- Email: [email protected]
A comparative study of the application of the method of least-squares in the potentiometric determination of protonation constants
0039.9140/82/050435~05103.00/0 Copyright 0 I982 Pergamon Prers Ltd
Talunru. Vol. 29. pp. 435 to 439. 1982 Printed in Great Britain. All rights reserved
ANNOTATION
A COMPARATIVE STUDY OF THE APPLICATION OF THE METHOD OF LEAST-SQUARES IN THE POTENTIOMETRIC DETERMINATION OF PROTONATION CONSTANTS M. CELIA GARCIA, G. RAMIS and C. MONGAY Department of Analytical Chemistry, Faculty of Chemistry, University of Valencia, Valencia. Spain (Receioed 13 May 1981. Accepted 5 October 1981) Summary-Methods of simple and multiple linear regression applied to the potentiometric determination of protonation constants of diprotic and triprotic acids are studied critically. The best way of fitting the data, according to the order of magnitude of the constants, is established. The conclusions are
checked by calculating the protonation constants of succinic and citric acids.
The potentiometric determination of the protonation constants of an acid-base system may utilize graphical or analytical methods. Graphical methods’-4 have been critic&d because of their lack of accuracy, but they may assist detection of incorrect points, which is usually not possible by using only analytical methods of calculation. Analytical methods, on the other hand, are quicker and more accurate and precise. They include the unweighted least-squares procedure, which requires only a simple calculation program, and has the advantage over other analytical methods that it can be done with programmable pocket calculators. In recent years high-speed computers have been widely used for calculation of constants.5-* The computer programs often include weighting factors, and are relatively sophisticated. If the successive protonation constants of an acid differ greatly in magnitude, they can be treated separately, but when they are close, simultaneous fitting of the data is necessary. This is the case discussed here. For diprotic acids, among the various methods described, the linear least-squares procedure has been widely used, because of its precision and ease and speed of application. For triprotic acids, the available analytical methods are much more laborious, so graphical curve-fitting methods are usually recommended.g Here, the leastsquares method has been considered as not entirely satisfactory by some authorslo In the present work the extension of the leastsquares method to triprotic acids is studied with the aid of an HP 67 calculator. The conditions of its applicability to diprotic and triprotic acids are examined in parallel, and examples are shown for acids for 435
which the successive logarithmic constants differ by unity. Systems with larger and smaller differences are also tested, with analogous results. Potentiometric determination of protonation constants involves two well-differentiated steps: first, measurement of the experimental points to get the set of values (Z, h) and secondly, an appropriate treatment of the data to obtain the constants. In this paper, hypothetical acids with known constants are assumed in order to be able to discuss not only the precision, but also the accuracy of the method. The theoretical titration curves of these acids are calculated, then “experimental” titration curves based on these are simulated by adding Gaussian errors to the volumes of reagent and the values of pH. Finally, the conclusions are applied to the determination of the protonation constants of two real systems, and the results are compared with others in the literature. THEORETICAL TITRATIONS
For an n-protic acid, H,A, titrated with a strong base, the titration curve is given by:
v=
v
(1)
where h is [H’]. V is the volume of acid of concentration CA, u is the volume of strong base of concentration C, and /It is the ith overall protonation con-. stant of the system, with /IO = 1 by definition.
436
ANNOTATION
