In the practice of simulation, the value K = 12 is recommended, since a certain computational advantage is achieved because the preceding equation is reduced to z= 2, (ri-6) and despite this value of K restricts the distribution to the limits +6, it leads to total valid values of z up to three times the standard deviation. TREATMENT OF THE DATA
Expression a plane
(1) may be written as the equation of
2
YZ=aox
Y+a,CXY+a2z
(4) Y*
leading to four regression planes of Z on X and Y. For each plane, the parameters ao, a, and a2 are obtained and therefrom the values of K,, K2 and E, are calculated. Similarly, the values of the different parameters are obtained from the regression planes of Y on X and Z, and of X on Y and Z. The mean values and standard deviations obtained from the 15 experimental determinations are shown in Table 2, where the theoretical curves corresponding to each determination are also plotted. The standard deviation of the protonation constants are given with two significant figures. DISCUSSION AND CONCLUSIONS
Z=a~+a*X+a*Y
(3)
where the parameters ao, a, and a2 are related to p,, p2 and E,, and X, Y and Z are functions of the values lo and e2 and of the variables h and E (being E = A/C). Equation (3) may lead to the four expressions shown in Table 1. From the 40 experimental points (A,, pH,) obtained for each curve, only those which fulfil the condition IA, - E&/ > 0.05 IA, - l 2Cl > 0.05 are used, due to the sensitivity
of the values of X,
Table 1
2
E* -E
E
hE
E
h-‘(E-E,)
h-2
_h-lE
E-E,
Eo-E E
-E,
The different ways of fitting the data should lead to the same set of constants if the experimental data would not be affected with error. The fact of having different values with each one of the possible fittings is due to the introduced errors, because in the resolution of the normal equations there are, in some cases, substraction operations between very close quantities, leading to a ‘loss of accuracy’ since the relative error of the difference is much larger than that one of the minuend or the subtrahend. For this reason, equations (I) and (IV) in Table 1 lead to much worse results than equations (II) and (III), giving rise occasionally to interchanged values of the constants (log K, < log K2) and inclusive to negative values, therefore the results obtained by means of these equations are not indicated in this paper. Another fact emphasized, and which as already known, constitutes a limitation of the spectrophotometric method, is that in order to get a better accuracy and precision, more pronounced spectral changes in passing from one acid-base species to another are required. If the values of the molar absorptivity coefficients of both species in an acidbase pair are similar, the method cannot be applied, the results being worse as these values are closer. It must be remarked that in the resolution of the normal equations for the regression of Y on X
A critical study of the linear least-squares
,,aY#.l * “,cul5 5*9”89 t “.W2 J,,V t 69
I&m8 = 0.01” 5,993” a “,“a1 ,111 = 108
i,l)y’)* 1 “&“I69 6.999, = “,m6, 6,0o,q t “,006’1 6,m01 A “,m?l 486 s 8” 995 * m
1,co,6 t. “,W,4 5.9976 I “,a,66 ~06 1 62
I,mo5 t “,Mg5 6.M,, f “.w,”
6.~9 f 0,010 6,00,8 = “,“094 $3, t ,a
6.9901 f “,uoB, 6.~1 * 0,010 ‘903f 115
1,m24 * “,W94 5.9964 A 0.~066 304” * 9,
?,mO * 0.01, 6,w** t “.mB, 4980 f 116
+ 0,011 6,m em2 d 13,
l&n4 f 0,012 5,999, t “,WP ,~*6 * 12,
‘,,<“,I f 0,011 ,,YYI’, f 0.~36 5,‘) f 128
1,M23 f o/m?? 5,996.l f “,W8 ,041 f 10,
?.“a, f “.co94 5rY918 = “,W?, ,101 f 96
7.““4 = “,“U 5.9961 * “P9” ,060 * 12,
7.a f “,“,, 5,995 f 0,011 Boo6 1 13,
7.0% : O,“,, 5.991, f “.mJ3n ,“,“9 f 114
6,999 t 0,010 ‘,M,, f o,‘w~ NIL! = 12,
6,99tyI * “,MM4 6,ooO f 0.0,” YD4 f 115
‘,,M,?
t “.W4
5,~;
: ;;“”
6,999 * 0,011 6~314 + “.@W ,980 * 116
?,“a3 6.“~ m,
c, = 12.5m
f
,I
III
-
249
method
4995 = 93 = 0,012
1,Ml
= 0,012 f 0.01, a UP
1~326t 121
7
”
apll
<, = ,“.uo
6,.
0
c, -
2.m
t,
,.m
-
= lS.oa
f, .
*o.Mo
6. t,
11
= l”.mxJ
‘,,wJ~ f “,m6* ti.999” t “,w%” 5 - 6Y
‘,.W” : 0,012 5,999, a “,m95 ,991 = 99
7,031 A 0,015 5,998 f “.“u !m01ta
6,993 * “.“2” 6,005 f “.“*, ,252, * 5,
t 0,018 ?,rn, t 0,017 6,o~ 14999= 106
7,cm f 0,012 5,998 t 0,016 ,7999 t 111
1.12 5,w -,‘I37
l,“,? f 0,014 5,913, = 0.012 2,‘)o f 121
1,013 f 0,015 5,981 f 0.012 508s f16
1.01, 10,021 5.906 i 0,022 12455 f 56
* 0.018 7.m 5,995 A “,“,‘I 14955* 10,
7.““5 * “,“,, 5.994 * 0,016 199e2f 1.51
6,996
(1, .)
