Linear Titration Plot for Resolving Mixtures of Weak Acids by Potentiometric Titration Colin McCallum and Derek Midgley" Department of Chemistry, The University, Southampton, England, and Cenfrai Electricity Research Laboratories, Kelvin Avenue, Leatherhead, Surrey, England
A linear titration plot method of resolving mixtures of two polybasic weak acids in solution by means of a potentiometric titration is described. Corrections for activity coefficients can be applled by an iterative procedure. The effects on the plot of errors in pH and in the acid equilibrium constants are dlscussed, together with ways of optimizing the calculation.
Several schemes have been proposed for the determination of mixtures of weak acids by potentiometric titration. Methods based on general error minimization techniques ( I , 2) are capable of wide application, but require a great deal of calculation.'Simpler methods (3,4) have been restricted to monobasic acids. Usually (2-4) no account has been taken of the need for activity coefficients, beyond modifying the experimental procedure to keep the ionic strength effectively constant. All the methods need accurately known dissociation constants and pH values and some (1,3) also require the total acidity to be known. Suggestions ( 1 , Z ) that i t is not always necessary to know the dissociation constants accurately are misleading to the extent that when that is so, i.e., when the acids are of very different strengths, the analysis can often be accomplished by much simpler methods such as Gran plots ( 5 )or even by taking the points of inflection of a pH-degree of titration curve. Apart from the additional complication of separately determining the total acidity, it is not always easy to do this accurately; Ricci (6) and Midgley and McCallum (7) have discussed some of the limitations for pH-volume of titrant curves and Gran plots, respectively. A good method should be capable of handling polybasic acids and of taking activity coefficients into account, while keeping calculation to a minimum. I t is desirable that one procedure encompass all the calculations, without needing separate determination of the total acidity; and it is a great advantage if the method is capable of graphical representation that will indicate what sort of systematic errors are present and possibly allow a better approximation to be made by selective use of the graph. A method with many of these qualities has been devised for binary mixtures of strong and polybasic weak acids ( 8 ) , but a separate determination of the total acidity was needed. The present method is more general, since it deals with mixtures of polybasic weak acids and does not use the total acidity. Activity coefficients can be calculated iteratively and graphical solutions are also possible. We have examined the method in the light of its susceptibility to errors in pH and in the acid dissociation constants and to the neglect of activity coefficients.
THEORY For convenience, charges have been omitted from ionic symbols. V is the volume of base added, V , that equivalent to the first acid, V, that equivalent to the second, and V , that equivalent to the total acidity. K, is the thermodynamic autoprotolysis constant for water and [OH] is given by K,/(H]fl, { H) being the experimentally observed hydrogen ion activity and f l the univalent ion activity coefficient. V , ml of a solution of a mixture of an acid H,XP, of con1232
ANALYTICAL CHEMISTRY, VOL. 48, NO. 8 , JULY 1976
centration C, (mol/l.) and an acid HhYQ, of concentration C y (molh.) are titrated with a strong base BOH of concentration CB (mol/l.). P and Q are univalent ions of the charge sign necessary to maintain electrical neutrality, h and j are the number of protons on each acid removable by titration, x and y are the negative charges on X and Y respectively, p = Ij - xl and q = Ih - yI. The ith association constants of each of the two acids, which have m and n dissociation steps respectively, are as follows:
K
= [HLXIFL/~Hl"XlF,
(1)
k, = [HtYlfi/(HILIYlfy
(2)
where F, and fy are the activity coefficients of X and Y and F, and fLare the activity coefficients of their respective 8th protonated forms. Titrations at constant ionic strength are treated by setting all activity coefficients equal to unity and using stoichiometric equilibrium constants. At any point in the titration, the mass halance equations are as follows: Total X,
+ . . . + [H,X] + ~.K,{HPF,/FA= [Xl(1+ P x ) (3) T, = [Y] + [HY] + . . . + [H,Y] + . . . + [HnY] = [Yl(1+ 2ki(HlLfy/fL)= Y (1+ P y ) (4)
T, = [XI + [HX] + . . . + [H,X] = [XIU
Total Y,
+
TB = [B] = CB.V/(V, V )
Total base,
(5)
From the equivalence relationships,
+ V) h*T, = CB-V,/(V , + V ) j.T, = CB.V,/(V,
(6)
(7)
If charges Zp and ZQ (equal to f l as appropriate) are assigned to P and Q respectively, then Z P P l = (x - j ) T x ZQ[QI = (Y
(8)
- h)T,
(9)
The charge balance equation is:
[B] + [HI+ Zp[P] + Z Q[Q] = [OH] + x [XI + (X - 1) [HX] + . . . + (x - L)[H,X] ( X - m)[H,X] y[Y] ( y - 1)[HY] . . ( y - i)[H,Y] + . + (Y - n)[H,YI (10)
+ + +. +
+
+
Substituting for [B], Zp[P], ZQ[Q] and all forms [H,X] and [H,Y] gives T B + [HI - 0' - x ) T , = [OH]
- (h - y ) T ,
+ x.T, - [X]OC,+ y*Ty- [Y].iY
In Equations 3 , 4 , and 11,p,, fly,
01,
and my are defined by
E K{HI'F,/F, 6, = F h,(WLf,/f,
Px =
1
(11)
(12)
(13)
0
ux
50
Figure 1. Effect of errors in association constants on plot for computer-generated titration of 10 ml of 0.01 M acid ”X” (K,= lo5 I./mol) 90 ml of 0.01 M acid “Y” ( k i = lo3 I./mol) with 0.1 M NaOH.