Values for the protonation constants of a hypothetical acid are selected and 40 “theoretical” points along its exact titration curve u =f(pH) are calculated, giving values to the parameters V, C, and C,. From this “exact” curve, 15 simulated “experimental” titration curves are obtained as follows. Each “theoretical” point is transformed into an “experimental” one by addition of a Gaussian error generated with the aid of a table of Tippett numbers,” which gives random deviates in units of standard deviation. To make use of this table, standard deviations of 0.01 ml in the volume of titrant and of 0.001 in pH are assumed. These values are about the lowest expected for the techniques most commonly used in the determination of stability constants. They illustrate the gradual differences, not only in precision but also in accuracy, found for the different methods of linear fitting discussed here. For the diprotic acids the values V = 40 ml, CA = 5 x lo-‘M and C, = 4 x 10m2M are assumed, except for the titration of the very weak acid with log Ki = 12 and log K2 = 11, for which C, was taken as O.lM. For the triprotic acids the values 1/ = 30 ml, CA = 5 x 10e3M and C, = 5 x 10e2M were used, except for the titration of the acid with log Ki = 12.5, log K2 = 11.5 and log K3 = 10.5 for which C, was 0.15M. TREATMENT
OF THE DATA
(a) Diprotic acids The expression
Ah + 2hh2
ii = 1 +
(2)
Blh + /32h2
may be written in the form: Y= f&-J+ a,X
(3)
where X and Y are functions of n and h, and the parameters a0 and a, are directly related to /LSiand /12. In this way, equation (3) may lead to the three forms shown in Table 1.’ For each of the 40 experimental points (u, pH) a pair of values @, h) is obtained. Only points with ii values given byO.l < ti 6 0.9 and 1.1 < n < 1.9 are used because of the sensitivity of the values of X and
Table 1. Y
X
I II III
.+,$,=I 1
h-(rs- 2) (1 -ii)
1
‘x+!%y=1 B2
82
xY=
a0
N
CXY = a&X
a,xX
+
+ a,CX2
(4)
leading to three regression lines of Y on X, from which the constants K, and K2 are estimated. In the same way, three regression lines of X on Y are fitted, leading to other values for the constants. This procedure is followed in the 15 titrations for each acid. The mean values and standard deviations of log K, and log K2 are given in Table 2. Standard deviations (s) are given with two significant figures. (b) Triprotic acids
In the same way the expression _
Ah + 2/%h2 + 3f13h3
n = 1 + j&h + j2h2 + j&h3
(5)
may be written as the equation of a plane 2 = a, + a, x + a2 Y
(6)
where X, Y and Z are functions of ii and h, and the parameters a,, a, and a, are functions of the constants /Ii, /I2 and /I3 as shown in Table 3. For the same reason as above, only the values 0.1 < ii < 0.9, 1.1 < ii < 1.9 and 2.1 < ii < 2.9 are used, reducing the number of useful data, N, to about 33 in each titration. Each of the four planes in Table 3 is fitted by the least-squares treatment by solving the system of equations &Z = aoN + a,CX
+ a2CY
CXZ = aoCX + a,CX2
+ a2CXY
CYZ = aoCY + a,CXY + a2CY2
(7)
to obtain four regression planes of Z on X and Y From each plane the values of the constants K1, K2 and K3 are calculated. Similarly the regression planes of Yon X and Z, and of X on Y and Z are obtained. The mean values and standard deviations obtained from the 15 simulated titrations are shown in Table 4.
DISCUSSION
Equation
Straight line
Y to slight experimental errors in n in the other ranges. For this reason, the number of useful points, N, is reduced to about 36 in each titration. Each of the three equations in Table 1 is fitted by the least-squares method, by using the usual equations:
ii-l
(2 - fi)h
-
-
(1 -nn)h
~
(2 $h2
As stated previously, the values for the standard deviations of the pH and volume are at the lower limit of those normally observed. This allows better observation of the gradual differences in precision and accuracy. In practice, the deviations are often a little bigger, leading to more dispersed values than the ones shown in Tables 2 and 4. In these tables, it can be seen that for equal deviations of pH and volume, slightly better precision is
437
ANNOTATlON
Table 2. Theoretical value
Fitting
Straight line I log K s
Straight line II. s log K
Straight line III log K s
Yon X
1.9950 1.055
0.0050 0.046
1.9956 1.048
0.0047 0.044
2.0023 0.94
0.0072 0.14
X on Y
2.0003 0.988
0.0046 0.049
2.0002 0.992
0.0046 0.050
2.0016 0.96
0.0070 0.12
Y on X
4.4985 3.5011
0.0038 0.0030
4.4999 3.4997
0.0015 0.0022
4.5004 3.497
0.0052 0.019
X on Y
4.4987 3.5010
0.0037 0.0030
4.4998 3.4996
0.0015 0.0022
4.5003 3.498
0.0053 0.019
log K, = 7.000
Y on X
7.aJO 6.0006
0.015 0.0054
7.001 6.0003
0.0016 0.0018
7.0011 5.996
0.0045 0.011
log K2 = 6.000
X on Y
7.001 6.0005
0.015 0.0054
7.0001 6.0003
0.0016 0.0018
7.0010 5.997
0.0045 0.011
log K, = 10.000
Y on X
9.998 9.0013
0.012 0.0040
9.9998 9.0002
0.0015 0.0019
X on Y
9.996 9.0013
0.019 0.040
9.9998 9.0002
0.0015 0.0019
9.9999 8.9999
0.0027 0.0063
11.991 11.004
