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Acid—base titration curves for acids with very small ratios of successive dissociation constants
ACID-BASE TITRATION CURVES FOR ACIDS VERY SMALL RATIOS OF SUCCESSIVE DISSOCIATION CONSTANTS BRUCE I-I. CAMPBELL
WITH
and LOUISMEITES
Department of Chemistry. Ciarkson College of Technology, Potsdam. N.Y. 13676. U.S.A (Recewed
4 Ju1.r 1973. Accepred 15 September
1973)
Summary-The shapes of the potentlomctric acid-base trtratron curves obtamed m the ncutrahzations of polyfunctronal acids or bases for whrch each successive dissociation constant is smaller than the following one are examined. In the region 0 < f i 1(wherefts the fractron of the equivalent volume of reagent that has been added) the slope of the titration curve decreases as the number 1of acidtc or has~c sites increases. The difference between the pH-values at f = 0.75 and f = 0.25 ILI\ ( I /Ilog 9 ab the‘ lower Imitt of1t5 maxm~um value.
In the derivation of equations that describe potentiometric acid-base titration curves for polyfunctional acids, it is generally assumed that pKi+, is larger than pKi. There are a number of difunctional acids for which the ratio of the thermodynamic constants K,/K2 is very little larger than the “statistical” ratio of 4.00, and this has prompted some authors to consider the problem of distinguishing such acids from monofunctional ones. Sturrock’ suggested a criterion based on the value of A(pH), which is defined as the difference between the values of pH at two points, one at 75% and the other at 25% of the way from the start of the titration to the single end-point that can be discerned. For a monofunctional acid A(pH) = log 9 = 0.954 if both the acid and base are moderately concentrated2 and if the acid is neither very strong nor very weak. If the solution is dilute, or if K, is either very large or very small, A(pH) < 09.54 for a monofunctional acid or base. As is well known, the titration curve for a monofun~tional acid is indistinguishable from that for a difunctional acid having K,/K2 = 4.00. If K,/K2 is larger than 430, the value of A(pH) for a difunctional acid exceeds that for a monofunctional acid having K, = (KI,'K2)*: if K,/K2 is smaller than 4.00, the value of A(pH) is less than that for such a monofunctional acid. So far, acids for which K,/K2 is very much smaller than the statistical value have not been considered, and it is with these that this paper is concerned. The following derivation describes how A(pH) depends upon the number of acid sites and upon the magnitude of the difference between pK, + 1 and pK,. Although very small values of K,/K, +1 are unusual. two cases that could exhibit such behaviour are described here. Consider a molecule containing a fairly strongly acidic group so placed that it is folded into the centre of the molecule and thus unable to dissociate. and a weakly acidic functional group that is exposed to the solution and hence to dissociation and attack by a base. When this outer group is neutralized. the change in charge on the molecule might cause a change in convolution of the molecule, bringing the more strongly acidic group to the outside and rendering this acid group susceptible to dissociation and attack. In this case. it will be the order in which the two protons are removed
BRUCEH. CAMPBELLand Low
118
MEITES
that will dictate which of the dissociation constants is taken as K, and which as Kz, whereas in the ordinary case the order with time is the same as, and is governed by, the order of the equilibrium constants. The second example would arise if addition of a proton yielded a protonated species that decomposed to form a base stronger than the original one. As will’be shown later, the resulting equilibrium titration curve not only has just one end-point but is also abnormally flat around its mid-point. This is just the behaviour that is displayed by titration curves for dialkyldithiocarbamates.3 The graphs shown in Figs. 1, 2 and 3 were obtained by means of a Hewlett-Packard Model 9 1OOAprogrammable calculator and a model 9125A plotter. The curves pertain
PH
Fig.
I. Plot of fraction
of species. 2. I’S pH for a trlfunctional pK, = 5.
acid with
pK,
= 3. pK,
= 4.
actd with
pK,
= 6. pK2 = 4.
PH Fig. 2. Plot of fraction’ of species, 9. cs. pH for a trifunctional pK3 = 2.
Acid-base trtration curves
Fig. 3 Titration curve of a trifunctional acid with a monofunctlonal 4. pK, = 2. c: = 1M. I’ = 3.