+
= Akj = 0; (x) AKj = -IO%, Akj = +IO%: (O)AKi = +IO%, Akj = - l O % ; ( O ) A K j = A k l = - l O % ; ( 0 ) A K 1 = A k l =+IO%;(-)trueplot.
(+) AKi
Some overlapping points near the origin are omitted for clarity. ux
m
a, =
1
i*K,{HJLFX/FI
(14)
n
my
=
E1 & { H ) i f y / f z
(15)
Substituting for [XI and [Y] in Equation 11, followed by rearrangement, gives:
([HI - [OH] + TB)/G- a,/(l + P x ) ) = Tx + T y [ h- a y / ( l+ Py)]/b - d ( 1 + &)I
(16)
Expressing [HI and [OH]in terms of the experimental quantity { H )and substituting for T,, Ty and T B gives:
({H)- & / ( H ) ) / f i + CB*V/(V,+ V ) j - aJ(1+ Px)
+
which is of the form y = a, a1.x and therefore a plot of U y against U, gives a straight line of intercept V,/j and slope V y / h ,where U, and U, are calculated from the known equilibrium constants K,, K,, and hi and the experimental data V,, V , CB and ( H )in Equations 19 and 20.
uy= ( V , + V ) W )- Kw/IHI)/fi + CB-V CBU
- ax/(1 -t P x ) ]
(19)
The formula is quite general, e.g., for a strong acid m = 1, K1 = 0; for an amino acid such as glycine, m = 2, x = j = 1, while for its hydrochloride m = j = 2, x = 1. Equations 21 and 22, analogous to Equation 18, can equally well be derived in terms of V , and V , or Ve and V,. (V,
+ V)(lHI- K,,>/lHi)/fi+ CwV
Flgure 2. Effect of neglecting activity coefficients on plots for com10 puter-generated titrations of ( A ) 90 ml of 0.01 M strong acid “X” ml of 0.01 M weak acid “Y”; ( B ) 10 ml of 0.01 M strong acid “X” 90 ml of 0.01 M weak acid “Y” with 0.1 M NaOH
+ +
Kj = 0; k i = (x)
IO6 I /mol, (0)IO2 I./mol. (-)true plots
( V , + V)({HI- K,/(Hl)/fi CBL - a x / ( l + P,)l
=-+-. V, Vy
+ CB*V
h.a,/(l
+ P x ) - j-ay/U + By)
(22)
j h-j j - a,/(l + P,) Effect of Errors in the Association Constants. Calculations were carried out on theoretically generated data representing the titration of 100-ml portions of 9:l and 1:9 mixtures of weak acids at a total acidity of 0.01 M with 0.1 M strong base. The following combinations of association constants (IJmoI) were used: 105/103,105/104,105/109.The apparent equivalent volumes indicated by the least squares fit of Equation 18 were obtained for all possible combinations of f10% errors in the constants. Although an erroneous association constant can cause large changes in the coordinates corresponding to each titration point, the effect on the slope and intercept of the plot is much smaller. This is shown in Figure 1 for the worst of all the cases we tested, the titration of a mixture of acids with association constants of lo5I./mol (1oOh)as “ X ” and IO3 l./mol(90%) as “Y”; the effect could only be made discernible on the graph by adopting the least accurate way of performing the calculations. The bias in the results is smaller if the error occurs in the smaller of the two association constants and if the acid with the inaccurate constant is the less concentrated, while it diminishes as the accurately known constant increases. Whatever the combination of association constants and concentrations, the bias is minimized by entering the stronger acid as the first, i.e., “X”, acid in Equation 18. In the examples tested as above, the bias given by the least squares straight line through ten approximately evenly spaced points was 2 4 % in the major component and the total acidity. Selection of points to give more accurate answers was possible in all cases. Effect of Neglecting Activity Coefficients. If activity coefficients are neglected, the calculations can be considerably simplified at the expense of some loss in precision and accuracy. The size of the errors depends on the strengths and relative proportions of the two acids, on the chemical structure of the acids, e.g., for a carboxylic acid with h = y = n = 1, acANALYTICAL CHEMISTRY, VOL. 48, NO. 8 , JULY 1976
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UX
Figure 3. Effect of a bias in pH on plots for computer-generatedtitrations of ( A )90 ml of 0.01M acid “X” (K1 = lo3 I./mol) 10 ml of 0.01M acid “Y” ( k , = lo5 IJmoI), ( B ) 10 ml of 0.01 M acid “X” (K1 = lo3 I./mol) 90 ml of 0.01 M acid “Y” (kl = lo5 I./mol) with 0.1 M NaOH
+
+
