James N. Butler
University of British Columbia Vancouver, British Columbia
:ahdating Titration Errors
Calculating titration errors is one of the most important applications of equilibrium principles to analytical chemistry. Unfortunately, most treatments of this problem have been unsatisfactory from one of two points of view. Either approximations were used which were not valid at the equivalence point, or else the equations obtained rigorously were cumbersome to evaluate. I n this paper some explicit expressions for the titration error in acid-base and precipitation titrations will be presented and also a simple method for deciding qualitatively whether or not a given titration is feasible.' The theoretical point in a titration where an amount of titrant exactly equivalent to the amount of substance being titrated has been added is called the equivalence point. The experimental approximation to this, obtained by using a colored indicator or a potentiometric measurement, is called the end point. The differencein volume added between the equivalence point and the end point, divided by the volume added a t the equivalence point, is the titration error. The titration error results from the failure to know the exact equivalence point, and is in addition to any errors resulting from volume measurements and weighing. Symmetrical Precipitation Titration
Note that a t the equivalence point, @ = 1. Eliminating P between (1) and (2), using (3) to express [GI-] in terms of [Ag+],and using (4) to eliminate CV, we obtain the equation of the titration curve (4):
For the purposes of calculating titration error, we can use an approximate relation for the volume factor which applies in the region near the equivalence point. Setting* = 1 in (4), we have:
When (6) is used to evaluate the volume factor in (5) we obtain the equation for the titration error: Titration error = @
- 1 = -(CoC
%)I
[Ag+j - [Ag
For the reverse titration of silver ion with chloride, the signs of the terms on the right hand side of (7) are reversed. At the equivalence point, = 1, and (7) gives
*
[Ag+12 = K.,
As an example of a precipitation titration where an insoluble 1-1 salt is formed consider the titration of chloride with silver ion. The same formalism applies to any 1-1 salt such as Hg2S04,PbSOI, AgBr, AgI, AgSCN, etc. If V od of chloride ion of molar concentration Cois titrated with V ml of silver ion of molar concentration C, the mass balances ( I , $ ) are [CI-]
cove + P = Vf Vo
(1)
[Ag+]
CV +P = v + vo
(2)
where P is the number of moles of AgCl that precipitate Der liter of solution. The only eauilibrium that need be considered in most of the u d a l analytical methods (3) is the solubility product:
(7)
(8)
which is the equation for [Ag+] in a solution of AgCl in pure water. If the concentration of free silver ion at the end point is known, it can be substituted in (7) to obtain the titration error. Emmple. In the Mohr method for chloride determination, 50 with 0,1 AgNOL. As an indiml of 0,1 NaCl is cator, 1.0 ml of 0.1 M K2Cr04is added. If the end paint is taken when the first red precipitate of Ag.CrO, forms, calculate the titration error. The red precipitate will farm when the solubility product of AgsCrO, is exceeded, that is when
[Ag+]'[CrOn']
> 1.1 X
10-'s
when roughly 50 ml of silver nitrate Since [CrO4-] = 1.0 X has been added, the end point will Occur when [Ag+] = 3.3
x
lo-"
The fraction titrated is defined to be the ratio of the number of of titwnt to the added a t the equivalence point:
When this value is substituted in (7), toget,herwith CO= C = 0.1 and K., = 1.8 X 10-lo we get @ - 1 = 5.5 X lo-'. This titration error (0.06%) is half the usual buret reading error (0.1%). In practice, the solubility product at a finite ionic strength will he larger than the value given, and an appreciable amount of the red precipitate must form before it ie visible. Thus the value cslculated representsthe minimum possible titration error.
'This treatment of titration errors will he included in the author's forthcoming hook, "Ionic Equilibrium," which will he published early in 1963 by Addison-Wesley Publishing Co., Reading, Mass.
A popular method for determining the end point in a potentiometric precipitation titration is to choose the point where the slope of the titration curve is maximum. For a symmetrical titration this is the best
[&+I [CI-I = Kso
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Journd of Chemical Education
(3)
way of detern~iningthe end point, since the point of maximum slope theoretically coincides with the equivalence point. This can be seen by differentiating (7) with respect to pAg (-log [Ag+]), which gives for the slope near the equivaleuce point
coc
Slope = dpAg dm = -0.434 ---- ( [ A ~ + ] + A)-' Co C LAP+]
+
The point of maximum slope occurs where the derivative of the slope is zero. By differentiatingthe above expression, we find that at the point of n~aximumslope [.4g+lr = KO.
This is the same as (8), which gave the value of [Ag+l a t the equivalence point. Unsymmetrical Precipitation Titrations
For titrations forming unsymn~etrical salts, MzX, such as silver oxalate or mercurous chloride, the titration curve equations are of slightly different form. For the titration of V,, ml of the divalent ion X of normality Co (molar concentration '/2 Co), with V ml of the monovalent ion M of molar concentrat,ion C, the mass balances are
The equivalence point, on the other hand, is obtained by setting @ = 1 in (13), and is a t [MI'
=
2K.,
(16)
An explicit expression for the titration error from this source is obtained by substituting (15) in (13): Titlation error
=
@ -
C o + C K.,'/I 1 = &-3 ---2 COC
(17)
The positive sign applies for the titration of a divalent ion with a monovalent ion, and the negative sign applies for the titration of a monovalent ion with a divalent ion. This expression should prove useful in pract,ieal potentiometric analysis because the error introduced by taking the end point a t the point of maximunl slope can be corrected for. Ezample. In the titration of 0.005 M Ph++ ion with 0.01 31 IOs- ion, since K.. = 2.6 X 10-IS, the titration error ealeulated from (17) is 1.9ye If the correct end point volume is desired, 1.9y0 must be subtracted from the volume at which the maximum slope occurs.
