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The use of approximation formulae in calculations of acid-base equilibria
0039.9140/80/1201-109910200/0
rallanro Vol 21. pp 1099 to I101 0 Pergamon Press Ltd 1980. Printed in Great Britain
ANNOTATION
THE USE OF APPROXIMATION FORMULAE IN CALCULATIONS OF ACID-BASE EQUILIBRIA A. A. S. C. MACHADO C.LQ.(U.P.), Chemistry Department, Faculdade de Ciincias, 4000 Porto, Portugal (Received 17 December 1979. Accepted 16 June 1980)
Summary-Details of diagrams recently presented for the regions where approximation formulae give the pH of mono- and diprotic acids with an error of less than 0.02 pH unit are discussed.
RESULTS AND DISCUSSION
Narasaki’ has recently reported calculations to identify the regions of concentration and dissociation constants where approximation formulae for calculation of [H’] and pH in acid-base equilibria give the pH with an error < 0.02. Similar calculations were done by me some time ago not only for pure solutions of single acids? but also for more complex systems3 A first comparison of Narasaki’s diagrams with mine showed broad agreement of the results, but suggested some differences in the shape of the curves. Full comparison was impossible because the units on the axes were different. In order to investigate the differences in the diagrams I recalculated some of my results, and found a very definite difference in some of the details of the diagrams. In this note, the differences are reported and discussed. CALCULATIONS The exact pH was calculated by computer to a precision of better than _+0.001 (the results were the same if the pH was calculated to kO.01) from the exact polynomial equations [Narasaki’s’ equations (10) and (18)] by using a general subroutine previously described4 for the calculation of pH. The only formula of those considered by Narasaki’ which had been included in my previous analysis of pure solutions of single acids’ was the simplest of all, i.e.,
CH’I =
m
pH = +pK,
- f log CA
(I’)
where KA is the dissociation constant of a monoprotic acid or the first dissociation constant of a diprotic acid, KA,. None of the second-degree approximations included in my previous study3 was considered by Narasaki.’ The difference between the pH value given by formula (I’) and the exact pH was calculated for successive sets of values of the variables (CA and K, for monoprotic acids, CA, K,, and K,, for diprotic acids) by a program which provided results as tables of calculation errors. The diagrams (Figs. 1 and 2) were drawn from these tables.
The results obtained for a monoprotic acid are presented in Fig. 1, which is a corrected version of Fig. 2 of my original work,’ turned upside down to allow it to be compared with Narasaki’s’ Fig. 1, case C. Inside the area limited by curves + 1 and - 1 the calculation error which resulted from replacement of the exact equation by formula (1’) is less than 0.01 pH unit. Similarly, inside the area limited by curves -5 and +5 the calculation error is smaller than 0.05 pH unit. On the central line or in the area marked zero the error is zero (when rounded off to two places of decimals); to the left it is negative and to the right it is positive. Comparison of Fig. 1 with Narasaki’s Fig. 1 case C, shows general agreement in the area where formula (1’) give results with a small error. However the area obtained by Narasaki’ for an error of less than +0.02 pH units is slightly smaller than that in the present diagram for an error of less than +O.Ol pH unit, and not slightly larger as expected. Moreover, the curves obtained in my work are continuous, and extended into the region of low concentration and p& (top left corner of the figure). This detail was not shown in Narasaki’s diagram, probably because it was drawn from a rather restricted set of calculated values or because the calculations were not precise enough. For comparing the results for diprotic acids, my previous plots (Figs. 3 and 4)’ were unsuitable because values of pKA, instead of pK,, (which was used in Narasaki’s Figs. 24) were plotted on the x-axis, so the results had to be recalculated and replotted. When the ratio KAZ/KA, was used as a variable, as in Narasaki’s Fig. 2, it was found that the agreement was not complete. The differences found were of the same type as above. The results are shown in Fig. 2, where the value of pKA2 (not the ratio
to99
1100
ANNOTATION
6-
0 0
2
4
6
8
IO
12
Fig. 1. Range of applicability of equation (I ‘) for monoprotic acids. The number near each line gives the calculation error multiplied by 100.
