D. A. Jenkins and J. L. Latham Harris College Preston, England
Estimation of Some
/
Journal o f Chemical Education
and K, by
Potentiometric Titration
Although a great deal of work has been published (1) on the analysis of potentiometric titration curves, much of it has been particularly concerned with the location of inflection points relative to equivalence points, rather than with the measurement of equilibrium constants. A particularly comprehensive theoretical treatment of acid-base potentiometric titrations is given by Ricci (2). Recently, Gold and Lowe (3) made a precise determination of the ionic product of deuterium oxide from a complete analysis of potentiometric titration curves. Their work, which allowed for the variation of activity coefficients with ionic strength, required the use of a digital computer. However, it does not seem to be generally realized that if ionic strength terms are assumed to be constant, a simple equation can be derived relating the emf change in the region of the equivalence point to ionization constants and solubility products for reactions in which all ions participating in the equilibrium are isovalent. Two cases are considered in this paper:
82
Ki
(a) Titrations involving strong electrolytes, where the system is determined by a single equilibrium constant, such as the ionic product of water or the solubility products of certain salts (examples are the neutralization of a strong acid with a strong base and the precipitation reaction between silver ions and halide ions) ; and (b) Titrations of a weak electrolyte with a strong electrolyte, where the system is defined by two equilibrium constants, namely the ionic product of water and an ionization constant (an example is the titration of a weak acid with a strong base). Titration of a Strong Monobasic Acid with a Strong Base
-
Consider the reaction represented by the equation H+X-
+ M+ OH-
M+X-
+ HIO,
and assume that the change of volume of the solution during titration is negligible. Let 6 = initial concentration of H + and X-, c = concentration of MOH added at any stage, and d =
concentration of H+ at this stage. Then by the condition of electrical neutrality: [MC1 [H+l = [OH-] [X-I (1) i.e.,
+
c
+
+d
=
(K&)
+b
(2)
where K , is the stoichiometric (classical) ionic product of water. If an electrode revenible to hydrogen ions is immersed in the solution, its electrode potential, E, varies during the titration according to the familiar relation:
Let El and Gz be respectively the electrode potentials when the quantity of base added is equivalent to 90 and 110% of the quantity of acid originally present and let dl and dz be the concentrations of hydrogen ions at these two points. Assuming that the activity coefficient of the hydrogen ion is constant within this range of titration, 8, -
EZ
= -A& =
RT d -In2 5 dl
(5)
stage. By the conservation of the chemical grouping Y it follows that [HY]
+ [Y-I [HY]
=
b
=
b -f
(7)
The stoichiometric ionization constant of HY is given by
Substituting equation (7) in equation (8) gives
By the condition of electrical neutrality: IM+l [H+l = [OH-] [Y-I (10) At the equivalence point the solution is alkaline and [H+] may be ignored compared with the remaining terns. Hence, in the region of the equivalence point, equation (10) approximates to
+
+
Substituting for f from equation (9) gives
Solving equation (3) for dl and by takimg c~ = 9b/lO and cz = llb/10, and taking positive roots gives
+ d b z + 400 K,)/20 ( - b + d b z + 400 ~ . ) / 2 0
Solving this quadratic in d and taking the positive root gives
dl = (b dl
=
Substituting for dl and dz in equation (5) gives bZe-A&3/RT Kw = 100(e-d65/RT - 1)1
This equation may be simplified if the solutions of strong acid and base are not too dilute and if unity may be ignored in comparison with e-&$jRT. The result is
Let 8, and d l be the electrode potential and hydrogen ion concentration respectively, when the acid is 90% neutralized, and Gz and dz be the corresponding values when alkali is present in 10% excess. Assuming that the activity coefficient of H+ is constant within this range of neutralization,
- &a
I
If the titration is being followed using a pH meter, this equation may be conveniently written as pK, = 2 - 2 log b
+ A(pH)
(6)
It should be noted that this equation, suitably transformed, is identical with the corresponding equation derived by Ricci (4). The same analysis applies to potentiometric titration curves obtained in the titration of silver ions by halide ions, using a silver indicator electrode. In this case the ionic product of water in neutralization titrations is replaced by the solubility product of the precipitated salt. Titration of a Weak Mobasic Acid with a Strong Base
-
Consider the reaction represented by the equation HY + M+OH- M+YHz0
+
and again assume that the change of volume of the solution during titration is negligible. Let b = initial concentration of HY, c = concentration of MOH added at any stage, d = concentration of H+ at this stage, and f = concentration of Y- at this
=
-A&
=
( R T / 3 )In(dl/dn)
Inserting values for dl and d. from equation (11) into equation (12) and solving the quadratic equation for K J K , it follows that
where
If y >> 9, i.e., d , / d z >> 11, this simplifies to TbK. K
=
+ 5 d 1 2 1 r 2 + 4 = 110y
(5)(11y)
In this case
log b - pK,
+ pK,
pK.