ApH = (+) 0.01, (0) -0.01, (0) 0.001,(-)true
plots
tivity coefficients appear in the CY and /3 terms in the ratio f l / f o while for ammonium ion ( h = n = 1, y = 0) the reciprocal ratio holds, and on the assignment as “X” and “Y” acids in Equation 18. Calculations with computer-generated data show that for 9:l and 1:9 mixtures (total acidity 0.01 M) of monocarboxylic acids with association constants of lo3 and lo5 l./mol, the advantage lies with assigning the stronger acid as “X”. Whichever way the assignment is made, the error and standard deviation of determining the weaker acid is the greater, but it is much smaller if determined as “Y”. V, was always within two standard deviations of the true value, but the error in V , could be greater than two standard deviations if V, > V,. The standard deviations were calculated from the least squares fit of the coordinates. For the case of mixtures of a strong acid (“X”)and a weak monobasic acid (“Y”) in the same conditions as above, the errors were much larger and greater than two standard deviations except when the weak acid was the major component. If the weak acid was itself moderately strong, e.g., lzl = lo2 l./mol, the errors were always large. Neglect of activity coefficients cannot be recommended if one of the components is a strong acid. The effect is illustrated in Figure 2 for mixtures of strong and weak acids, but could not be demonstrated graphically in the case of two weak acids. Effect of a Bias in pH. A constant error in pH produces errors in the equivalent volumes, the magnitudes depending on the association constants of the acids, on which acid is the major component, on the assignment as “X” and “Y” acids, and on the error in pH itself. Positive errors in pH lead to an overestimate of the weaker acid and an underestimate of the stronger. A negative error in pH produces equal deviations in the opposite sense. The precision and accuracy are much better if the stronger acid is “X” in Equation 18 and if the weaker acid is the major component. Calculations with computer-generated data show that for 9:l and 1:9 mixtures (total acidity 0.01 M) of monocarboxylic acids with association constants of lo3 and IO5 l./mol, a systematic error of 0.001 pH unit produces errors of a t most 2% in the equivalent volume 1234
ANALYTICAL CHEMISTRY, VOL. 48, NO. 8, JULY 1976
(of the weaker acid as the minor constituent) when the least squares fit is made over approximately equally spaced points between V = 0 and V = V,. An error of 0.01 pH unit increases the error in the same equivalent volume to about 20%, but in both cases the apparent value is less than two standard deviations from the true one. The errors in the total acidity and in the concentration of the major component are about 1%for a pH error of 0.01 unit. In all cases, the plot was linear over a large enough part of the curve to enable an accurate result to be obtained (Figure 3). For similar theoretical titrations of mixtures of a strong acid and a weak acid, the errors are smaller but the apparent values are more than two standard deviations from the true ones. Figure 3 shows that the deviations from linearity are greatest near the origin and that by plotting only the points away from the origin, the accuracy and precision would be much improved over those obtained from a wide spread of points. I t should be emphasized that random p H errors, while increasing the standard deviation of the fit, have a negligible effect on the accuracy. Combined pH and Association Constant Errors. Errors in pH can occur simultaneously with errors in the association constants, but as the two factors cause deviations in different parts of the curve, accumulation or cancellation of errors in V , and V, is negligible compared with the standard deviation produced by either error alone. Theoretical titrations of 1:9 and 9 1 mixtures of weak acids as in the previous sections were calculated with all possible combinations of f10% error in the association constant of the “Y” acid and f O . O 1 unit error in the pH. In all cases, the error in one of V , or V, was reduced compared with that produced as a result of a single type of error.