111
7.03,
0.012
‘,,m*
2 ;;““’
6~” ‘Jrn? * “,“,4 8,
6,009 ,252, f 0,020 54
106 6,~ 14999 I= 0,017
6.~0 *oxa .= 0,016 171
.,,a~, + “,ar,” ,.999” f o,Mqq
‘,,“a ‘,wJ,
* 0,016 * 0.012
190, f UY
1,002 * o,“,‘, 6,wx f 0,015 W2f9”
6.996 f 0,020 6.~8 * 0,020 ,*v* f ,,
= 0.011 1,“w 6,mo t 0.016 ,4997 f 91
1.m* z “&WY 6&%X = o,“,, *cc03 f ,q,
‘,,co~” * “,wm ,,992* t o&n@ 6” f 96
?.“a? * 0,016 5,993 = 0,012 *“uz * ,,9
7,W * 0,016 5,991 f 0.04 )064tn5
1,016 * 0.02, >,964 2 “,W2 12448 t 56
?,a39 * 0.011 5.99, = 0.016 1494” t 95
?.rn, = 0.01, 5,994 a 0,014 ,992, = 1%
?,<*ur2 f “,“078 ),9’)9” t 0,~~) , f 9”
6,999 f 0,016 6,,XK, f “.“,2 WY, = ,111
~,WJ t o,“,., 6,~” t 0,015 yn* * 9”
6,994 * 0,020 6,W + 0.02” ,*,,9 f 54
1,cO* f 0,011 6.W I”,“,6 14991 2 I)?
6.9995 f “PW, 5,993 = 0,014 *a*, f 14,
z = 9”
= 0,020
?,mo
r 0,012
6,~;
* “,MJ6,
f “,a,5
* 0,018
?,rn,
b.99,:; t L$XM”
?,a*:,
l
f, =,o.aa 62.
III
“2
0
f*
- 1.w”
2.500
62 - 5.cm
(2 = 12.500
r* - ,5.(x0 6,996 t 0,015 6,002 t 0,014 9916 t 16
6,911 f 0,034 6,021 t “,“,6 ,“4”” t 606
6.9966 1 “,uOVV 6,998 * 0,021 6,m,“* “.CW,? 5,991 f 0,020 ,cxwY* 216 ,M4? * 117
6,991 4 0.01, 6,“q * 0,012 1~52 * 61
,,‘W i 0,021 5.996 * 0,023 ,“mD * 57
6,987 t “.“34 6.012 t 0,014 lu224 * 516
1,m*3 t “,m34 5.9903 t “,WfM 9962 * 10,
?,a35 f 0,021 5.992 : 0.02, 9933 t 215
7,mo * 0,012 6,m0 k 0,011 9996 * 6”
‘,,“26 5,vll ,Wl,
6,979 i 0,034 6,“2P * “,“,5
6,99’,‘, A 0.~96 6&“19 f “,w’,,
10104 Et 60.3
100.~ t 116
1,ooO * 0,021 5,999 f 0,020 ,ooO’, * 216
6,992 : 0.01, 6.~08 f 0,012 ,M52 t 61
?,rn? * 0.02, 5,99+ * “.M, ,oooq*>7
?,mO * “,“,4 5,999 b 0,014 ,uDs t 129
6,994 * 0,011 6,m61* 0.0394 ,m4, 1 5,
l.c.38 * 0.02, 5,998 + “,“*3 ,““,” * 58
*,m* * 0.01, 5,995 = 0.012 9936 : 112
.I.“066 t “,“co9 5,99,8 f o&06, 9902 f 32
?,m6 k 0,014 5,992 I”,“,4 9927 f 129
,,““* * 0,011 589;;: “&a
1,024 f 0,022 5,919 t “so24 ,“a5 t 59
?,“a = 0,015 5,995 f “8014 1~26 = 19
1,996 t “o”,” L,W, i “,“,2 1~29 = ~6
?,m,5 5,;;;;
6,998 f 0,015 5,998 * “,“,4 ,mc6 t 129
6.99, l 0,011 6~354 = “,W* lM4, = 5,
7,““~ t 0,021 5,996 f 0,021 ,omg*50
6,996 f 0,016 6,022 * 0.015 991, = a5
5 = 2.w
Ll -
‘,
c, - *0.co”
6,998 2 0.0,” 6.C.729tf “JxW 912 83
6.995 t 0,021 6&w” + * “,“I91 2456 15
?,rn? f “,M, 5,911 * f 0,061 ,209 174
6.992 = 0,018 6,011 * = O&W, 215 ,5,“,
1&o* t 0,015 5,997 t* 0,020 ,991, 296
1.06 i “,40 6,“Z f “,28 -1642 t ,Joo
1,“,4 t 0,016 5,901” 2 “,“V4 ,165 f 7,
1,029 i 0.02, 5.992, i “,‘WM 2519 i ,6
?,“,, t “,“2, 5,961 t “,“1, 12195 l ,94
6.996 * 0,018 6,~) t 0.028 ,5034 t 211
7,036 f 0,016 5,992 * 0.02, ,987l f xv
1.002 t “,“,I ,,Y,,,; : ;;W*
7,w 6,mU ,m,
‘,,alt) i 0,024 5.9992 2512 * I 0.~~8, 7”
6,981 f 0,019 6,005 12‘94 t* “,ooO 234
6,982 * 0,018 220 6.04, f= 0,028 15295
6,991 * “,“,5 6,011 t* “,oz, m,,, ,,”
1,cn> i “,“,tl 5,996” A “,wF~ 34 f 1,”
7,cw i 0.021 6,~ = 0,015 963 i ,4,
?,uol i “.“I2 6.~3 t 0,012 2w3’99
6,996 a O,“,, 6,012 k “,“,6 12>6’, t 10,
6.99, * 0.012 6,018 f 0.011 15,217. 12,
1.m* 6,“~ 2~2,
‘,.“,2
7.0,”
7,“,9 i “.“29 5,996 I”.“,, *WI f 85
7.0,” I O,“,” 5,968 L 0,020 12401 A 1,”
7,“ca t “,“,2 5.991 f “,“,? 14981t 122
1,005 t “.“,” 5199, * “,“,4 ,989, * 117
6,9>2
1,“% I”.“,6 5.902 * 0.042 ,245” * ,?”