0.052 0.014
12.0005 11.0004
0.0048 0.0032
12.0015 10.9998
0.0061 0.0042
log K2 = 11.000 X on Y 11.998 11.003
0.054 0.014
12.0002 11.0003
0.0048 0.0032
12.ooO8 1l.WO8
0.0062 0.0041
log K,
= 2.000
log K2 = 1.000
log K, = 4.500 log K2 = 3.500
log K2 = 9.000
log K, = 12.ooO
YonX
obtained in the determination of the constants of diprotic acids than for triprotic acids. It is also clear that the four methods of fitting for triprotic acids lead to much larger differences in precision and accuracy than do the three methods for diprotic acids. The different methods of fitting would lead to the same set of constants if the experimental data were not subject to errors. From Table 2 it can be seen that, for diprotic acids with pK-values of about 3.5-l 1.5, the values of X and Yare interchangeable, leading to negligible differences in the values of the constants and standard deviations. For strong acids, the regression of Y on X appears to lead to results with similar standard deviations but less accurate mean values. This is also true for triprotic acids, where the greater differences show that the
10.OOOo 0.0027 8.9997 0.0063
regression of Z on X and Y is clearly the least accurate. Generally, straight line II was found to give better precision and accuracy than lines I and III. Some authors have distrusted this equation,” but others have accepted it.” For strong acids, the results from straight line I are nearer to those from straight line II than those of straight line III are. For these acids, evaluation of Kr is more accurate and precise than that of K2 with both lines I and II. For strong bases, the results from straight line III are close to those of the straight line II, but straight line I gives poor agreement. For the triprotic acids (Table 4) results from plane I are not very useful, and they become worse for weaker acids, eventually resulting in interchanged
Table 3. Plane
IV
Equation
Z
Y
X
438
ANNOTATION
Table 4. Plane I log K s
Plane II log K s
Plane III log K
Z on X and Y
3.0058 1.966 1.190
0.0041 0.018 0.086
3.0079 0.0037 1.969 0.015 1.208 0.080
3.01 1.37 1.69
Yon X and Z
3.0006 I.996 I .030
0.0034 0.013 0.080
3.0009 0.0031 1.998 0.012 0.996 0.087
3.0031 1.97 I.14
0.0089 0.11 0.48
3.01 1.65 2.48
(4*)
3.0017 1.991
3.0015 0.0031 1.995 0.012 1.019 0.086
3.0040 1.968 1.16
0.008 1 0.098 0.43
3.01 1.20 2.70
(5:)
1.059
0.0036 0.014 0.083
Z on X and Y
5.01 4.001 3.0009
0.10 0.036 0.0083
4.9995 0.0028 4.0001 0.0031 3.0010 0.0037
4.9995 4.006 3.022
0.0062 0.013 0.022
YonXandZ
4.997 4.004 3.0005
0.098 0.036 0.0083
4.9994 0.0028 4.0003 0.0030 3.0009 0.0037
4.9999 3.999 3.003
0.0062 0.013 0.010
5.00 3.89 3.76
Xon
5.10 3.987 3.0029
0.15 0.038 0.0087
4.9992 0.0029 3.9999 0.003 1 3.0010 0.0037
4.9996 4.000 3.0013
0.0062 0.013 0.0098
5.00 3.91 3.63
Z on X and Y
7.09 7.35 5.9872
0.70 0.33 0.0065
7.99 17 0.0082 7.016 0.015 5.9939 0.0079
7.9963 7.018 6.011
0.008 1 0.023 0.015
Yon X and Z
7.24 7.32 5.99
l6*)
7.9909 0.0082 7.017 0.015 5.9939 0.0071
7.9969 7.007 5.995
0.0080 0.019 0.012
8.00 6.85 6.73
(3*)
7.978 0.012 7.010 0.023 5.9976 0.0081
7.9966 7.008 5.994
0.0081 0.020 0.013
8.00 6.87 6.74
(4’)
Theoretical values
Fitting
log K, = 3.000 log K2 = 2.CQO log K3 = 1.000 Xon
log
YandZ
K, = 5.000
log K2 = 4.000 log Kj = 3.000
YandZ
log K, = 8.000 log K2 = 7.000 log K1 = 6.000 Xon
YandZ
t
(7*)
t
9.36 9.97 8.49
(5*)
10.504 0.011 9.493 0.017 8.5032 0.0090
10.4997 9.5008 8.5013
0.0041 0.005 1 0.0049
10.50 9.53 7.86
(8*)
Yon X and Z
9.28 IO.01 8.49
(5*)
10.504 0.011 9.494 0.017 8.5030 0.0089
10.4997 9.5004 8.5007
0.0041 0.0052 0.0048
10.496 9.497 8.35
0.021 0.083 0.55
t
10.4954 0.0089 9.487 0.021 8.5033 0.0091
10.4996 9.5006 8.5005
0.0041 0.0051 0.0048
10.5004 0.0094 9.494 0.076 (14*) 8.39 0.45
12.464 11.541 10.481
0.036 0.045 0.020
12.497 11.5006 10.5011
0.020 0.0055 0.0055
12.477 0.049 11.58 0.13 (12*) 10.11 0.3i
12.457 11.549 10.480
0.042 0.052 0.021
12.498 11.4997 10.4990
0.020 0.0055 0.0056
12.498 11.36 10.51
0.053 0.50 0.23
12.383 11.521 10.481
0.096 0.026 0.020
12.495 11.5019 LO.4974
0.019 0.0053 0.0057
12.488 11.520 10.39
0.052 0.086 0.27
log K3 = 8.500 Xon
YandZ
Z on X and Y log K, = 12.500 log K2 = II.500
t
(22)
Z on X and Y log K, = 10.500 log KL = 9.500
Plane IV log K s
s
Yon X and Z
t
log KJ = 10.500 Xon
YandZ
* Among the I5 titrations. only the number indicated in parenthesis lead to positive values of the constants. t This fitting gives rise to negative values of the constants in all the I5 titrations.
