available high-speed digital data acquisition systems (22) to the existing minicomputer system. Because of considerations and results such as the foregoing, as well as those arrived at in related studies on frequency multiplexing (23-25), our commitment to digital signal processing in applications of ac measurements to
quantitative kinetic-mechanistic studies is now total. Ac polarographic instrumentation based on analog signal conditioning ( I , 24) is now utilized in our laboratory only for qualitative or semiquantitative survey work.
ACKNOWLEDGMENT
(22) "Model 8100 Transient Recorder," Biomation Corp., Palo Alto, Calif., 1971.
The authors are indebted to S. C. Creason, K . R. Bullock, J. W. Hayes, and A. M. Bond for their interest and helpful suggestions.
123) . , S. C. Creason and D. E. Smith. J. Electroanal. Chem.. 36. A1 (1972); 40, A1 (1972). (24) 6. J. Huebert and D. E. Smith, Anal. Chem., 44, 1179 (1972). (25) S. C. Creason, J. W. Hayes, and D. E. Smith, J. Electroanal. Chem., in press.
Received for review February 5, 1973. Accepted April 12, 1973. Work supported by NSF Grant GP-28748X. D.E.G. was a NASA Graduate Fellow, 1971-1972.
I
Determination of Composition of Mixtures of Weak Acids by Potentiostatic Titration Jouko J. Kankare Department of Chemistry, University of Turku, Turku 50, Finland
A simple linear relationship between the degree of deprotonation of an acid mixture and the mole fractions, and degrees of deprotonation of the components has been derived. I t is shown that the recently developed potentiostatic titration can be successfully employed for determining the degrees of deprotonation and the composition of acid mixtures. The use of the method is illustrated by analyzing binary mixtures of acetic, tartaric, citric, phthalic, and isophthalic acids with an accuracy better than f2%. A similar analysis of the ternary mixture of acetic, tartaric, and citric acids gave inferior results.
Difficulties are often encountered in the analyses of mixtures of organic acids. Gas, liquid, or ion-exchange chromatographic methods are often employed if appropriate conditions and columns can be found. However, the conventional potentiometric titration is seldom possible because of similar strengths of the acids. Recently, Purdie, Tomson, and Cook ( I ) described two methods by which the composition of a two-component mixture of weak acids was determined by pH titration. One method was based on computer analysis of the titration curves and the other was purely empirical, utilizing titration curves measured for mixtures of known compositions. In the former method, a rather complicated mathematical formulation was necessary, because the authors preferred the use of thermodynamic ionization constants. The purpose of this paper is to show that working in constant ionic strength media considerably simplifies the mathematics and that the recently developed potentiostatic titration ( 2 ) is also applicable to the analysis of mixtures of weak acids.
THEORY A general mathematical treatment of the equilibrium system is possible without unduly complicated expres-
sions. Let the mixture contain n acids denoted by HmlA(,,.The following equilibria prevail in the solution:
+
Hmt-,&211- j H + i = l , ...,n ; j = l , ..., m, H+ OHH,O (1) The analytical mole fraction of an m,-basic acid HmLA(,, is denoted by x , , and its cumulative acidity constants by pl,, j = 1, . . . , m,. The total amount of acids in the mixture is a (in moles), and the total volume V. The mixture is assumed to be titrated with a solution of a strong base. Let the added amount of the base be b (in moles). Then we get the following set of equations:
H,,A,,,
+
The mathematical complexity of the protonation equilibria is considerably reduced by introducing a quantity Z, the "degree of deprotonation" ( 2 ) :
(4) Another expression for 2 is obtained from Equation 3
(5) Equations 2 give (6)
Substitution into Equation 5 gives
(1) N. Purdie, M . B. Tomson, and G. K. Cook, Anal. Chem., 44, 1525 (1972). (2) J. J. Kankare. Ana/. Chem., 44, 2376 (1972).
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The quotients in Equation 7 are degrees of deprotonation for the individual acids and thus we may use the notation
In the matrix form this set of equations can be presented as
(,' :>(-\) (s>
m .
1..1 0
Thus, a simple linear relationship is obtained between the degree of deprotonation of the mixture and the mole fractions and degrees of deprotonation of its components:
(18)
=
The inverse of the matrix can be computed by the method of bordering ( 3 ) .Actually, we need only the last column of the inverted matrix, because this column contains the v a h e s of the mole fractions. It is easily shown that
n
z= &z, V=l
J =!
In case of polybasic acids the total number of moles of acids a is not directly measurable. Titration to the inflection point yields the total number of equivalents ii. It is easily seen that the following equation holds: n
si = a
Crn,x,
(10)
1=1
(19)
If a high speed computer is available, the standard deviations of the mole fractions x i can be conveniently calculated by Quenouille's method (4, 5). In this method, an estimate for the variance of x i is given by
Substitution into Equation 4 gives where N denotes the total number of experiments and x i ( y ) the calculated mole fraction of component i obtained when the data containing the results of the uth experiment are omitted.