=
pK,
=
lag 90 - A&3/2.303RT
+ log b - log 90 + A&3/2.303 RT
Volume 43, Number 2, February 1966
/
(13)
83
If the titration is followed by measuring change of pH, then in aqueous solution at 25OC, equation (13) simplifies to pK. = 12.05
+ lag b - A(pH)
(14)
Comparison of this equation with the corresponding equation of Ricci (5) shows a discrepancy. Ricci's equation suitably transformed, would correspond to a pK value 0.050 units lower than that calculated from equation (13). It should he noted that although equations (6) and (14) give reasonably accurate results for pK (see Experimental Results, below), they involve the following simplifying assumptions: (a) The mathematical derivations are based on equations (I), (2), and (5). Equation (1) is exact; equation (5) is accurate at constant ionic strength. However, the use of the classical constant K, in equation (2) involves the approximation that for the hydroxyl ions, activity may he replaced by concentration. (b) The simple mathematical forms of equations (6) and (13) are due to the approximation that e-AES'RT>> 1. This condition is satisfied in the titration of acids of ionization constant greater than 0 in which case -A& > 0.2 volt. Phenol is an example of an acid whose ionization constant (lo-'? is too low to be estimated by this method. (c) I n the experimental determination of pK values, A& (the change in potential of the indicator electrode) is assumed to equal the magnitude of the change in emf of the experimental cell. This implies that the reference electrode and liquid junction potentials have constant values. The experimental determination of A& does not require the exact location of the equivalence point since the slopes of the potentiometric titration curve are small and approximately equal a t the 90 and 110% points. It has been shown by Roller (6) and Meites and Goldman (1) that the difference between the equivalence point and the point of inflection is too small to be detected in the cases considered in the paper. Consequently the point of inflection is taken as the 100% point and from i t the 90 and 110% points may be measured. Experimental Results
Equations (6) and (14) were checked by studying the titrations listed in the table below. The titrations were carried out at 25.0°C + O.l0C using solutions mainly prepared from general purpose grade reagents. I n all cases 50 ml of the reactant -A&
Reactant M/41 HCl M/41 HCl M/41 KBr M/41 K I M/41 CHsCOOH M/41 CHaCHzCOOH M/41 n-CHrCH&H&OOH MI410 C.H.COOH
-
84
Titrant M NaOH M A~NOI M A~NOI M AgNOt M NaOH M NaOH M NaOH M/10 NaOR
Pye No. 11126 alkali glass electrode.
/ Journal of Chemical Education
(volt) 0.516 0.264 0.406 0.620 0.336 0.327 0.330 0.306
(M/10 or M/100) was diluted with 150 ml of water and titrated with M/10 or M/100 titrant. In should be noted that the value of the concentration of the reactant used in the above tahle has been corrected for the change in volume during titration; the value quoted corresponds to that a t the equivalence point. The reference electrode was a saturated calomel electrode incorporating a salt bridge of saturated potassium chloride or ammonium nitrate solution. The silver/silver halide indicator electrodes were prepared electrolytically accordmg to the procedure adopted by Hornibrook, Janz, and Gordon (8). The use of a hydrogen electrode to follow the titration of benzoic acid consistently gave an emf change which was approximately 5% less than the value correspondmg to the accepted pK value of henzoic acid. Similar results were obtained with electrodes on which the minimum of platinization had been made. A glass indicator electrode was found to give a much improved value for the emf change. The unsatisfactory behavior of the hydrogen electrode in this instance may be due to the partial reduction of the benzoic acid at the electrode (9). Examination of the tahle shows that equations (6) and (14) may be used to obtain pK values to within 0.1 unit of the accepted value for most of the examples examined. It is therefore suggested that this method is specially suitable for the rapid determination of the pK value of a weak monohasic acid. Acknowledgment. The authors would like to express their thanks to Mrs. H. Mary Latham for the mathematical derivations of equations (6) and (14). Literature Cited
(1) HIBBITS,J. PHERSON,
o., J. CHEM.EDUC.,35, 201 (1958); CHRISTOH. L., J. CHEM.EDUC.,40, 63 (1963); BUTLER,
J. N., J. CEBM.E ~ u c . ,40, 66 (1963); MEITES,L.,
AND
GOLDMAN, J. A,, Anal. Chim. Acta, 29, 472 (1963); STOKES,R. H., Au~tmliimJ. Chem., 16, 759 (1963). (2) RICCI, J. E., "Hydrogen Ion Concentratian," Princeton University Press, Princeton, New Jersey, 1952. (3) GOLD,V., AND LOWE,B. M., Proc. C h m . Soc., 140, May (1963). (4) RICCI,J. E., op. d l . , p. 140, equation 12. (5) R~ccr,J. E., op. eit., p. 161, equation 43. P. S., J. Am. Chem. Sac., 50, l(1928). (6) ROLLER, ( 7 ) PARSONS,R., "Handbook of Electrochemical Constants," Butterworth Scientific Publications, London, 1959. W. J., JANZ,G. J., AND GORDON, A. R., J. (8) HORNIBROOK, Am. C h . Soc., 64,513 (1942); JANZ, G. J., AND GORDON, A. R., J. Am. Chem. Soc., 65,218 (1943). D. A,, BELCRER,D., AND SHEDLOVSKY, T., (9) MACINNES, J. Am. Chem. Soc., 60, 1094 (1938); KOLTHOFF, I. M., AND LAITENEN, H. A,, '(pH and Electrc-Titrations," John Wiley& Sons, New York, 1941, p. 91. PK,, P&, or PK. Determined Accepted (7) 13.9 14.01 9.7 9.75 12.3 12.1 15.7 16.0 4.76 4.76 4.91 4.87 4.82 4.86 4.27 4.20
Indicator Electrode GlassAg/AgCI Ag/AgBr &/&I Pt/& (1 atm) Pt/H1 (1 atm) Pt/H2 (1 atm) Glass.