6.995 * “0012 ~/XI’, f 0,011 15049 * 12,
1,CO,* * O&O96 5,991 = “,“,4 ,9w9 f 168
L,
s
10
-
6,998 *“,“,5 6,m, 10,014 9916 f 18
‘,,a)** f “,mti 5,999, f 0.005” 991, f 19
t “,wB : ;W*
6.u)
* 0,021 * 0,024 * 57
5,997 f 0,010 i.w* f. 0,012 ,w,” 1 116
6. =
I
a
7
=
0
f,
-
l.wm
12.m
- ,5.c.xl
t.w
8aae. I,uo, 5.9,“;
II
f 0,015 : ‘J;W4
t “.“,I f 0.~92 15
-
Ill
li a,“,?
1.99~ i “~~192 6.~0 0, I 10, ,“,, 6,?Y,
apH
5-w
f, = ,o.m
-
L* -
t “,“2”
6.Wll 1 1”,“,” -22 i ,!!J
L 0.02, 1 “,“,I I 12”
6,911 t 0.034 6.1~” f I).“,? 8Y 1 192
t 0,051
6.01, = 0,011 *WI t 11.4
* 0.0096 * “~11, * 161
,
C. MONGAYetal.
250
*Among the 15 determinations,
only those indicated in parentheses lead to positive values of the constants.
and 2 by equation (II), the parameter a0 takes the value l/e,, and when E, approximates to zero, the error of this fitting becomes very large. This fact is clearly observed in the first acid-base system in series 2 and 4, where E, = 0, which give rise to unacceptable results; it may also be observed in these series how the results become better as the value of e, increases. Another fact which has a general character is that when the curve A = f(pH) is U-shaped, the fitting by means of equation (II) (except for the regression of Y on X and 2, specially as c, is smaller) leads to a little more precise and accurate results than the fitting by means of equation (III), being the results of the former clearly much better as the difference between the value of el and the values of l0 and l2 increases. On the other hand, when the curve has the shape of an inverted U, the fitting by means of equation (III) is preferable, leading the regression of Y on X and 2 to somewhat more unaccurate values than the regressions of X on Y and 2, and Z on X and Y. This can be observed in the series l-4 and it has also been corroborated for a series equivalent to series 1, where l, = l2 < E, and another equivalent to series 4, where E,,> l2. Finally, when the curve is ascending or
descending (e2 < E, < E,,or co < E, < e2), the method leads to results less precise because of the only inflection point, while when the curve has a maximum or a minimum, it presents two inflection points. In these cases, the best results may correspond to either of equations (II) and (III), in accordance with the gradual transformation from a U-shaped to an inverted U-shaped curve, as observed in series 4 or in its equivalent where co > l2. From the three different fitting possibilities for each plane, the regression of Y on X and 2 gives the best values (though other regressions may lead to total acceptable values), except when either of co or e2 trends to zero, where only the treatment based upon equation (III) leads to valid results. This can be observed in series 5 and 6 and it has been also observed with a series of curves corresponding to values of the molar absorptivity constants, where l2 - E, = l1- co. Although the results in Table 2 correspond to a diprotic acid-base system with protonation constants log K, = 7.000 and log K2 = 6.000, the conclusions have been checked for systems with protonation constants log K, = 3.000, log Kz = 2.000 and log K, = 11BOO, log K2 = 10.000, leading to similar results, not depending therefore on the conclusions for the value of these constants.
A critical study of the linear least-squares method Acids, which logarithms of their constants differ in 0.25 and 3 unities have been also studied, leading to the same conclusions, though the precision of the results increases as the difference between the values of the protonation constants of the acids increases. Finally, the data in Table 2 have been obtained assuming an analytical concentration of lo-‘M and values of the standard deviation for the absorbance and pH of 0.003 and 0.002, respectively, values that are usual in these spectrophotometric determinations. Series for other values of the standard deviation, up to 0.01 in the absorbance and 0.005 in pH have been also studied, and though in these cases the results are less precise, the conclusions on the best way of fitting are the same. APPLICATIONOF THE METHOD The preceding conclusions have been applied to the determination of the protonation constants of the 3-aminobenzoic acid, the benzidine and the maleinimidodioxime. The determination of the constants of the 3aminobenzoic acid and the benzidine were performed at 25°C with a PYE Unicam u.v.-visible SP S-100 spectrophotometer and a Radiometer PHM 84 with a combined glass electrode Radiometer GK 2401 C. The standardization was made with 0.05 M potassium hydrogen phtalate (pH 4.005) and 0.025 M KHzP04/0.025 M Na2HP04 (pH 6.858). The absorbance of five series of solutions 8.989 x lo-‘M 3-aminobenzoic acid and of five series of solutions 5.315 x lo-‘M benzidine was measured in l.OOO-cm cells at 280 and 3OOnm, respectively. The solutions were buffered with HSP04/KH2P04 between pH 2.5 and 3.5 and C&COOH/NaOH between pH 3.5 and 5.5 at 0.05 M ionic strength[lO]. All the reagents were R.A.
251
For the maleinimidodioxime the data obtained by ALBELDA[ 1I] have been used. The 3-aminobenzoic acid presents at the wavelength of 280nm the values of the molar absorptivity coefficients of the nonprotonated and diprotonated species l0 = 795 and 4 = 692; at the wavelength of 300 nm the benzidine has the values l0 = 16 520 and e2 = 0; and at 320 nm (wavelength corresponding to the isosbestic point) e. = e2 = 10867 for the maleinimidodioxime. In Table 3 the results obtained from the different ways of fitting by means of equations (II) and (III) are shown. In the case of the 3-aminobenzoic acid, because cI is smaller than e. and 4, the curve is U-shaped, therefore the regressions of X on Y and Z, and of Z on X and Y by means of equation (II) give the best results. In Table 4 the mean values of the constants are given; the values of the protonation constants have been corrected to ionic strength null by using the Debye-Htickel equation, to be compared with other values from the literature. The application of the method to the data cited by ALBERT and SERJEANT[6] leads to the values log K, = 4.788, log K2 = 3.086 and E, = 278. For the benzidine, the value of eI is between the values of lo and l2; being the experimental curve ascending-shaped and because l2 = 0 the best fitting corresponds to the regression of Y on X and Z by means of equation (III). The results corrected to ionic strength null are given in Table 4, together with those obtained by other authors. The protonation constants and molar absorptivity coefficient of the monoprotonated species obtained making use of the data cited by ALBERT and SERJEANT[~] are log K, = 4.681, log K2 = 3.438 and l1 = 9981. In this case, the values from the different fittings are more dissimilar (Table 3) because of the only inflection point of the curve. In Table 5 the results from 15 simulated determinations are indicated,