Equation 8 now becomes (12) The values of Z j - Zmj can be evaluated from the experimental data. Because of the constraint
Ex1= 1
(13)
1-1
a t least n - 1 determinations of Z and [H+] must be made in order to obtain the mole fractions x i . Preferably more than the minimum number of determinations should be performed in order to allow a statistical averaging of the results. In case of binary mixtures, Equations 12 and 13 can be easily solved (see Equation 29) and the standard deviations can be estimated in the usual way, provided that the errors are reasonably small. A more complicated mathematical analysis is necessary when dealing with mixtures of three or more components or with analytical data of low precision. The problem can be most easily treated by matrix algebra. Let us define a matrix (14) where the superscript i refers to the number of the experiment. If the column vector of mole fractions is denoted by x,we obtain a matrix equation cx =0 (15) If the overall number of experiments is greater than n 1, there is usually no solution x for Equation 15. The "best" solution is obtained by seeking the minimum for the positive definite form xTCTCx subject to constraint 13. The problem of conditional extremum can be solved by the method of Lagrangian multipliers, which leads to the set of equations:
"-1
which gives (CTCx), -
-
2
1878
h =0
DETERMINATION OF 2 AND [H+] The conventional method for determining [H+] is to use a glass-electrode p H meter. T o obtain accurate results, rather delicate and expensive instruments are usually required, and the electrodes must be frequently recalibrated using solutions of known hydrogen ion concentration. However, it has been previously shown that no calibration is necessary and potential need not be measured if potentiostatic titration is used ( 2 ) . The pertinent theory is outlined below. By rearranging Equation 11, we obtain
b = EZ - V([H+I - [OH-]) (21) If the additions of acid and base are performed in such a way that the hydrogen ion concentration is kept constant, we see that a simple linear relationship prevails between a and b. A small correction is necessary because the volume Vincreases during the titration. At any point we have v = v , > +T r i -b (22) cn
+
cb
where V , is the initial volume, Ea the normality of the acid mixture, and c b the concentration of the titrant base. Substituting into Equation 21, and assuming that [OH-] is negligible as compared with [H+],we obtain
or with new notations
b=rii+s
(24)
By making alternate additions of a strong base and the mixture under study so that after each addition of the base the hydrogen ion concentrations are equal, and by plotting the base consumption us. acid consumption, a straight line is obtained. The values of [H+] and 2 can be calculated from the slope r and intercepts of this line
i = 1,...,n (17)
(3) V. N. Faddeeva, "Computational Methods of Linear Algebra," Dover Publications. New York, N.Y.. 1959, Chap. 2. ( 4 ) M . H . Quenouille, Biometrika, 43, 353 (1956). (5) R . A . Dammkoehler. J. Bioi. Chem., 241, 1955 (1966).
A N A L Y T I C A L C H E M I S T R Y , VOL. 45, N O . 11, SEPTEMBER 1973
(25) -
Z=
r - s/(Vocn) 1 S/(V,C,)
+
(26)
If the ionic strength of the solution is maintained reasonably constant during the additions by using small aliquots of the titrants and a sufficiently high concentration of an inert electrolyte, Equation 24 is strictly obeyed. At least two points are needed before the values of r and s can bp calculated. As a matter of fact, only one potentiostatic titration is necessary. First, a small amount d o of the acid mixture is added into the titration medium followed by a larger amount d l , and titration is then carried out with the strong base until the potential after the first addition is achieved. Let the consumption of the base be b l . We obtain
(27) This titration may be made using any automatic titrator. However, depending on the instrument, it may be advantageous to reduce the "backlash" of the end-point setting of the titrator by performing one pretitration after the first addition of the acid mixture to a close, but otherwise arbitrary end point. The subsequent titration(s) is (are) run to this end point. An alternative method is to add first a small quantity a, of a strong acid, then a quantity cil of the acid mixture, and titration is run to the same potential as after the first addition. In this case, the hydrogen ion concentration is calculated directly from a,, and 2 is obtained in the obvious way. This method has the drawback that two standard solutions and dispensers are needed. However, it has an advantage not possessed by the other methods. As shown by Equation 12, only the degrees of deprotonation 2, for individual acids are required, and the cumulative acidity constants PI, serve only as interpolation parameters by means of which the values of 2, can be calculated a t given hydrogen ion concentrations. If the values of 2, for pure acids are determined a t the same hydrogen ion concentrations as for the acid mixtures, the extra step of calculating the acidity constants is avoided. The equal hydrogen ion concentrations are easily effected if the adjustments are made by a strong acid. If the acidity constants are to be determined, it is important that their values reproduce the values of 2, as well as possible. The acidity constants may be calculated (2) from the measured values of 2 and [ H f ] by transforming Equation 8 into a linear form and using the method of least squares to solve the resulting set of equations. However, without proper weighting, this calculational procedure does not yield the "best" values for the acidity constants unless very reliable data have been used. A better method is to use a nonlinear least squares program, which minimizes the function
It should be noted that this method has been recommended (6) for calculating the equilibrium constants also in those cases in which the main purpose has not been to obtain a good interpolation formula for the degree of deprotonation. Anderegg in "Coordination Chemistry," Vol. I , W. E. Martell, E d . , Van Nostrand-Reinbold Company, New Y o r k , N . Y . , 1971, D 427.