Table 5. Temp.. ‘C 25 20 30 20
Ionic strength O.lS(NaCIO,) O.lO(NaCIO,) 0.1 (KCI) 0.10(NaC104)
log K, 4.88 5.28 5.116 5.205 (s = 0.015
log
K2
4.05 4.00 3.952 3.995 (s = 0.015)
Reference 14 15 16 This work
439
ANNOTATION
Table 6. Temp., “C Ionic strength 25 20 20 25 30 25 20
0.15 0.1 (NaClOJ 0.1 (NaClOJ 0.1 (KN03) 0.1 (KCI) 0.1 (KN09) 0.10 (NaClOJ
log
Kl
loi3
K2
4.34 5.62 4.35 5.68 5.68 (s = 0.02) 4.38 (s = 0.02) 4.30 5.65 4.395 5.750 4.39 5.72 5.660(s = 0.015) 4.335(s = 0.015)
values of the constants (log K1 log K2 etc.). Plane IV also does not give valid results for all acids. KI is always obtained with a slightly better precision than K2 and K1, which may become interchanged (log K2 < log K3). Plane II leads to good results for all acids, but for the strongest acids, K3 is less precise, and K1 is the best constant. For medium-strength acids, this plane leads to acceptable values of all three constants. As the acid strength decreases, KS is determined with better precision (but plane III gives better precision for weak acids). Plane III gives good results for weak and mediumstrength acids (comparable to those of plane II for the latter). For strong acids, the results are not very good. Results were also obtained for acids with greater and smaller differences between their protonation constants, and similar conclusions were reached.
CONCLUSIONS
For diprotic acids, least-squares fitting by means of straight line II is to be preferred. For strong acids, it is better to use the regression of X on Y For other acids, both regressions should be performed, and the results with smaller dispersion, or the mean value, if both regressions have a similar precision, accepted. For triprotic acids, the data should be fitted by means of plane II or III for strong and weak acids, respectively. For strong acids, the regression of 2 on X and Y should not be used. For other acids, the three possible regressions should be calculated, and the one with the smallest dispersion, or the mean value of those with the smallest dispersion, accepted.
Titrations were done at 20” with the aid of a Radiometer PHM 84, (resolution 0.001 pH unit) and a combination glass electrode, Radiometer GK2401 C, standardized against 0.05M potassium hydrogen phthalate (PH 4.001) and 0.025M potassium dihydrogen phosphate/O.O25M disodium hydrogen phosphate (pH 6.874). The ionic
K3
Reference
17 2.94 18 2.87 19 2.96 (s = 0.03) 20 2.79 16 3.023 8 2.92 2.97 (S = 0.03) This work
strength was adjusted to 0.10M by addition df an appropriate amount of sodium perchlorate. All the reagents were
of analytical grade. (a) Succinic acid. Five titrations were done of 50 ml of 1.656 x 10b3M disodium succinate with 0.0212M hydrochloric acid. The mean values of the constants and standard deviations obtained by means of straight line II are shown in Table 5, where they are also compared with some literature constants obtained in similar conditions. Both regressions (X on Y and Y on X) were used because their standard deviations were of the same order of magnitude. (b) Citric acid. Five titrations of 50 ml of 1.025 x lo-“M trisodium citrate were done with 0.019OM hydrochloric acid. The mean values of the constants and standard deviations calculated by means of plane II are shown in Table 6, where some values from the literature are also given. The three fitting possibilities were used. REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
EXPERIMENTAL
h3
17. 18. 19. 20.
J. Speakman, J. Chem. Sot., 1940, 855. L. G. Sill&, Acta Chem. Stand., 1954. 8. 318. Idem. ibid., 1956. 10, 186. F. J. C. Rossotti. H. Rossotti and L. G. Sill& ibid.. 1956. IO, 203. D. Litchinsky, N. Purdie, M. Tomson and W. White, Anal. Chem., 1969, 41, 1726. F. J. C. Rossotti, H. Rossotti and R. J. Whewell, J. Inorg. Nucl. Chem., 1971, 33, 2051. T. Briggs and J. Stuehr, Anal. Chem.. 1974,46, 1517. Idem, ibid., 1975. 41, 1916. P. J. C. Rossotti and H. Rossoiti. Determination of Stabiliry Constants, p. 90. McGraw-Hill, New York, 1961. S. S. Tate. A. K. Grzybowski and S. P. Datta, J. Chem. SOL, 1965. 3905. W. E. Deming, Statistical Adjustment of Data, Dover, New York, 1964. G. L. Gumming, J. S. Rollett, F. J. C. Rossotti, and R. J. Whewell, J. Chem Sot. Dalton Trans., 1972, 2652. H. Irving and H. S. Rossotti. J. Chem. Sot.. 1953.3397. H. J. deiruin, D. Kaitis ahd R. B. Temple, k$ J. Chem., 1962, 15,457. E. Campi, Ann. Chim.(Rome). 1963,53, 96. P. Niebergall, R. Schnaare and E. Sugita, J. Pharm. Sci., 1973, 64, 656. N. C. Li, 4. Lindenbaum and J. M. White, J. Inorg. Nucl. Chem., 1959, 12, 122. E. Campi, G. Ostacoli, M. Meirone and G. Saini, ibid., 1964, 26, 553. C. F. Timberlake, J. Chem., Sot., 1964. 5078. K. S. Rajan and A. E. Martell. Inorg. Chem., 1965, 4, 462.