119
strong base. pK, = 6, pK2 =
to a trifunctional acid. Although those in Fig. 2 have the same general shape as those for a monofunctional acid, they are much steeper in the vicinity off = 05, where pH 5 pK, for the trifunctional one. for the monofunctional acid while pH w l/3 [p(K,K,K,)] Because values of A(pH) taken from plotted titration curves, such as the one in Fig. 3, are subject to fairly large graphical errors, those quoted below were obtained from numerical calculations made with a digital computer. The titration curve is described by the equation
(l/j) f=
i i=l
4)CH~A- C(H+)- (OH-)1
Cn+ + r [(H+) - (OH-)]
(1)
’
wherefis proportional to the amount of titrant added and is equal to 1.00 at the last (jth) equivalence point. j is the number of acidic sites, CHjAis the initial formal concentration of H,A. I’is the dilution parameter defined as the ratio of CnjA to the formal concentration of the titrant (which is assumed to be monofunctional), and ai is the fraction of the total A that is present as the ith species. Equation (1) is combined with the familiar expression for xi xi = [H+]‘+
ith term [H+]‘-‘K, + . . . + K,K2...Kj
’
to obtain an implicit relationship between [H ‘1 and 1: This was solved at f = 0.25 and 0.75 and A(pH) was obtained from the resulting values. Figures 4 and 5b are drawn for acids of different j but with (K,K,...Kj)' ‘j always equal to lo-‘. For a monobasic acid this value would correspond to the maximum on a plot of A(pH) against PK,~,~. As Ki/Ki+ I decreases the titration curve becomes flatter and A(pH) decreases, but if Ki/Ki+ 1 is as small as lo-’ a further decrease in it has no perceptible further effect on A(pH), which is also insensitive to variations of I’ in this range of acid strength. This limiting value of A(pH) is plotted against j in Fig. 4. and may be shown to be proportional to l/j. In the following paragraphs an algebraic proof of this proportionality is given. Figure 5 shows how the limiting value of A(pH) depends on (l/j)XpK, which is equal to the negative logarithm of the geometric mean dissociation constant. and on the concentration of the acid titrated.
BRWE
H. CAMPBELL~~~ LOUISMEITES
I
I,
0.5
4
0
I
I 6
I
I
I
3
I
I
1
0
Fig. 4. Dependence of the limiting value (see text) of A(pH) on j for the titration of a polyfunctional acid with a monofunctional strong base with (K,K2.. .K,) 1 ’ = IO-‘. Values Indistinguishable kom those shown here are obtained whenever C: 2 IO-‘M and K, K,_, I IO-’
0.6 ‘1
(I/J)IpK
h
Fig. 5. Dependence of A(pH) on (a) (I/j)XpK, for (0) a difunctional acid with K,/K2 = IO- ’ and (A) a trifunctional acid with K,/K2 = K,/K, = IO-‘, and(h) the concentration of a trifunctional acid with pK, = 9. pK, = 7, and pK3 = 5. with r = 2.
AC&base
trtration
curves
121
If dilution. dissociation of H iA. and dissociation of water are all neglected, equation (1) becomes f=
(I.ij) i i=l
iz,.
We define a parameter 111,as equal to KJK, + , and assume all the lili to be very small; A(pH) is then independent of 177. Under these conditions only the species H,A and AJ-’ are important and therefore every x except CQ,and x, is equal to zero. Of these, the former does not appear in equation (3) because the index has 1 rather than 0 as its smallest value. Hence every term in equation (3) is either zero or excluded, with the sole exception of x,. and equation (3) can therefore be written as ,f’= i(jg,) = 2,.
(4)
KIK2K,...Kj 'I= [H+]j+ K,K2K,...Ki'
(5)
equation (4) may now be written as K,K,K3...KJ .f= [H+]‘+ K,K,K,...Ki' From the definition above of
(6)
177~
K, = K,/n1,
(7)
K, = K,,i1772 = K,/(777,771,)
(8)
Ki = K,
1-I n(l/777,),
(9)
i=l
and. upon substitution into equation (6). K 1's 1/771i i=l
f=
j-
[H ‘1’ + K 1n
(10)
1
l/rrZi
which can be solved for [H ‘1 giving
:1 -m-t n
(11)
hence pH = pK, - flog[(l
-_f-)/” - (l/j)log
(12)
To examine the dependence of A(pH) upon j.f can be set equal to 0.75 and then to 0.25. The difference between the resulting pH-values gives A(pH) = (1 ‘j)log9 = 0.954/j
(13)
BRUCEH. CAMPBELLand Louts MEITES
122
The corresponding titration curve has only one point of maximum slope (see Fig. 3). no matter how many protons are on the original acid. In the region 0 c f c 1 the curve is symmetrical aroundf = 0.5 and in this interval the sign of the second derivative changes only once. Because the value of (log 9)/j can be thought of as a limit of A(pH) as 111goes to zero. it can also be used in general to aid in developing an empirical formulation of the dependence of A(pH) upon m. For a difunctional acid having tn < 5, Meites’ gave the empirical equation A(pH) = O-478 + O-253 rn) + OGO741~. The intercept may be compared with the theoretical value of 0.4772. It is suggested that the general form for a j-functional acid is A(pH) - (l/j)log 9 = a& + bm + . . . , and is valid as long as nz is not too much greater than the statistical value.