CALCULATION The coordinates of the plot can easily be calculated with a desk calculator if activity coefficients are either constant or neglected, but for the most general use, a computer is desirable. We have written a FORTRAN program MIXAC (available on request), which calculates the coordinates in a subroutine DOUBLE using activity coefficients calculated by subroutine ACTCO from the ionic strength obtained by a function subroutine MU. The equivalent volumes are obtained from an unweighted least squares fit of the coordinates by a subroutine LEAST. The program also includes the subroutines GRAN and DM used to resolve mixtures of a strong and a weak acid by another method (8) and which also call ACTCO, MU, and LEAST. Both types of procedure may be carried out on the same run if desired. ACTCO calculates activity coefficients from the equation
+
- log f z = A * ~ ~ [ 1 ~ ’ ~W / (2 1 )0.211 (23) where z is the ionic charge, I the ionic strength, and A the Debye-Huckel coefficient, but other equations could be used instead. When activity coefficients are calculated, the program operates iteratively until both the equivalent volumes differ from their values on the previous cycle by no more than 0.01%. If spurious data result in a negative equivalent volume, iteration stops and a warning message is printed before control is transferred to the next operation. The program prints the total and individual equivalent volumes and their respective standard deviations determined from the least squares f i t of the points. The program allows both acids to have up to six functional groups, but we have not tested it with more than four groups per acid. All forms of acids may be accommodated, e.g., amino acids and amine hydrochlorides as well as carboxylic acids. Since the equilibrium constants are supplied as overall association constants, strong acids are entered with j = x = m = 1 a n d K 1 = 0. Although there are no theoretical limitations on the plot, experimental error restricts the calculations to the use of data
~~
Table I. Results with Experimental Data References Acid X Nitric Nitric “Nitric”c Hydrochloric Hydrochloric Hydrochloric Hydrochloric Tartaric Succinic Malic Tartaric Tartaric
Acid Y Oxalic Oxalic Tartaric Oxalic Citric TADf TAD f Oxalic Lactic Citric Acetic Malic
a
Taken
V,, ml 10.10 1.01 0.00 9.20 2.12 0.2086 0.1313 2.02 7.30 4.05 4.10 4.18
b
Found
V,, ml 1.01 10.10 12.45 9.65 4.59 0.1713 0.1735 10.10
3.28 4.46 3.85 3.86
V,, ml 10.17 1.14 0.02 9.14 2.14 0.2083 0.1508 2.14 7.14 4.05 4.04 4.18
v,, ml 0.89 9.90 12.28 9.72 4.85 0.1717 0.2039 10.11 3.28 4.54 3.84 3.86
a Source of data. Source of association constants. Only tartaric acid present, I = 0.1. e Quinhydrone electrode. f 1, 5,8, 12 tetraazadodecane. g I = 0.04.
from only part of a titration curve, if a given mixture of two acids is to be resolved. This can be seen most easily for the case of two monobasic weak acids (h = j = m = n = x = y = 1)with association constants such that K1 = r-kl, where 0 < r < 1.In Equation 18,as the p H increases, the abscissa tends to unity and the ordinate to V , V, = V,. Since the latter converges the faster, the condition for a plot to resolve the acids is that V, V , ( 1 KI{H]fl)/(l hl(Hjf1) is significantly different from V,. Substituting for K1, this can be expressed as
+
+
+
+
[ V , + V , (1+ r-hl{H)fd/(l+ hi(H]fi)- Ve]/Ve6 - D (24) where D is a positive discrimination factor. Hence (1.
- 1)kl{H]f1/(1+ hl(Hlf1) 6 -D.Ve/Vy
(25)
In order to resolve the mixture, therefore, if 0 = V J V , { H ]2 DB/(1 - 1.