Talunru. Vol. 29. pp. 435 to 439. 1982 Printed in Great Britain. All rights reserved
ANNOTATION
A COMPARATIVE STUDY OF THE APPLICATION OF THE METHOD OF LEAST-SQUARES IN THE POTENTIOMETRIC DETERMINATION OF PROTONATION CONSTANTS M. CELIA GARCIA, G. RAMIS and C. MONGAY Department of Analytical Chemistry, Faculty of Chemistry, University of Valencia, Valencia. Spain (Receioed 13 May 1981. Accepted 5 October 1981) Summary-Methods of simple and multiple linear regression applied to the potentiometric determination of protonation constants of diprotic and triprotic acids are studied critically. The best way of fitting the data, according to the order of magnitude of the constants, is established. The conclusions are
checked by calculating the protonation constants of succinic and citric acids.
The potentiometric determination of the protonation constants of an acid-base system may utilize graphical or analytical methods. Graphical methods’-4 have been critic&d because of their lack of accuracy, but they may assist detection of incorrect points, which is usually not possible by using only analytical methods of calculation. Analytical methods, on the other hand, are quicker and more accurate and precise. They include the unweighted least-squares procedure, which requires only a simple calculation program, and has the advantage over other analytical methods that it can be done with programmable pocket calculators. In recent years high-speed computers have been widely used for calculation of constants.5-* The computer programs often include weighting factors, and are relatively sophisticated. If the successive protonation constants of an acid differ greatly in magnitude, they can be treated separately, but when they are close, simultaneous fitting of the data is necessary. This is the case discussed here. For diprotic acids, among the various methods described, the linear least-squares procedure has been widely used, because of its precision and ease and speed of application. For triprotic acids, the available analytical methods are much more laborious, so graphical curve-fitting methods are usually recommended.g Here, the leastsquares method has been considered as not entirely satisfactory by some authorslo In the present work the extension of the leastsquares method to triprotic acids is studied with the aid of an HP 67 calculator. The conditions of its applicability to diprotic and triprotic acids are examined in parallel, and examples are shown for acids for 435
which the successive logarithmic constants differ by unity. Systems with larger and smaller differences are also tested, with analogous results. Potentiometric determination of protonation constants involves two well-differentiated steps: first, measurement of the experimental points to get the set of values (Z, h) and secondly, an appropriate treatment of the data to obtain the constants. In this paper, hypothetical acids with known constants are assumed in order to be able to discuss not only the precision, but also the accuracy of the method. The theoretical titration curves of these acids are calculated, then “experimental” titration curves based on these are simulated by adding Gaussian errors to the volumes of reagent and the values of pH. Finally, the conclusions are applied to the determination of the protonation constants of two real systems, and the results are compared with others in the literature. THEORETICAL TITRATIONS
For an n-protic acid, H,A, titrated with a strong base, the titration curve is given by:
v=
v
(1)
where h is [H’]. V is the volume of acid of concentration CA, u is the volume of strong base of concentration C, and /It is the ith overall protonation con-. stant of the system, with /IO = 1 by definition.
436
ANNOTATION