6-
4-
2-
O0
2
4
6
6
IO
12
Fig. 2. Range of applicability of equation (1’) for diprotic acids. Lines are labelled with the appropriate values of pKAI. Areas ofapplicability are indicated by arrows, and vertical lines refer to the condition pKA, < PK,,~ - 0.60. KA2/KA1) is used as the variable, for the sake of convenience (tables of acid dissociation constants list values of successive constants and not ratios). For each value of pKAz the calculation error introduced by the use of formula (1’) is smaller than 0.05 pH units inside the area hmited by the two lines marked with the value of pK,, (the line on the left corresponds to negative errors, that on the right to positive errors). The vertical lines refer to the condition pK,, < pKA, - 0.60, which is true for all known real dibasic acids. As with monoprotic acids, the areas extend into the region of low concentration and pKA, which again was not found by Narasaki.’ It is interesting to note that when the value of PKA~ is increased to ca. 8, there is enlargement of the area where formula (1’) can be used without introducing
appreciable error. This is expected because, when the second dissociation becomes weaker, its contribution to the production of protons in solution decreases. Thus, the first dissociation dominates protolytic equilibria and the use of formula (I), where only the first dissociation is considered, leads to a smaller error. For values of pKAz larger than 8 the second dissociation is completely negligible so the area stops widening. For such large values of pKA2 the diprotic acid behaves like a monoprotic acid and the lines in Fig. 2 coincide with those of Fig. 1. CONCLUSIONS
Diagrams like those presented by Narasaki’ should be used with caution, because the lines which limit the
1101
ANNOTAflON
areas seem to be very sensitive to calculation procedures: precise diagrams seem to be obtained only when the calculated values are sufficiently precise and numerous. Ackrlo~ledyertlrnt-I thank Mrs. M. AssunCio C. Lima for running the programs for me.
TM. 27 12-F
REFERENCES
I. H. Narasaki, Talnnta, 1979, 26, 605. 2. A. A. S. C. Machado, Rev. Port. Quim., _
1971. 13, 19; 1973, 15, 133. 3. Idem, ibid., 1971, 13, 236; 1972, 14, I, 7; 1974, 16, 65. 4. Idem, ibid., 1971, 13, 14,
rallanro Vol 21. pp 1099 to I101 0 Pergamon Press Ltd 1980. Printed in Great Britain
ANNOTATION
THE USE OF APPROXIMATION FORMULAE IN CALCULATIONS OF ACID-BASE EQUILIBRIA A. A. S. C. MACHADO C.LQ.(U.P.), Chemistry Department, Faculdade de Ciincias, 4000 Porto, Portugal (Received 17 December 1979. Accepted 16 June 1980)
Summary-Details of diagrams recently presented for the regions where approximation formulae give the pH of mono- and diprotic acids with an error of less than 0.02 pH unit are discussed.
RESULTS AND DISCUSSION
Narasaki’ has recently reported calculations to identify the regions of concentration and dissociation constants where approximation formulae for calculation of [H’] and pH in acid-base equilibria give the pH with an error < 0.02. Similar calculations were done by me some time ago not only for pure solutions of single acids? but also for more complex systems3 A first comparison of Narasaki’s diagrams with mine showed broad agreement of the results, but suggested some differences in the shape of the curves. Full comparison was impossible because the units on the axes were different. In order to investigate the differences in the diagrams I recalculated some of my results, and found a very definite difference in some of the details of the diagrams. In this note, the differences are reported and discussed. CALCULATIONS The exact pH was calculated by computer to a precision of better than _+0.001 (the results were the same if the pH was calculated to kO.01) from the exact polynomial equations [Narasaki’s’ equations (10) and (18)] by using a general subroutine previously described4 for the calculation of pH. The only formula of those considered by Narasaki’ which had been included in my previous analysis of pure solutions of single acids’ was the simplest of all, i.e.,
CH’I =
m
pH = +pK,
- f log CA
(I’)
where KA is the dissociation constant of a monoprotic acid or the first dissociation constant of a diprotic acid, KA,. None of the second-degree approximations included in my previous study3 was considered by Narasaki.’ The difference between the pH value given by formula (I’) and the exact pH was calculated for successive sets of values of the variables (CA and K, for monoprotic acids, CA, K,, and K,, for diprotic acids) by a program which provided results as tables of calculation errors. The diagrams (Figs. 1 and 2) were drawn from these tables.