Table 3
C. MONGAYet al.
252
Table 4 r
T
,-anLnob~neo*0 aci.d l 0
benzidins
L0.sKZ
4,x 4.198
3,015
283
+o
25'
4,155 = 0,009
3,056 = c,cm
284 = 6
+o
25'
4,91
3,15
POQ
4,648
3,426
25'
4,815 = 0,022
3,406 * 0,008
0
3.08
(12)
25' 25'
l
= $03
&I.zw”.a~~
I
+o
+o maleintiid* dlorima
I.08K1
= 0,04
(6) mir
XOPk
03) 9.733 11.4x
(6) = 117
TbiS rork
0,:o I
25'
11,*4
10,10
(11) =
0,50 n
25'
II,14
10,20
(11F
$50
25Q
11,165 + 0,045
10,w
I
+ 0,015
17.37, = 105
XJis wa*
*Method proposed by ALBEDA[ll]. **Method of ANG. Table 5. Benzidine, log K, = 4.680; log KZ = 3.550; l0 = 16 440; l, = 9500;
Fitting
PlEZW3
x on Y adz II *
III
Log Kl 4136
El 5.127
Y on x adz
4,74
3157
9.960
4.76
3,58
10.125
x OIIY and z
4,697 * 0,027
3,561 = 0,014
9.688 C-245
Y onx anaz
4,685 * 0,025
3,553 * 0,014
9.549 = 234
z onx andY
4,669 * 0,024
3,542 = 0,014
9.351= 232
lead to positive values of the constants.
deviations the same as in Table 2, and the values of the protonation constants and molar absorptivity coefficients very close to those of the benzidine. It may be observed that the fitting by means of the regression of Y on X and 2 by equation (III) is preferable. Finally, because of the inverted U-shaped experimental curve of the maleinimidodioxime, the regressions of X on Y and Z, and of Z on X and Y by means of equation (III) give the best fitting. The mean values obtained are shown in Table 4, together with others from the literature. being the standard
REFERENCES IRVING,
3123
z onx andY
*Only 14 determinations
[l] H.
LwK2
J. A. D. EWART and J. T. WILSON, .I.
Chem. Sot. 2672 (1949). [2] R. NKS~NEN, P. LUMME and A. L. MUKULA, Acta
Chem. Stand. 5, 1199 (1951).
[31 A. K. LUNN and R. A. MORTON, Analyst 77, 718 (1952). [41 C. V. BANKS and A. B. CARLSON, Anal. Chim. Acta 7, 291 (1952). 151 E. W. WENTWORTHand W. HIRSCH. J. Phvs. Chem. 71, 218 (1%7). [61 A. ALBERT and E. P. SERJEANT,The Determination of Ionization Constants. Chavman. London (1971). w. A. E. MCBRYDE, Talanta il, 979 (1974). . ’ t;; R. HANUS, Anal. Chim. Acta 90, 167 (1977). r91 T. H. NAYLOR, J. L. BALINTFY, D. S. BURDICKand K. CHU, Computer Simulation Techniques. John Wiley, New York (1%6). rw V. CERDA and C. MONGAY, An. Quin. 76B, 154 (1980). u11 M. L. ALBELDA, Licentiature Thesis, University of .-
Barcelona (1980).
WI R. M. SMITH and A. E. MARTELL, Critical Stability Constants. Plenum Press, New York (1976). r131 J. LURIE, Handbook of Analytical Chemistry. Mir, Moscow (1975).
0584-8539/82/020247-06503.00/0 @ 1982 Pergamon Press Ltd
38A. No. 2, pp. 247-252, 1982
Printed in Great Britain.
A critical study of the linear least-squares method applied to the spectrophotometric determination of protonation constants of diprotic acids
Department
C. MONGAY, G. RAMIS and M. C. GARCIA of Analytical Chemistry, Faculty of Chemistry, University of Valencia, Valencia, Spain (Received 25 July 1981)
Abstract-The method of multiple linear regression applied to the spectrophotometric determination of the protonation constants and the molar absortivity coefficient of the monoprotonated species of diprotic acids, is studied critically. The best way of fitting the data, according to the shape of the absorbance vs pa-curve, is established. The conclusions are applied to real systems.
INTRODUCTION Spectrophotometry is, after potentiometry, the most widely-used technique in the determination of protonation constants of acid-base systems. Although spectrophotometry is more time-consuming and is considered as less precise, it is an ideal method for the study of sparingly soluble substances or those substances, which constants are particularly low or high. Many diverse methods of calculation have been proposed for the spectrophotometric determination of the two protonation constants of a diprotic acid, when they lie far enough, since in this case the same treatment used for the monoprotic acids may be applied [14]. However, when the two protonation constants are too close, the problem becomes more complex because not only the two protonation constants are unknown, but also the molar absortivity coefficient of the monoprotonated species, which cannot be directly determined experimentally. The methods proposed in this case[S-81 use to be very laborious, often restrictive and they are usually based upon iterative procedures, including occasionally weighting factors. These methods require the use of high speed computers and very sophisticated programs. In this work, the application of the linear leastsquares method in the determination of the protonation constants of diprotic acids with overlapping equilibria, is studied critically. This method has the advantage over other methods of requiring only a simple program, of being very rapid, noniterative and of being overcome by using programmable pocket calculators. In order to be able to discuss not only the precision, but also the accuracy of the method, hypothetical acids of prefixed protonation constants and molar absorptivity coefficients are assumed. Since the different ways of fitting the data are not equivalent, the conditions of applicability of each one are studied.