(6) G.
EXPERIMENTAL Apparatus. A Radiometer TTTl titrator was used for all titrations. In order to improve the stability of the instrument, its voltage regulator tubes were replaced by solid state devices pA723 (Fairchildj with subsidiary circuits. The one-turn potentiometer used for adjusting the end point was replaced by a precision tenturn potentiometer. Line voltage was regulated by a Philips stabilizer. Titrations were performed in a jacketed Metrohm titration vessel thermostated a t 25 f 0.05 "C and equipped with a Metrohm combined glass-silver-silver chloride electrode, an inlet tube for nitrogen, and a magnetic stirring bar. Two Agla micrometer syringes (Burroughs Wellcome & Co.) and a Radiometer ABUl2 automatic buret unit were used for dispensing solutions. With the Agla syringes, volumes could be read with an accuracy of 0.02 pl. The total volume of the ABU12 buret which was used for dispensing the standard base solution was 2.5 ml, and its reading accuracy 1pl. Reagents. The stock solutions of acetic, (+)-tartaric, and citric acids were prepared from Merck analytical grade reagents in distilled water. The solutions with known proportions of acids were made up by mixing the stock solutions. Phthalic acid (E. Merckj was recrystallized from water and isophthalic acid (Fluka) from alcohol. Solutions of these acids were made by weighing and dissolving in 70% (v/v) dioxane-water. Standard solutions of potassium hydroxide and hydrochloric acid were prepared by diluting from Merck ampoules. The titration medium was in all cases a 0.1M solution of Merck analytical grade potassium chloride in distilled water. Procedures. The normality of the mixture whose composition was to be determined was obtained by titrating potentiometrically to the inflection point. A small portion ( 0 . 5 1 ml) of the acid mixture was added from a micrometer syringe to 25 ml of 0.1M KCI. Titration with the standard base was carried out either by recording the curve directly to a potentiometric recorder or by adding the titrant until the end potential estimated from the known acidity constants was reached. Continuous flow of nitrogen was maintained throughout the titrations. The potentiostatic titrations were performed using three slightly different procedures outlined in the theoretical part of this paper. In two of the methods (A and B), the acidity constants were first determined using potentiostatic titration (2). Procedures A and B differed in the method of adjusting the initial hydrogen ion concentration. A small portion (10-300 plj of the acid mixture (A) or standard solution of HC1 (B) was added from the micrometer syringe to 25 ml of 0.1M KCI and the potential was allowed to settle for 5-15 minutes. The end-point setting was adjusted to this potential and 300 to 500 pl of the acid mixture was added. Titration was now conducted to the pre-set end point. In the third method (C), the values of the degree of deprotonation 2 were determined for pure acids and their mixtures using equal initial additions of HCI. In this case, the acidity constants were not evaluated.
RESULTS AND DISCUSSION The binary mixtures of acetic and tartaric acids, tartaric and citric acids, and phthalic and isophthalic acids were selected for testing the analytical method. An attempt was also made to determine the composition of a ternary mixture of acetic, tartaric, and citric acids. The choice of the compounds was arbitrary, although some attention was paid to the possible difficulties encountered in conventional analytical methods. As described in the experimental part, three slightly different procedures were followed in the potentiostatic titrations. In two of the methods, the acidity constants of the components were needed. The pK of acetic acid, 4.562, was obtained from the previous results (2). The three cumulative acidity constants of citric acid were recalculated using the data from the same paper (2) by minimizing the function F (Equation 28). This minimization was carried out using Chandler's (7) efficient direct search program on a Univac 1108 computer. The acidity constants of tartaric, phthalic, and isophthalic acids were Chandler, "Mlnimum of a Function of Several Variables," Program 66.1, Quantum Chemistry Program Exchange, Indiana University. Bloomington, Ind.
( 7 ) J. P.
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Table I . Determination of Composition of Mixtures of Acetic and Tartaric Acids Known mole fraction of acetic acid
0.663 0.663 0.663 0.663 0.663 0.663 0.663 0.663 0.887 0.887 0.887 0.887 0.887
Normality
fa
0.1990 0.1990 0.1990 0.1990 0.1990 0.1990 0.1990
0.368 0.364 0.366 0.366 0.517 0.411 0.336
[H+]M
X
lo4
2.192 2.169 2.177 2.181 0.823 1.632 2.666
...