+ D0)kfl
(26)
Experimental data from various sources gives ordinates that deviate by 0.1-1% from the least squares line. A conservative value for D would be in the range 0.003-0.03, depending on the precision with which measurements are made. Graphical Solution. Figures 1-3 show that errors in the parameters of Equation 18 can distort the plot from linearity, which leads to errors in V , and V , calculated by least squares and is indicated by large standard deviations in the equivalent volumes. The shape of the plot can show the origin of the error and good results may be obtainable from a linear portion of the curve. Deviation from linearity near the origin indicates an error in pH: a low p H causes an upturn in the graph and a high pH a downturn. Deviations due to errors in the association constants appear away from the origin and are caused almost wholly by the “Y” acid constants as errors in the “X” acid constants elongate or compress the line but do not cause curvature. An overestimate of the “Y” acid association constant produces an upturn in the line and an underestimate has the opposite effect. In order to facilitate analysis of the results, the program prints the coordinates of each point together with the deviation of the point from the least squares line. Experimental Data. The method was tested with experimental data taken from the literature and with data collected as described previously (8).The results in Table I show good agreement between the values found and the theoretical ones. In most cases the error in the total equivalent volume is less than the error in the individual ones. The apparent large discrepancy in the results for the hydrochloric acid-TAD mixture a t I = 0.1 presumably arises from an error in Hedvig and Powell’s original paper ( 1 4 ) , since their tabulated values for the degree of formation (n) cannot be derived from their quoted concentrations a t ionic strengths of 0.1,0.15, and 0.2,
-
-
but are in good agreement with those calculated usina the concentrations we found. At I = 0.04, there is close agreement on all counts. Auerbach and Smolczyk’s (13) p H data were recalculated using modern ( 1 5 ) values for the standard potentials of the quinhydrone and calomel electrodes. Scope of the Method. The method has’worked well for titrations performed at constant and variable ionic strengths and with both glass and quinhydrone electrodes. Although the program provides for up to six functional groups on each acid, it has not so far been tested experimentally with more than four groups on one acid or with mixtures containing more than bi- and tri-functional acids. Its range is nevertheless far wider than that of graphical methods and the amount of calculation required is less than with error minimization techniques. For mixtures containing a strong acid, however, our earlier method (8) is less prone to error as a result of uncertainties in the pH or equilibrium constants, provided the total acidity can be independently determined. The accuracy obtainable by using literature values for the equilibrium constants is good, but could be improved if the analyst first determined his own values with his own apparatus. Even when a systematic error is present, graphical analysis of the results can indicate the source of error and will often provide a good approximation to the true results. Unless the system is very well defined, calculating the results from only two points can lead to errors, since no check is possible through the shape of the plot or the standard deviations of the calculated equivalent volumes, cf. Ivaska ( 4 ) . I
ACKNOWLEDGMENT We thank G. M. Armitage for making his data available to us. LITERATURE C I T E D (1) N. Purdie, M. B. Tornson, and G. K. Cook, Anal. Chem., 44, 1525 (1972). (2) F. Ingman, A. Johansson. S. Johansson, and R. Karlsson. Anal. Cbim. Acta, 64, 113(1975). (3) A. Frisque and V. W. Meloche, Anal. Chem., 26, 468 (1954). (4) A. Ivaska, Jalanta, 21, 1167 (1974). (5) G. Gran, Analyst (London), 77, 661 (1952). (6) J. E. Ricci, “Hydrugen Ion Concentration”, Princeton University Press, Princeton, N.J., 1952. (7) D. Midgley and C. McCallum, Jalanta, 21, 723 (1974). (8) C. McCallurn and D. Midgley, Anal. Chim. Acta, 78, 171 (1975). (9) R. S. Robinson and R. A. Stokes, “Electrolyte Solutions”, 2nd ed. revised, Butterworths. London, 1965. (10) H. S. Dunsrnore and 0. Midgley, J. Chern. SOC.,Dalton Trans., 64 (1972) (11) G. M. Armitage, unpublishedwork. (12) G. M. Armitage and H. S. Dunsrnore, J. lnorg. Nucl. Chem., 35,8 17 (1973). (13) F. Auerbach and E. Srnolczyk, 2.Phys. Chern., 110, 65 (1924). (14) G. R. Hedvig and H. K. J. Powell, Anal. Chem., 43, 1206 (1971). (15) D. J. G. lves and G. J. Janz, “Reference Electrodes,Theory and Practice”, Academic Press, New York and London, 1961.
RECEIVEDfor review October 24, 1975. Accepted March 10, 1976. ANALYTICAL CHEMISTRY, VOL. 48, NO. 8, JULY 1976
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