Finally, the conclusions are applied to the determination of the protonation constants and molar absorptivity coefficient of the monoprotonated species of three real systems, and the results are compared with others from the literature.
SIMULATION OF THE EXPERIMENTAL CURVES The absorbance of a solution of a diprotic acidbase system measured in a cell of l.OOOcm of optical pathlength, at a determined wavelength, is given by
eo+ e&h + e2P2h2 A=
c
l+P,h+&h*
where h is [H’], C the analytical concentration of the acid-base system and eO, c, and l2 the molar absorptivity coefficients of the nonprotonated, monoprotonated and diprotonated species, respectively. The cumulative protonation constants are PI and p2. In order to generate the ‘experimental’ curves, a diprotic system with determined values of its protonation constants and molar absorptivity coefficient of its different forms is assumed. About 40 points of the theoretical curve A, = f(pH,) given by equation (1) are calculated for a certain value of C, and from this ‘exact’ curve, 15 simulated ‘experimental’ curves are obtained, transforming the different theoretical points (A,, pH,) into experimental points (A,, pH,), assigning to the former gaussian errors by means of the expressions A. = A, +
uAz
(2) PI-L = pf-5 +
CpHZ
being (T,, and gpR the standard deviations associated to the absorbance and pH, which values are 241
248
C. MONGAY et
also assumed, and z an aleatory number, which follows the normal distribution law. The different values of z have been generated making use of the central limit theorem, which postulates that the distribution of probability of the sum of K independent and identically probable values of a certain aleatory variable approximates to a normal distribution as K increases [9]. Therefore a computer has been used, which provides uniformly distributed aleatory numbers, ri, between null and the unity, and the equation
al.
Y and 2 to slight experimental errors in the absorbance in the rejected zones. For this reason, the number of useful points, N, is reduced to about 30 in each determination. Each one of the four equations in Table 1 is fitted by the least-squares procedure by solving the system of normal equations CZ=a,N+a,
xX+a2x
Y
C,Xz=ao~x+~*~XZ+a2~xY
has been applied. In the practice of simulation, the value K = 12 is recommended, since a certain computational advantage is achieved because the preceding equation is reduced to z= 2, (ri-6) and despite this value of K restricts the distribution to the limits +6, it leads to total valid values of z up to three times the standard deviation. TREATMENT OF THE DATA
Expression a plane
(1) may be written as the equation of
2
YZ=aox
Y+a,CXY+a2z
(4) Y*
leading to four regression planes of Z on X and Y. For each plane, the parameters ao, a, and a2 are obtained and therefrom the values of K,, K2 and E, are calculated. Similarly, the values of the different parameters are obtained from the regression planes of Y on X and Z, and of X on Y and Z. The mean values and standard deviations obtained from the 15 experimental determinations are shown in Table 2, where the theoretical curves corresponding to each determination are also plotted. The standard deviation of the protonation constants are given with two significant figures. DISCUSSION AND CONCLUSIONS
Z=a~+a*X+a*Y
(3)
where the parameters ao, a, and a2 are related to p,, p2 and E,, and X, Y and Z are functions of the values lo and e2 and of the variables h and E (being E = A/C). Equation (3) may lead to the four expressions shown in Table 1. From the 40 experimental points (A,, pH,) obtained for each curve, only those which fulfil the condition IA, - E&/ > 0.05 IA, - l 2Cl > 0.05 are used, due to the sensitivity
of the values of X,
Table 1
2
E* -E
E
hE
E
h-‘(E-E,)
h-2
_h-lE
E-E,
Eo-E E
-E,
The different ways of fitting the data should lead to the same set of constants if the experimental data would not be affected with error. The fact of having different values with each one of the possible fittings is due to the introduced errors, because in the resolution of the normal equations there are, in some cases, substraction operations between very close quantities, leading to a ‘loss of accuracy’ since the relative error of the difference is much larger than that one of the minuend or the subtrahend. For this reason, equations (I) and (IV) in Table 1 lead to much worse results than equations (II) and (III), giving rise occasionally to interchanged values of the constants (log K, < log K2) and inclusive to negative values, therefore the results obtained by means of these equations are not indicated in this paper. Another fact emphasized, and which as already known, constitutes a limitation of the spectrophotometric method, is that in order to get a better accuracy and precision, more pronounced spectral changes in passing from one acid-base species to another are required. If the values of the molar absorptivity coefficients of both species in an acidbase pair are similar, the method cannot be applied, the results being worse as these values are closer. It must be remarked that in the resolution of the normal equations for the regression of Y on X
A critical study of the linear least-squares
,,aY#.l * “,cul5 5*9”89 t “.W2 J,,V t 69
I&m8 = 0.01” 5,993” a “,“a1 ,111 = 108
i,l)y’)* 1 “&“I69 6.999, = “,m6, 6,0o,q t “,006’1 6,m01 A “,m?l 486 s 8” 995 * m
1,co,6 t. “,W,4 5.9976 I “,a,66 ~06 1 62
I,mo5 t “,Mg5 6.M,, f “.w,”
6.~9 f 0,010 6,00,8 = “,“094 $3, t ,a
6.9901 f “,uoB, 6.~1 * 0,010 ‘903f 115
1,m24 * “,W94 5.9964 A 0.~066 304” * 9,
?,mO * 0.01, 6,w** t “.mB, 4980 f 116
+ 0,011 6,m em2 d 13,
l&n4 f 0,012 5,999, t “,WP ,~*6 * 12,
‘,,<“,I f 0,011 ,,YYI’, f 0.~36 5,‘) f 128
1,M23 f o/m?? 5,996.l f “,W8 ,041 f 10,
?.“a, f “.co94 5rY918 = “,W?, ,101 f 96
7.““4 = “,“U 5.9961 * “P9” ,060 * 12,
7.a f “,“,, 5,995 f 0,011 Boo6 1 13,
7.0% : O,“,, 5.991, f “.mJ3n ,“,“9 f 114
6,999 t 0,010 ‘,M,, f o,‘w~ NIL! = 12,
6,99tyI * “,MM4 6,ooO f 0.0,” YD4 f 115
‘,,M,?
t “.W4
5,~;
: ;;“”
6,999 * 0,011 6~314 + “.@W ,980 * 116
?,“a3 6.“~ m,
c, = 12.5m
f
,I
III
-
249
method
4995 = 93 = 0,012
1,Ml
= 0,012 f 0.01, a UP
1~326t 121
7
”
apll
<, = ,“.uo
6,.