...
0.1997 0.1997 0.1997 0.1997
0.415 0.297 0.258 0.232
0.662 1.184 1.540 1.843
, . .
...
. . .
, . .
Zlb
ZZh
0.111 0.112 0.112 0.112 0.250 0.144 0.093
1.249 1.253 1.252 1.251 1.570 1.353 1.179
...
. . .
0.293 0.188 0.151 0.129
1.630 1.459 1.372 1.311
Mole fraction of acetic acid found
0.666 0.675 0.672 0.671 0.667 0.666 0.676 0.670f 0.002c 0.868 0.888 0.889 0.892 0.885f 0.005c
'
a The degree of deprotonation of the mixture determined by method A. The degrees of deprotonation of acetic and tartaric acids, respectively, calculated using values pp = 4.562for acetic acid and ppl = 2.820and p p = ~ 6.721 for tartaric acid. The value of mole fraction and its standard error were calculated using Equations 19 and 20.
Table 11. Determination of Composition of a Mixture of Tartaric and Citric Acids Known mole fraction of tartaric acid
0.495 0.495 0.495 0.495 0.495
[H+]M
lo4
'
Normality
ZQ
0.2517 0.2517 0.2517 0.2517
0.534 0.446 0.407 0.380
1.074 2.239 3.062 3.809
1.490 1.242 1.129 1.050
1.217 1.015 0.933 0.875
...
...
...
...
.
.
X
z1
Z2
Mole fraction of tartaric acid found
0.476 0.479 0.477 0.477 0.477 f O.O0lL
*
'The degree of deprotonation of the mixture determined bv method A. The degrees of deprotonation of tartaric and citric acids, respectively, calculated using values pp, = 2.873,p p = ~ 7.258.and pp3 = 12.907for citric acid (for tartaric acid, see Table I ) . The value of mole fraction and its standard error were calculated u s h g Equations 19 and 20.
Table I l l . Determination of Composition of Mixtures of Phthalic and lsophthalic Acids Known mole fraction of phthalic acid
Normality
0.870 0.870 0.692 0.692 0.395 0.395 0.128 0.128
0.2237 0.2237 0.2495 0.2495 0.1860 0.1860 0.1303 0.1303
No of determiMole fraction of nations Methoda phthalic acid foundb
6 6 4 4 4 4 2 2
0
C 0
C 0
C B
C
0.858 f 0.005 0.873 f 0.007 0.691 f 0.009 0.694 f 0.009 0.366 f 0.005 0.374 f 0.004 0.149 0.159
a In method B, values pp, = 2.743 and pp2 = 7.641for phthalic acid and p p ~= 3.425 and p/& = 7.672for isophthalic acid were used. b T h e value of mole fraction and its standard error were calculated using Equations 19 and 20.
determined by potentiostatic titration ( 2 ) . The values of the constants are given in Tables I, 11, and 111. In case of binary mixtures, Equation 12 can be solved for X I : (29)
This equation was applied to calculate the compositions given in Tables I, 11, and 111. It can be easily shown that the right-hand side of Equation 29 has an indefinite value 010 when Z1/Zz = m1frn2. Hence if ml = m2 and all the corresponding acidity constants of the two acids are equal, it is not possible to determine the composition of the mixture (a result which can be also deduced without any mathematics). If ml # m2, it is always possible to find the range of hydrogen ion concentration where the composition of the mixture can be determined. 1880
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Tables I, 11, and I11 show that, in most cases, the estimated compositions are within f 2 % of the actual composition. The somewhat greater deviations a t low proportions of phthalic acid in Table I11 are understandable in view of the rather small differences between the acidity constants of phthalic and isophthalic acids. Rather inaccurate results were obtained for the composition of a ternary mixture of acetic, tartaric, and citric acids. The actual composition (in mole fractions) was 0.330, 0.332, and 0.338, but the analysis gave 0.396 f 0.018, 0.342 f 0.005, and 0.262 f 0.023, respectively. This analysis was carried out by method B using seven different hydrogen ion concentrations. Comparison of the results obtained by methods A, B, and C does not reveal any remarkable differences between the attainable accuracies. From a computational point of view, method C has the attractive feature that the acidity constants need not be evaluated.
CONCLUSIONS This study indicates that potentiostatic titration is a useful method for determining the composition of mixtures of weak acids, unless very high accuracy is required. It must be noted that the results in this study were obtained with rather modest equipment. There is every reason to believe that more accurate results can be obtained, if a more stable null-point detector and more precise dispensers were available. Construction of a precision instrument is in progress in our laboratory.
Received for review January 4, 1973. Accepted February 20, 1973.
45, NO. 1 1 , SEPTEMBER 1973