Values for the protonation constants of a hypothetical acid are selected and 40 “theoretical” points along its exact titration curve u =f(pH) are calculated, giving values to the parameters V, C, and C,. From this “exact” curve, 15 simulated “experimental” titration curves are obtained as follows. Each “theoretical” point is transformed into an “experimental” one by addition of a Gaussian error generated with the aid of a table of Tippett numbers,” which gives random deviates in units of standard deviation. To make use of this table, standard deviations of 0.01 ml in the volume of titrant and of 0.001 in pH are assumed. These values are about the lowest expected for the techniques most commonly used in the determination of stability constants. They illustrate the gradual differences, not only in precision but also in accuracy, found for the different methods of linear fitting discussed here. For the diprotic acids the values V = 40 ml, CA = 5 x lo-‘M and C, = 4 x 10m2M are assumed, except for the titration of the very weak acid with log Ki = 12 and log K2 = 11, for which C, was taken as O.lM. For the triprotic acids the values 1/ = 30 ml, CA = 5 x 10e3M and C, = 5 x 10e2M were used, except for the titration of the acid with log Ki = 12.5, log K2 = 11.5 and log K3 = 10.5 for which C, was 0.15M. TREATMENT
OF THE DATA
(a) Diprotic acids The expression
Ah + 2hh2
ii = 1 +
(2)
Blh + /32h2
may be written in the form: Y= f&-J+ a,X
(3)
where X and Y are functions of n and h, and the parameters a0 and a, are directly related to /LSiand /12. In this way, equation (3) may lead to the three forms shown in Table 1.’ For each of the 40 experimental points (u, pH) a pair of values @, h) is obtained. Only points with ii values given byO.l < ti 6 0.9 and 1.1 < n < 1.9 are used because of the sensitivity of the values of X and
Table 1. Y
X
I II III
.+,$,=I 1
h-(rs- 2) (1 -ii)
1
‘x+!%y=1 B2
82
xY=
a0
N
CXY = a&X
a,xX
+
+ a,CX2
(4)
leading to three regression lines of Y on X, from which the constants K, and K2 are estimated. In the same way, three regression lines of X on Y are fitted, leading to other values for the constants. This procedure is followed in the 15 titrations for each acid. The mean values and standard deviations of log K, and log K2 are given in Table 2. Standard deviations (s) are given with two significant figures. (b) Triprotic acids
In the same way the expression _
Ah + 2/%h2 + 3f13h3
n = 1 + j&h + j2h2 + j&h3
(5)
may be written as the equation of a plane 2 = a, + a, x + a2 Y
(6)
where X, Y and Z are functions of ii and h, and the parameters a,, a, and a, are functions of the constants /Ii, /I2 and /I3 as shown in Table 3. For the same reason as above, only the values 0.1 < ii < 0.9, 1.1 < ii < 1.9 and 2.1 < ii < 2.9 are used, reducing the number of useful data, N, to about 33 in each titration. Each of the four planes in Table 3 is fitted by the least-squares treatment by solving the system of equations &Z = aoN + a,CX
+ a2CY
CXZ = aoCX + a,CX2
+ a2CXY
CYZ = aoCY + a,CXY + a2CY2
(7)
to obtain four regression planes of Z on X and Y From each plane the values of the constants K1, K2 and K3 are calculated. Similarly the regression planes of Yon X and Z, and of X on Y and Z are obtained. The mean values and standard deviations obtained from the 15 simulated titrations are shown in Table 4.
DISCUSSION
Equation
Straight line
Y to slight experimental errors in n in the other ranges. For this reason, the number of useful points, N, is reduced to about 36 in each titration. Each of the three equations in Table 1 is fitted by the least-squares method, by using the usual equations:
ii-l
(2 - fi)h
-
-
(1 -nn)h
~
(2 $h2
As stated previously, the values for the standard deviations of the pH and volume are at the lower limit of those normally observed. This allows better observation of the gradual differences in precision and accuracy. In practice, the deviations are often a little bigger, leading to more dispersed values than the ones shown in Tables 2 and 4. In these tables, it can be seen that for equal deviations of pH and volume, slightly better precision is
437
ANNOTATlON
Table 2. Theoretical value
Fitting
Straight line I log K s
Straight line II. s log K
Straight line III log K s
Yon X
1.9950 1.055
0.0050 0.046
1.9956 1.048
0.0047 0.044
2.0023 0.94
0.0072 0.14
X on Y
2.0003 0.988
0.0046 0.049
2.0002 0.992
0.0046 0.050
2.0016 0.96
0.0070 0.12
Y on X
4.4985 3.5011
0.0038 0.0030
4.4999 3.4997
0.0015 0.0022
4.5004 3.497
0.0052 0.019
X on Y
4.4987 3.5010
0.0037 0.0030
4.4998 3.4996
0.0015 0.0022
4.5003 3.498
0.0053 0.019
log K, = 7.000
Y on X
7.aJO 6.0006
0.015 0.0054
7.001 6.0003
0.0016 0.0018
7.0011 5.996
0.0045 0.011
log K2 = 6.000
X on Y
7.001 6.0005
0.015 0.0054
7.0001 6.0003
0.0016 0.0018
7.0010 5.997
0.0045 0.011
log K, = 10.000
Y on X
9.998 9.0013
0.012 0.0040
9.9998 9.0002
0.0015 0.0019
X on Y
9.996 9.0013
0.019 0.040
9.9998 9.0002
0.0015 0.0019
9.9999 8.9999
0.0027 0.0063
11.991 11.004
0.052 0.014
12.0005 11.0004
0.0048 0.0032
12.0015 10.9998
0.0061 0.0042