0
c, -
2.m
t,
,.m
-
= lS.oa
f, .
*o.Mo
6. t,
11
= l”.mxJ
‘,,wJ~ f “,m6* ti.999” t “,w%” 5 - 6Y
‘,.W” : 0,012 5,999, a “,m95 ,991 = 99
7,031 A 0,015 5,998 f “.“u !m01ta
6,993 * “.“2” 6,005 f “.“*, ,252, * 5,
t 0,018 ?,rn, t 0,017 6,o~ 14999= 106
7,cm f 0,012 5,998 t 0,016 ,7999 t 111
1.12 5,w -,‘I37
l,“,? f 0,014 5,913, = 0.012 2,‘)o f 121
1,013 f 0,015 5,981 f 0.012 508s f16
1.01, 10,021 5.906 i 0,022 12455 f 56
* 0.018 7.m 5,995 A “,“,‘I 14955* 10,
7.““5 * “,“,, 5.994 * 0,016 199e2f 1.51
6,996
(1, .)
111
7.03,
0.012
‘,,m*
2 ;;““’
6~” ‘Jrn? * “,“,4 8,
6,009 ,252, f 0,020 54
106 6,~ 14999 I= 0,017
6.~0 *oxa .= 0,016 171
.,,a~, + “,ar,” ,.999” f o,Mqq
‘,,“a ‘,wJ,
* 0,016 * 0.012
190, f UY
1,002 * o,“,‘, 6,wx f 0,015 W2f9”
6.996 f 0,020 6.~8 * 0,020 ,*v* f ,,
= 0.011 1,“w 6,mo t 0.016 ,4997 f 91
1.m* z “&WY 6&%X = o,“,, *cc03 f ,q,
‘,,co~” * “,wm ,,992* t o&n@ 6” f 96
?.“a? * 0,016 5,993 = 0,012 *“uz * ,,9
7,W * 0,016 5,991 f 0.04 )064tn5
1,016 * 0.02, >,964 2 “,W2 12448 t 56
?,a39 * 0.011 5.99, = 0.016 1494” t 95
?.rn, = 0.01, 5,994 a 0,014 ,992, = 1%
?,<*ur2 f “,“078 ),9’)9” t 0,~~) , f 9”
6,999 f 0,016 6,,XK, f “.“,2 WY, = ,111
~,WJ t o,“,., 6,~” t 0,015 yn* * 9”
6,994 * 0,020 6,W + 0.02” ,*,,9 f 54
1,cO* f 0,011 6.W I”,“,6 14991 2 I)?
6.9995 f “PW, 5,993 = 0,014 *a*, f 14,
z = 9”
= 0,020
?,mo
r 0,012
6,~;
* “,MJ6,
f “,a,5
* 0,018
?,rn,
b.99,:; t L$XM”
?,a*:,
l
f, =,o.aa 62.
III
“2
0
f*
- 1.w”
2.500
62 - 5.cm
(2 = 12.500
r* - ,5.(x0 6,996 t 0,015 6,002 t 0,014 9916 t 16
6,911 f 0,034 6,021 t “,“,6 ,“4”” t 606
6.9966 1 “,uOVV 6,998 * 0,021 6,m,“* “.CW,? 5,991 f 0,020 ,cxwY* 216 ,M4? * 117
6,991 4 0.01, 6,“q * 0,012 1~52 * 61
,,‘W i 0,021 5.996 * 0,023 ,“mD * 57
6,987 t “.“34 6.012 t 0,014 lu224 * 516
1,m*3 t “,m34 5.9903 t “,WfM 9962 * 10,
?,a35 f 0,021 5.992 : 0.02, 9933 t 215
7,mo * 0,012 6,m0 k 0,011 9996 * 6”
‘,,“26 5,vll ,Wl,
6,979 i 0,034 6,“2P * “,“,5
6,99’,‘, A 0.~96 6&“19 f “,w’,,
10104 Et 60.3
100.~ t 116
1,ooO * 0,021 5,999 f 0,020 ,ooO’, * 216
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6,994 * 0,011 6,m61* 0.0394 ,m4, 1 5,
l.c.38 * 0.02, 5,998 + “,“*3 ,““,” * 58
*,m* * 0.01, 5,995 = 0.012 9936 : 112
.I.“066 t “,“co9 5,99,8 f o&06, 9902 f 32
?,m6 k 0,014 5,992 I”,“,4 9927 f 129
,,““* * 0,011 589;;: “&a
1,024 f 0,022 5,919 t “so24 ,“a5 t 59
?,“a = 0,015 5,995 f “8014 1~26 = 19
1,996 t “o”,” L,W, i “,“,2 1~29 = ~6
?,m,5 5,;;;;
6,998 f 0,015 5,998 * “,“,4 ,mc6 t 129
6.99, l 0,011 6~354 = “,W* lM4, = 5,
7,““~ t 0,021 5,996 f 0,021 ,omg*50
6,996 f 0,016 6,022 * 0.015 991, = a5
5 = 2.w
Ll -
‘,
c, - *0.co”
6,998 2 0.0,” 6.C.729tf “JxW 912 83
6.995 t 0,021 6&w” + * “,“I91 2456 15
?,rn? f “,M, 5,911 * f 0,061 ,209 174
6.992 = 0,018 6,011 * = O&W, 215 ,5,“,
1&o* t 0,015 5,997 t* 0,020 ,991, 296
1.06 i “,40 6,“Z f “,28 -1642 t ,Joo
1,“,4 t 0,016 5,901” 2 “,“V4 ,165 f 7,
1,029 i 0.02, 5.992, i “,‘WM 2519 i ,6
?,“,, t “,“2, 5,961 t “,“1, 12195 l ,94
6.996 * 0,018 6,~) t 0.028 ,5034 t 211
7,036 f 0,016 5,992 * 0.02, ,987l f xv
1.002 t “,“,I ,,Y,,,; : ;;W*
7,w 6,mU ,m,
‘,,alt) i 0,024 5.9992 2512 * I 0.~~8, 7”
6,981 f 0,019 6,005 12‘94 t* “,ooO 234
6,982 * 0,018 220 6.04, f= 0,028 15295
6,991 * “,“,5 6,011 t* “,oz, m,,, ,,”
1,cn> i “,“,tl 5,996” A “,wF~ 34 f 1,”
7,cw i 0.021 6,~ = 0,015 963 i ,4,
?,uol i “.“I2 6.~3 t 0,012 2w3’99
6,996 a O,“,, 6,012 k “,“,6 12>6’, t 10,
6.99, * 0.012 6,018 f 0.011 15,217. 12,
1.m* 6,“~ 2~2,
‘,.“,2
7.0,”
7,“,9 i “.“29 5,996 I”.“,, *WI f 85
7.0,” I O,“,” 5,968 L 0,020 12401 A 1,”
7,“ca t “,“,2 5.991 f “,“,? 14981t 122
1,005 t “.“,” 5199, * “,“,4 ,989, * 117
6,9>2
1,“% I”.“,6 5.902 * 0.042 ,245” * ,?”