The results obtained for a monoprotic acid are presented in Fig. 1, which is a corrected version of Fig. 2 of my original work,’ turned upside down to allow it to be compared with Narasaki’s’ Fig. 1, case C. Inside the area limited by curves + 1 and - 1 the calculation error which resulted from replacement of the exact equation by formula (1’) is less than 0.01 pH unit. Similarly, inside the area limited by curves -5 and +5 the calculation error is smaller than 0.05 pH unit. On the central line or in the area marked zero the error is zero (when rounded off to two places of decimals); to the left it is negative and to the right it is positive. Comparison of Fig. 1 with Narasaki’s Fig. 1 case C, shows general agreement in the area where formula (1’) give results with a small error. However the area obtained by Narasaki’ for an error of less than +0.02 pH units is slightly smaller than that in the present diagram for an error of less than +O.Ol pH unit, and not slightly larger as expected. Moreover, the curves obtained in my work are continuous, and extended into the region of low concentration and p& (top left corner of the figure). This detail was not shown in Narasaki’s diagram, probably because it was drawn from a rather restricted set of calculated values or because the calculations were not precise enough. For comparing the results for diprotic acids, my previous plots (Figs. 3 and 4)’ were unsuitable because values of pKA, instead of pK,, (which was used in Narasaki’s Figs. 24) were plotted on the x-axis, so the results had to be recalculated and replotted. When the ratio KAZ/KA, was used as a variable, as in Narasaki’s Fig. 2, it was found that the agreement was not complete. The differences found were of the same type as above. The results are shown in Fig. 2, where the value of pKA2 (not the ratio
to99
1100
ANNOTATION
6-
0 0
2
4
6
8
IO
12
Fig. 1. Range of applicability of equation (I ‘) for monoprotic acids. The number near each line gives the calculation error multiplied by 100.
6-
4-
2-
O0
2
4
6
6
IO
12
Fig. 2. Range of applicability of equation (1’) for diprotic acids. Lines are labelled with the appropriate values of pKAI. Areas ofapplicability are indicated by arrows, and vertical lines refer to the condition pKA, < PK,,~ - 0.60. KA2/KA1) is used as the variable, for the sake of convenience (tables of acid dissociation constants list values of successive constants and not ratios). For each value of pKAz the calculation error introduced by the use of formula (1’) is smaller than 0.05 pH units inside the area hmited by the two lines marked with the value of pK,, (the line on the left corresponds to negative errors, that on the right to positive errors). The vertical lines refer to the condition pK,, < pKA, - 0.60, which is true for all known real dibasic acids. As with monoprotic acids, the areas extend into the region of low concentration and pKA, which again was not found by Narasaki.’ It is interesting to note that when the value of PKA~ is increased to ca. 8, there is enlargement of the area where formula (1’) can be used without introducing
appreciable error. This is expected because, when the second dissociation becomes weaker, its contribution to the production of protons in solution decreases. Thus, the first dissociation dominates protolytic equilibria and the use of formula (I), where only the first dissociation is considered, leads to a smaller error. For values of pKAz larger than 8 the second dissociation is completely negligible so the area stops widening. For such large values of pKA2 the diprotic acid behaves like a monoprotic acid and the lines in Fig. 2 coincide with those of Fig. 1. CONCLUSIONS
Diagrams like those presented by Narasaki’ should be used with caution, because the lines which limit the
1101
ANNOTAflON
areas seem to be very sensitive to calculation procedures: precise diagrams seem to be obtained only when the calculated values are sufficiently precise and numerous. Ackrlo~ledyertlrnt-I thank Mrs. M. AssunCio C. Lima for running the programs for me.
TM. 27 12-F
REFERENCES
I. H. Narasaki, Talnnta, 1979, 26, 605. 2. A. A. S. C. Machado, Rev. Port. Quim., _
1971. 13, 19; 1973, 15, 133. 3. Idem, ibid., 1971, 13, 236; 1972, 14, I, 7; 1974, 16, 65. 4. Idem, ibid., 1971, 13, 14,