log K2 = 11.000 X on Y 11.998 11.003
0.054 0.014
12.0002 11.0003
0.0048 0.0032
12.ooO8 1l.WO8
0.0062 0.0041
log K,
= 2.000
log K2 = 1.000
log K, = 4.500 log K2 = 3.500
log K2 = 9.000
log K, = 12.ooO
YonX
obtained in the determination of the constants of diprotic acids than for triprotic acids. It is also clear that the four methods of fitting for triprotic acids lead to much larger differences in precision and accuracy than do the three methods for diprotic acids. The different methods of fitting would lead to the same set of constants if the experimental data were not subject to errors. From Table 2 it can be seen that, for diprotic acids with pK-values of about 3.5-l 1.5, the values of X and Yare interchangeable, leading to negligible differences in the values of the constants and standard deviations. For strong acids, the regression of Y on X appears to lead to results with similar standard deviations but less accurate mean values. This is also true for triprotic acids, where the greater differences show that the
10.OOOo 0.0027 8.9997 0.0063
regression of Z on X and Y is clearly the least accurate. Generally, straight line II was found to give better precision and accuracy than lines I and III. Some authors have distrusted this equation,” but others have accepted it.” For strong acids, the results from straight line I are nearer to those from straight line II than those of straight line III are. For these acids, evaluation of Kr is more accurate and precise than that of K2 with both lines I and II. For strong bases, the results from straight line III are close to those of the straight line II, but straight line I gives poor agreement. For the triprotic acids (Table 4) results from plane I are not very useful, and they become worse for weaker acids, eventually resulting in interchanged
Table 3. Plane
IV
Equation
Z
Y
X
438
ANNOTATION
Table 4. Plane I log K s
Plane II log K s
Plane III log K
Z on X and Y
3.0058 1.966 1.190
0.0041 0.018 0.086
3.0079 0.0037 1.969 0.015 1.208 0.080
3.01 1.37 1.69
Yon X and Z
3.0006 I.996 I .030
0.0034 0.013 0.080
3.0009 0.0031 1.998 0.012 0.996 0.087
3.0031 1.97 I.14
0.0089 0.11 0.48
3.01 1.65 2.48
(4*)
3.0017 1.991
3.0015 0.0031 1.995 0.012 1.019 0.086
3.0040 1.968 1.16
0.008 1 0.098 0.43
3.01 1.20 2.70
(5:)
1.059
0.0036 0.014 0.083
Z on X and Y
5.01 4.001 3.0009
0.10 0.036 0.0083
4.9995 0.0028 4.0001 0.0031 3.0010 0.0037
4.9995 4.006 3.022
0.0062 0.013 0.022
YonXandZ
4.997 4.004 3.0005
0.098 0.036 0.0083
4.9994 0.0028 4.0003 0.0030 3.0009 0.0037
4.9999 3.999 3.003
0.0062 0.013 0.010
5.00 3.89 3.76
Xon
5.10 3.987 3.0029
0.15 0.038 0.0087
4.9992 0.0029 3.9999 0.003 1 3.0010 0.0037
4.9996 4.000 3.0013
0.0062 0.013 0.0098
5.00 3.91 3.63
Z on X and Y
7.09 7.35 5.9872
0.70 0.33 0.0065
7.99 17 0.0082 7.016 0.015 5.9939 0.0079
7.9963 7.018 6.011
0.008 1 0.023 0.015
Yon X and Z
7.24 7.32 5.99
l6*)
7.9909 0.0082 7.017 0.015 5.9939 0.0071
7.9969 7.007 5.995
0.0080 0.019 0.012
8.00 6.85 6.73
(3*)
7.978 0.012 7.010 0.023 5.9976 0.0081
7.9966 7.008 5.994
0.0081 0.020 0.013
8.00 6.87 6.74
(4’)
Theoretical values
Fitting
log K, = 3.000 log K2 = 2.CQO log K3 = 1.000 Xon
log
YandZ
K, = 5.000
log K2 = 4.000 log Kj = 3.000
YandZ
log K, = 8.000 log K2 = 7.000 log K1 = 6.000 Xon
YandZ
t
(7*)
t
9.36 9.97 8.49
(5*)
10.504 0.011 9.493 0.017 8.5032 0.0090
10.4997 9.5008 8.5013
0.0041 0.005 1 0.0049
10.50 9.53 7.86
(8*)
Yon X and Z
9.28 IO.01 8.49
(5*)
10.504 0.011 9.494 0.017 8.5030 0.0089
10.4997 9.5004 8.5007
0.0041 0.0052 0.0048
10.496 9.497 8.35
0.021 0.083 0.55
t
10.4954 0.0089 9.487 0.021 8.5033 0.0091
10.4996 9.5006 8.5005
0.0041 0.0051 0.0048
10.5004 0.0094 9.494 0.076 (14*) 8.39 0.45
12.464 11.541 10.481
0.036 0.045 0.020
12.497 11.5006 10.5011
0.020 0.0055 0.0055
12.477 0.049 11.58 0.13 (12*) 10.11 0.3i
12.457 11.549 10.480
0.042 0.052 0.021
12.498 11.4997 10.4990
0.020 0.0055 0.0056
12.498 11.36 10.51
0.053 0.50 0.23
12.383 11.521 10.481
0.096 0.026 0.020
12.495 11.5019 LO.4974
0.019 0.0053 0.0057
12.488 11.520 10.39
0.052 0.086 0.27
log K3 = 8.500 Xon
YandZ
Z on X and Y log K, = 12.500 log K2 = II.500
t
(22)
Z on X and Y log K, = 10.500 log KL = 9.500
Plane IV log K s
s
Yon X and Z
t
log KJ = 10.500 Xon
YandZ
* Among the I5 titrations. only the number indicated in parenthesis lead to positive values of the constants. t This fitting gives rise to negative values of the constants in all the I5 titrations.