It is not possible to apply a similar treatment to polyfunctional acids for which Ki is larger than Ki+i* This is because all the terms in u are then important and the resulting equations are impossible to solve in a closed form. One last aspect of equation (13) is that acids of the type considered here could be used to prepare buffers of much higher buffer capacity than those that can be made from monofunctional ones or the familiar polyfunctional ones. Acknowledgement-We wish to express our thanks to the National Science Foundation and the Eastman Kodak Company for the departmental grants that made possible the purchase of the PDPI/I computer used in this study. REFERENCES 1. 2. 3. 4.
P. E. Sturrock, J. Chem. Educ., 1968,45,268. L. Meites, ibid., 1972,49,682, 850. C. L. Chakrabarti. Anal. Chem., 1969.41, 1441. A. Lomax and A. J. Bard. J. Chrrn. Edrrc.. 1968. 45. 621 Form der potentiometrischen &ire-Basen-Titrationskurven bei der Neutralisation polyfunktioneller Sauren oder Basen. bei dcncnJede individuelle Dissoziationskonstante kleiner ist als die der nachfolgenden Stufe. wird untersucht. Im Bereich 0 < j < I (wo f’der zugegebene Bruchteil des aquivalenten Reagensvolumens ist) nimmt die Steigung der Titrationskurve ab, wenn die Anzahlj saurer oder basischer Funktionen zunimmt. Die untere Grenze fur die Differenz der pH-Werte beif = 0,75 undf= 0,25 betragt (l/j) log 9. Zusammenfassung-Die
R&me-On examine Ies formes des courbes de titrage potentiometrique acid+base dans les neutralisations d’acides ou bases polyfonctlonnels pour lesquels chaque constante de dissociation successive est plus petite que la suivante. Dans la region 0 < f’< I (oh fest la fraction du volume equivalent de reactif qui a ite aJoutee). la pente de la courbe de titrage decroit quand le nombre j despositionsacidesou basiquescroit. Ladifferenceentre les valeurs de pH aj’ = 0.75 etf = 0.25 a (l/j) log 9 pour limite infirieure.
WITH
and LOUISMEITES
Department of Chemistry. Ciarkson College of Technology, Potsdam. N.Y. 13676. U.S.A (Recewed
4 Ju1.r 1973. Accepred 15 September
1973)
Summary-The shapes of the potentlomctric acid-base trtratron curves obtamed m the ncutrahzations of polyfunctronal acids or bases for whrch each successive dissociation constant is smaller than the following one are examined. In the region 0 < f i 1(wherefts the fractron of the equivalent volume of reagent that has been added) the slope of the titration curve decreases as the number 1of acidtc or has~c sites increases. The difference between the pH-values at f = 0.75 and f = 0.25 ILI\ ( I /Ilog 9 ab the‘ lower Imitt of1t5 maxm~um value.
In the derivation of equations that describe potentiometric acid-base titration curves for polyfunctional acids, it is generally assumed that pKi+, is larger than pKi. There are a number of difunctional acids for which the ratio of the thermodynamic constants K,/K2 is very little larger than the “statistical” ratio of 4.00, and this has prompted some authors to consider the problem of distinguishing such acids from monofunctional ones. Sturrock’ suggested a criterion based on the value of A(pH), which is defined as the difference between the values of pH at two points, one at 75% and the other at 25% of the way from the start of the titration to the single end-point that can be discerned. For a monofunctional acid A(pH) = log 9 = 0.954 if both the acid and base are moderately concentrated2 and if the acid is neither very strong nor very weak. If the solution is dilute, or if K, is either very large or very small, A(pH) < 09.54 for a monofunctional acid or base. As is well known, the titration curve for a monofun~tional acid is indistinguishable from that for a difunctional acid having K,/K2 = 4.00. If K,/K2 is larger than 430, the value of A(pH) for a difunctional acid exceeds that for a monofunctional acid having K, = (KI,'K2)*: if K,/K2 is smaller than 4.00, the value of A(pH) is less than that for such a monofunctional acid. So far, acids for which K,/K2 is very much smaller than the statistical value have not been considered, and it is with these that this paper is concerned. The following derivation describes how A(pH) depends upon the number of acid sites and upon the magnitude of the difference between pK, + 1 and pK,. Although very small values of K,/K, +1 are unusual. two cases that could exhibit such behaviour are described here. Consider a molecule containing a fairly strongly acidic group so placed that it is folded into the centre of the molecule and thus unable to dissociate. and a weakly acidic functional group that is exposed to the solution and hence to dissociation and attack by a base. When this outer group is neutralized. the change in charge on the molecule might cause a change in convolution of the molecule, bringing the more strongly acidic group to the outside and rendering this acid group susceptible to dissociation and attack. In this case. it will be the order in which the two protons are removed
BRUCEH. CAMPBELLand Low
118
MEITES
that will dictate which of the dissociation constants is taken as K, and which as Kz, whereas in the ordinary case the order with time is the same as, and is governed by, the order of the equilibrium constants. The second example would arise if addition of a proton yielded a protonated species that decomposed to form a base stronger than the original one. As will’be shown later, the resulting equilibrium titration curve not only has just one end-point but is also abnormally flat around its mid-point. This is just the behaviour that is displayed by titration curves for dialkyldithiocarbamates.3 The graphs shown in Figs. 1, 2 and 3 were obtained by means of a Hewlett-Packard Model 9 1OOAprogrammable calculator and a model 9125A plotter. The curves pertain
PH
Fig.