6.995 * “0012 ~/XI’, f 0,011 15049 * 12,
1,CO,* * O&O96 5,991 = “,“,4 ,9w9 f 168
L,
s
10
-
6,998 *“,“,5 6,m, 10,014 9916 f 18
‘,,a)** f “,mti 5,999, f 0.005” 991, f 19
t “,wB : ;W*
6.u)
* 0,021 * 0,024 * 57
5,997 f 0,010 i.w* f. 0,012 ,w,” 1 116
6. =
I
a
7
=
0
f,
-
l.wm
12.m
- ,5.c.xl
t.w
8aae. I,uo, 5.9,“;
II
f 0,015 : ‘J;W4
t “.“,I f 0.~92 15
-
Ill
li a,“,?
1.99~ i “~~192 6.~0 0, I 10, ,“,, 6,?Y,
apH
5-w
f, = ,o.m
-
L* -
t “,“2”
6.Wll 1 1”,“,” -22 i ,!!J
L 0.02, 1 “,“,I I 12”
6,911 t 0.034 6.1~” f I).“,? 8Y 1 192
t 0,051
6.01, = 0,011 *WI t 11.4
* 0.0096 * “~11, * 161
,
C. MONGAYetal.
250
*Among the 15 determinations,
only those indicated in parentheses lead to positive values of the constants.
and 2 by equation (II), the parameter a0 takes the value l/e,, and when E, approximates to zero, the error of this fitting becomes very large. This fact is clearly observed in the first acid-base system in series 2 and 4, where E, = 0, which give rise to unacceptable results; it may also be observed in these series how the results become better as the value of e, increases. Another fact which has a general character is that when the curve A = f(pH) is U-shaped, the fitting by means of equation (II) (except for the regression of Y on X and 2, specially as c, is smaller) leads to a little more precise and accurate results than the fitting by means of equation (III), being the results of the former clearly much better as the difference between the value of el and the values of l0 and l2 increases. On the other hand, when the curve has the shape of an inverted U, the fitting by means of equation (III) is preferable, leading the regression of Y on X and 2 to somewhat more unaccurate values than the regressions of X on Y and 2, and Z on X and Y. This can be observed in the series l-4 and it has also been corroborated for a series equivalent to series 1, where l, = l2 < E, and another equivalent to series 4, where E,,> l2. Finally, when the curve is ascending or
descending (e2 < E, < E,,or co < E, < e2), the method leads to results less precise because of the only inflection point, while when the curve has a maximum or a minimum, it presents two inflection points. In these cases, the best results may correspond to either of equations (II) and (III), in accordance with the gradual transformation from a U-shaped to an inverted U-shaped curve, as observed in series 4 or in its equivalent where co > l2. From the three different fitting possibilities for each plane, the regression of Y on X and 2 gives the best values (though other regressions may lead to total acceptable values), except when either of co or e2 trends to zero, where only the treatment based upon equation (III) leads to valid results. This can be observed in series 5 and 6 and it has been also observed with a series of curves corresponding to values of the molar absorptivity constants, where l2 - E, = l1- co. Although the results in Table 2 correspond to a diprotic acid-base system with protonation constants log K, = 7.000 and log K2 = 6.000, the conclusions have been checked for systems with protonation constants log K, = 3.000, log Kz = 2.000 and log K, = 11BOO, log K2 = 10.000, leading to similar results, not depending therefore on the conclusions for the value of these constants.