Table 5. Temp.. ‘C 25 20 30 20
Ionic strength O.lS(NaCIO,) O.lO(NaCIO,) 0.1 (KCI) 0.10(NaC104)
log K, 4.88 5.28 5.116 5.205 (s = 0.015
log
K2
4.05 4.00 3.952 3.995 (s = 0.015)
Reference 14 15 16 This work
439
ANNOTATION
Table 6. Temp., “C Ionic strength 25 20 20 25 30 25 20
0.15 0.1 (NaClOJ 0.1 (NaClOJ 0.1 (KN03) 0.1 (KCI) 0.1 (KN09) 0.10 (NaClOJ
log
Kl
loi3
K2
4.34 5.62 4.35 5.68 5.68 (s = 0.02) 4.38 (s = 0.02) 4.30 5.65 4.395 5.750 4.39 5.72 5.660(s = 0.015) 4.335(s = 0.015)
values of the constants (log K1 log K2 etc.). Plane IV also does not give valid results for all acids. KI is always obtained with a slightly better precision than K2 and K1, which may become interchanged (log K2 < log K3). Plane II leads to good results for all acids, but for the strongest acids, K3 is less precise, and K1 is the best constant. For medium-strength acids, this plane leads to acceptable values of all three constants. As the acid strength decreases, KS is determined with better precision (but plane III gives better precision for weak acids). Plane III gives good results for weak and mediumstrength acids (comparable to those of plane II for the latter). For strong acids, the results are not very good. Results were also obtained for acids with greater and smaller differences between their protonation constants, and similar conclusions were reached.
CONCLUSIONS
For diprotic acids, least-squares fitting by means of straight line II is to be preferred. For strong acids, it is better to use the regression of X on Y For other acids, both regressions should be performed, and the results with smaller dispersion, or the mean value, if both regressions have a similar precision, accepted. For triprotic acids, the data should be fitted by means of plane II or III for strong and weak acids, respectively. For strong acids, the regression of 2 on X and Y should not be used. For other acids, the three possible regressions should be calculated, and the one with the smallest dispersion, or the mean value of those with the smallest dispersion, accepted.
Titrations were done at 20” with the aid of a Radiometer PHM 84, (resolution 0.001 pH unit) and a combination glass electrode, Radiometer GK2401 C, standardized against 0.05M potassium hydrogen phthalate (PH 4.001) and 0.025M potassium dihydrogen phosphate/O.O25M disodium hydrogen phosphate (pH 6.874). The ionic
K3
Reference
17 2.94 18 2.87 19 2.96 (s = 0.03) 20 2.79 16 3.023 8 2.92 2.97 (S = 0.03) This work
strength was adjusted to 0.10M by addition df an appropriate amount of sodium perchlorate. All the reagents were
of analytical grade. (a) Succinic acid. Five titrations were done of 50 ml of 1.656 x 10b3M disodium succinate with 0.0212M hydrochloric acid. The mean values of the constants and standard deviations obtained by means of straight line II are shown in Table 5, where they are also compared with some literature constants obtained in similar conditions. Both regressions (X on Y and Y on X) were used because their standard deviations were of the same order of magnitude. (b) Citric acid. Five titrations of 50 ml of 1.025 x lo-“M trisodium citrate were done with 0.019OM hydrochloric acid. The mean values of the constants and standard deviations calculated by means of plane II are shown in Table 6, where some values from the literature are also given. The three fitting possibilities were used. REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
EXPERIMENTAL
h3
17. 18. 19. 20.
J. Speakman, J. Chem. Sot., 1940, 855. L. G. Sill&, Acta Chem. Stand., 1954. 8. 318. Idem. ibid., 1956. 10, 186. F. J. C. Rossotti. H. Rossotti and L. G. Sill& ibid.. 1956. IO, 203. D. Litchinsky, N. Purdie, M. Tomson and W. White, Anal. Chem., 1969, 41, 1726. F. J. C. Rossotti, H. Rossotti and R. J. Whewell, J. Inorg. Nucl. Chem., 1971, 33, 2051. T. Briggs and J. Stuehr, Anal. Chem.. 1974,46, 1517. Idem, ibid., 1975. 41, 1916. P. J. C. Rossotti and H. Rossoiti. Determination of Stabiliry Constants, p. 90. McGraw-Hill, New York, 1961. S. S. Tate. A. K. Grzybowski and S. P. Datta, J. Chem. SOL, 1965. 3905. W. E. Deming, Statistical Adjustment of Data, Dover, New York, 1964. G. L. Gumming, J. S. Rollett, F. J. C. Rossotti, and R. J. Whewell, J. Chem Sot. Dalton Trans., 1972, 2652. H. Irving and H. S. Rossotti. J. Chem. Sot.. 1953.3397. H. J. deiruin, D. Kaitis ahd R. B. Temple, k$ J. Chem., 1962, 15,457. E. Campi, Ann. Chim.(Rome). 1963,53, 96. P. Niebergall, R. Schnaare and E. Sugita, J. Pharm. Sci., 1973, 64, 656. N. C. Li, 4. Lindenbaum and J. M. White, J. Inorg. Nucl. Chem., 1959, 12, 122. E. Campi, G. Ostacoli, M. Meirone and G. Saini, ibid., 1964, 26, 553. C. F. Timberlake, J. Chem., Sot., 1964. 5078. K. S. Rajan and A. E. Martell. Inorg. Chem., 1965, 4, 462.