I. Plot of fraction
of species. 2. I’S pH for a trlfunctional pK, = 5.
acid with
pK,
= 3. pK,
= 4.
actd with
pK,
= 6. pK2 = 4.
PH Fig. 2. Plot of fraction’ of species, 9. cs. pH for a trifunctional pK3 = 2.
Acid-base trtration curves
Fig. 3 Titration curve of a trifunctional acid with a monofunctlonal 4. pK, = 2. c: = 1M. I’ = 3.
119
strong base. pK, = 6, pK2 =
to a trifunctional acid. Although those in Fig. 2 have the same general shape as those for a monofunctional acid, they are much steeper in the vicinity off = 05, where pH 5 pK, for the trifunctional one. for the monofunctional acid while pH w l/3 [p(K,K,K,)] Because values of A(pH) taken from plotted titration curves, such as the one in Fig. 3, are subject to fairly large graphical errors, those quoted below were obtained from numerical calculations made with a digital computer. The titration curve is described by the equation
(l/j) f=
i i=l
4)CH~A- C(H+)- (OH-)1
Cn+ + r [(H+) - (OH-)]
(1)
’
wherefis proportional to the amount of titrant added and is equal to 1.00 at the last (jth) equivalence point. j is the number of acidic sites, CHjAis the initial formal concentration of H,A. I’is the dilution parameter defined as the ratio of CnjA to the formal concentration of the titrant (which is assumed to be monofunctional), and ai is the fraction of the total A that is present as the ith species. Equation (1) is combined with the familiar expression for xi xi = [H+]‘+
ith term [H+]‘-‘K, + . . . + K,K2...Kj
’
to obtain an implicit relationship between [H ‘1 and 1: This was solved at f = 0.25 and 0.75 and A(pH) was obtained from the resulting values. Figures 4 and 5b are drawn for acids of different j but with (K,K,...Kj)' ‘j always equal to lo-‘. For a monobasic acid this value would correspond to the maximum on a plot of A(pH) against PK,~,~. As Ki/Ki+ I decreases the titration curve becomes flatter and A(pH) decreases, but if Ki/Ki+ 1 is as small as lo-’ a further decrease in it has no perceptible further effect on A(pH), which is also insensitive to variations of I’ in this range of acid strength. This limiting value of A(pH) is plotted against j in Fig. 4. and may be shown to be proportional to l/j. In the following paragraphs an algebraic proof of this proportionality is given. Figure 5 shows how the limiting value of A(pH) depends on (l/j)XpK, which is equal to the negative logarithm of the geometric mean dissociation constant. and on the concentration of the acid titrated.
BRWE
H. CAMPBELL~~~ LOUISMEITES
I
I,
0.5
4
0
I
I 6
I
I
I
3
I
I
1
0
Fig. 4. Dependence of the limiting value (see text) of A(pH) on j for the titration of a polyfunctional acid with a monofunctional strong base with (K,K2.. .K,) 1 ’ = IO-‘. Values Indistinguishable kom those shown here are obtained whenever C: 2 IO-‘M and K, K,_, I IO-’
0.6 ‘1
(I/J)IpK
h
Fig. 5. Dependence of A(pH) on (a) (I/j)XpK, for (0) a difunctional acid with K,/K2 = IO- ’ and (A) a trifunctional acid with K,/K2 = K,/K, = IO-‘, and(h) the concentration of a trifunctional acid with pK, = 9. pK, = 7, and pK3 = 5. with r = 2.