A critical study of the linear least-squares method Acids, which logarithms of their constants differ in 0.25 and 3 unities have been also studied, leading to the same conclusions, though the precision of the results increases as the difference between the values of the protonation constants of the acids increases. Finally, the data in Table 2 have been obtained assuming an analytical concentration of lo-‘M and values of the standard deviation for the absorbance and pH of 0.003 and 0.002, respectively, values that are usual in these spectrophotometric determinations. Series for other values of the standard deviation, up to 0.01 in the absorbance and 0.005 in pH have been also studied, and though in these cases the results are less precise, the conclusions on the best way of fitting are the same. APPLICATIONOF THE METHOD The preceding conclusions have been applied to the determination of the protonation constants of the 3-aminobenzoic acid, the benzidine and the maleinimidodioxime. The determination of the constants of the 3aminobenzoic acid and the benzidine were performed at 25°C with a PYE Unicam u.v.-visible SP S-100 spectrophotometer and a Radiometer PHM 84 with a combined glass electrode Radiometer GK 2401 C. The standardization was made with 0.05 M potassium hydrogen phtalate (pH 4.005) and 0.025 M KHzP04/0.025 M Na2HP04 (pH 6.858). The absorbance of five series of solutions 8.989 x lo-‘M 3-aminobenzoic acid and of five series of solutions 5.315 x lo-‘M benzidine was measured in l.OOO-cm cells at 280 and 3OOnm, respectively. The solutions were buffered with HSP04/KH2P04 between pH 2.5 and 3.5 and C&COOH/NaOH between pH 3.5 and 5.5 at 0.05 M ionic strength[lO]. All the reagents were R.A.
251
For the maleinimidodioxime the data obtained by ALBELDA[ 1I] have been used. The 3-aminobenzoic acid presents at the wavelength of 280nm the values of the molar absorptivity coefficients of the nonprotonated and diprotonated species l0 = 795 and 4 = 692; at the wavelength of 300 nm the benzidine has the values l0 = 16 520 and e2 = 0; and at 320 nm (wavelength corresponding to the isosbestic point) e. = e2 = 10867 for the maleinimidodioxime. In Table 3 the results obtained from the different ways of fitting by means of equations (II) and (III) are shown. In the case of the 3-aminobenzoic acid, because cI is smaller than e. and 4, the curve is U-shaped, therefore the regressions of X on Y and Z, and of Z on X and Y by means of equation (II) give the best results. In Table 4 the mean values of the constants are given; the values of the protonation constants have been corrected to ionic strength null by using the Debye-Htickel equation, to be compared with other values from the literature. The application of the method to the data cited by ALBERT and SERJEANT[6] leads to the values log K, = 4.788, log K2 = 3.086 and E, = 278. For the benzidine, the value of eI is between the values of lo and l2; being the experimental curve ascending-shaped and because l2 = 0 the best fitting corresponds to the regression of Y on X and Z by means of equation (III). The results corrected to ionic strength null are given in Table 4, together with those obtained by other authors. The protonation constants and molar absorptivity coefficient of the monoprotonated species obtained making use of the data cited by ALBERT and SERJEANT[~] are log K, = 4.681, log K2 = 3.438 and l1 = 9981. In this case, the values from the different fittings are more dissimilar (Table 3) because of the only inflection point of the curve. In Table 5 the results from 15 simulated determinations are indicated,
Table 3
C. MONGAYet al.
252
Table 4 r
T
,-anLnob~neo*0 aci.d l 0
benzidins
L0.sKZ
4,x 4.198
3,015
283
+o
25'
4,155 = 0,009
3,056 = c,cm
284 = 6
+o
25'
4,91
3,15
POQ
4,648
3,426
25'
4,815 = 0,022
3,406 * 0,008
0
3.08
(12)
25' 25'
l
= $03
&I.zw”.a~~
I
+o
+o maleintiid* dlorima
I.08K1
= 0,04
(6) mir
XOPk
03) 9.733 11.4x
(6) = 117
TbiS rork
0,:o I
25'
11,*4
10,10
(11) =
0,50 n
25'
II,14
10,20
(11F
$50
25Q
11,165 + 0,045
10,w
I
+ 0,015
17.37, = 105
XJis wa*
*Method proposed by ALBEDA[ll]. **Method of ANG. Table 5. Benzidine, log K, = 4.680; log KZ = 3.550; l0 = 16 440; l, = 9500;
Fitting
PlEZW3
x on Y adz II *
III
Log Kl 4136
El 5.127
Y on x adz
4,74
3157
9.960
4.76
3,58
10.125
x OIIY and z
4,697 * 0,027
3,561 = 0,014
9.688 C-245
Y onx anaz
4,685 * 0,025
3,553 * 0,014
9.549 = 234
z onx andY
4,669 * 0,024
3,542 = 0,014
9.351= 232
lead to positive values of the constants.
deviations the same as in Table 2, and the values of the protonation constants and molar absorptivity coefficients very close to those of the benzidine. It may be observed that the fitting by means of the regression of Y on X and 2 by equation (III) is preferable. Finally, because of the inverted U-shaped experimental curve of the maleinimidodioxime, the regressions of X on Y and Z, and of Z on X and Y by means of equation (III) give the best fitting. The mean values obtained are shown in Table 4, together with others from the literature. being the standard
REFERENCES IRVING,
3123
z onx andY
*Only 14 determinations
[l] H.
LwK2
J. A. D. EWART and J. T. WILSON, .I.
Chem. Sot. 2672 (1949). [2] R. NKS~NEN, P. LUMME and A. L. MUKULA, Acta
Chem. Stand. 5, 1199 (1951).
[31 A. K. LUNN and R. A. MORTON, Analyst 77, 718 (1952). [41 C. V. BANKS and A. B. CARLSON, Anal. Chim. Acta 7, 291 (1952). 151 E. W. WENTWORTHand W. HIRSCH. J. Phvs. Chem. 71, 218 (1%7). [61 A. ALBERT and E. P. SERJEANT,The Determination of Ionization Constants. Chavman. London (1971). w. A. E. MCBRYDE, Talanta il, 979 (1974). . ’ t;; R. HANUS, Anal. Chim. Acta 90, 167 (1977). r91 T. H. NAYLOR, J. L. BALINTFY, D. S. BURDICKand K. CHU, Computer Simulation Techniques. John Wiley, New York (1%6). rw V. CERDA and C. MONGAY, An. Quin. 76B, 154 (1980). u11 M. L. ALBELDA, Licentiature Thesis, University of .-
Barcelona (1980).
WI R. M. SMITH and A. E. MARTELL, Critical Stability Constants. Plenum Press, New York (1976). r131 J. LURIE, Handbook of Analytical Chemistry. Mir, Moscow (1975).