AC&base
trtration
curves
121
If dilution. dissociation of H iA. and dissociation of water are all neglected, equation (1) becomes f=
(I.ij) i i=l
iz,.
We define a parameter 111,as equal to KJK, + , and assume all the lili to be very small; A(pH) is then independent of 177. Under these conditions only the species H,A and AJ-’ are important and therefore every x except CQ,and x, is equal to zero. Of these, the former does not appear in equation (3) because the index has 1 rather than 0 as its smallest value. Hence every term in equation (3) is either zero or excluded, with the sole exception of x,. and equation (3) can therefore be written as ,f’= i(jg,) = 2,.
(4)
KIK2K,...Kj 'I= [H+]j+ K,K2K,...Ki'
(5)
equation (4) may now be written as K,K,K3...KJ .f= [H+]‘+ K,K,K,...Ki' From the definition above of
(6)
177~
K, = K,/n1,
(7)
K, = K,,i1772 = K,/(777,771,)
(8)
Ki = K,
1-I n(l/777,),
(9)
i=l
and. upon substitution into equation (6). K 1's 1/771i i=l
f=
j-
[H ‘1’ + K 1n
(10)
1
l/rrZi
which can be solved for [H ‘1 giving
:1 -m-t n
(11)
hence pH = pK, - flog[(l
-_f-)/” - (l/j)log
(12)
To examine the dependence of A(pH) upon j.f can be set equal to 0.75 and then to 0.25. The difference between the resulting pH-values gives A(pH) = (1 ‘j)log9 = 0.954/j
(13)
BRUCEH. CAMPBELLand Louts MEITES
122
The corresponding titration curve has only one point of maximum slope (see Fig. 3). no matter how many protons are on the original acid. In the region 0 c f c 1 the curve is symmetrical aroundf = 0.5 and in this interval the sign of the second derivative changes only once. Because the value of (log 9)/j can be thought of as a limit of A(pH) as 111goes to zero. it can also be used in general to aid in developing an empirical formulation of the dependence of A(pH) upon m. For a difunctional acid having tn < 5, Meites’ gave the empirical equation A(pH) = O-478 + O-253 rn) + OGO741~. The intercept may be compared with the theoretical value of 0.4772. It is suggested that the general form for a j-functional acid is A(pH) - (l/j)log 9 = a& + bm + . . . , and is valid as long as nz is not too much greater than the statistical value.
It is not possible to apply a similar treatment to polyfunctional acids for which Ki is larger than Ki+i* This is because all the terms in u are then important and the resulting equations are impossible to solve in a closed form. One last aspect of equation (13) is that acids of the type considered here could be used to prepare buffers of much higher buffer capacity than those that can be made from monofunctional ones or the familiar polyfunctional ones. Acknowledgement-We wish to express our thanks to the National Science Foundation and the Eastman Kodak Company for the departmental grants that made possible the purchase of the PDPI/I computer used in this study. REFERENCES 1. 2. 3. 4.
P. E. Sturrock, J. Chem. Educ., 1968,45,268. L. Meites, ibid., 1972,49,682, 850. C. L. Chakrabarti. Anal. Chem., 1969.41, 1441. A. Lomax and A. J. Bard. J. Chrrn. Edrrc.. 1968. 45. 621 Form der potentiometrischen &ire-Basen-Titrationskurven bei der Neutralisation polyfunktioneller Sauren oder Basen. bei dcncnJede individuelle Dissoziationskonstante kleiner ist als die der nachfolgenden Stufe. wird untersucht. Im Bereich 0 < j < I (wo f’der zugegebene Bruchteil des aquivalenten Reagensvolumens ist) nimmt die Steigung der Titrationskurve ab, wenn die Anzahlj saurer oder basischer Funktionen zunimmt. Die untere Grenze fur die Differenz der pH-Werte beif = 0,75 undf= 0,25 betragt (l/j) log 9. Zusammenfassung-Die
R&me-On examine Ies formes des courbes de titrage potentiometrique acid+base dans les neutralisations d’acides ou bases polyfonctlonnels pour lesquels chaque constante de dissociation successive est plus petite que la suivante. Dans la region 0 < f’< I (oh fest la fraction du volume equivalent de reactif qui a ite aJoutee). la pente de la courbe de titrage decroit quand le nombre j despositionsacidesou basiquescroit. Ladifferenceentre les valeurs de pH aj’ = 0.75 etf = 0.25 a (l/j) log 9 pour limite infirieure.