+
I n applying (17) in practice, it must be remembered that K., is the concentration equilibrium constant, which holds a t the ionic strength of the solution at the equivalence point. The equilibrium is governed by the solubility product [Mla[Xl = Ks,
(11)
Eliminating P between (9) awl (lo), and using (11) and (4), we have the equation for the titration curve
Near the equivalence point, the volume factor can be approximated by (6), and the titration error is given by
Acid-Base Titrations
The titration of a strong acid with a strong or weak base and the titration of a strong base with a strong or weak acid, can all be considered as special cases of the titration of a weak acid with a weak base. We shall derive an expression for the titration error in a weak acid-weak base titration, and then simplify to give the other cases ( 5 , 6 ) . If Vo ml of an acid HA of molar concentration Co is titrated with V ml of a base B of molar concentration C,the mass balances are [HA] + [A-I
For the reverse titration of a monovalent ion with a divalent ion, the signs on the right hand side of (13) are reversed. It is well known that the point of maximum slope in an unsymmetrical titration does not coincide with the equivalence point, but explicit expressions for the titration error from this source are not usually given.l Such an expression is very simple t o derive from (13). Differentiating (13) with respect to pM gives for the slope Slope = @ = -0.434 dm
coc Co ([MI +
+ 4 [MI %)-I
(14)
The point of maximum slope is found by differentiating this expression and setting the result equal to zero, which gives the condition
=
cove V + Vo
There are three equilibria involved in this system:
The final relation required is the charge balance, or electroneutrality condition (I,8 ) : [H+l + [BHf1 = [OH-] [A-I (23) Eliminating [HA]between (20) and (18) gives
+
Eliminating [B] between (22) and (19) gives a J. 0. HIBBITS,J. CHEM.EDZ'C., 35,201
(1058) gives an eexpres sianfor thia titrationerror whleh differs from our result by about a frtctor of two, because he defines titration error in an unconventional way. See H. L. C~ISTOPHERSON, J. CHEMEDUC., 40, 63 (1965) for an alternative derivation which gives the same result as OUT8.
Substituting (21), (24), (25), and (4) in (23) gives the equation for the titration curve: Volume 40, Number 2, February 1963
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67
activity coefficientscan be assumed to be unity, hut for more accurate calculations (if these should become desirable for some reason) an empirical activity coefficient expression such as the Davies equation (7) is useful. I n aqueous solution at 2 5 T the activity coefficient of an ion of charge z is given by
This may he rearranged to give:
- log
This eqnation represents an exact solutiou of the six equations in six unknowns, but is cumbersome to use in this form. For calculating titration error in practical acid-base titrations certaiu simplifying assumptions can he made. If Tve are close to the equivalence point, the volume fact,or can be approximated by (6). If [H+l is small compared to K. yet large compared to K,/Kb. \re can reduce (26) to the simpler form Titration prmr
-1=
=
+ (& coc
- IH+I)
- 1 = --
=
(28)
If K , is very large, we have the titration of a strong acid by a weak base. This titration is usually performed the other may around, so we shall give the expression (from (27) with opposite signs) for the titration of a weak base with a strong acid: Titration error
=
rn -
+ C (lH+l coc
1 = Co
= 9, -
Co + 1 =-
coc
where I is ionic strength. I n converting from pH to [H+]remember that pH = -log [Ht] y e
The Sharpness Index
A convenient quantity to deal with in considering titration errors qualitatively in acid-base titrations is the sharpness index, defined to be the magnitude of the slope of the titration curve at the equivalence point (6) :
By differentiating the expressions derived above for the titration curve in the region near the equivalence point, it is a straightforward matter to obtain explicit expressions for the sharpness index. For the titration of a strong acid with a strong base, differentiation of (30) and application of the equivalence point condition [H+I2= Kn.gives n
=
coc 1 coC O.?li ---- -= 2.17 X 106 co C K,'h Co C
+
+
(:D)
wbere the value of K , a t 25'C has been used. For the titration of a weak acid with a strong base, differentiation of (28) and application of the equivalence point condition @ = 1gives
where [H+]' is the value of [H+] obtained by setting a = 1 in (28). If the equivalence point is more basic than pH = 8, the sharpness index is given by =
(- coc %)"'
0.217 Co f C K ,
For the titration of a weak hase &I1 a strong acid, differentiationof (29) and application of the equivalence point condition @ = 1gives
-
If both K, and Kb are very large, we have the titration of a strong acid with a strong hase, whose titration error is given by Titration error
f
O.R(l!I z2 (1----';v, 0.2 I)
+
which will give the titration error for the titration of a weak acid with a weak base nuder any conditions where a practical end point could he ob~erved.~For the titration of a weak base with a weak acid, the opposite signs are taken for all the terms on the right hand side of (27). By differentiation, it can be shown that the poiut of maximum slope of the titration curve coincides with the equivalence point, so that the maximum slope m~thodcan be used for determining the end ~ o i n t under conditions where indicators are inapplicable. If Kb is very large, we have the titration of a weak acid with a stroug hase, and the titration error is given by Titration error
y =
(&- [H'])
(30)
As in the case of the precipitation titrations, the equilibrium constants to he used are concentration constants applying at the ionic strength of the solution a t the equivalence point. For rough calculations, the
where [H+]' is obtained by setting @ = 1 in (29). If the equivalence point is more acid thau pH = G, the sharpness index is given by
For the titration of a weak acid with a weak base, differentiation of (27) under the assumption that both [H+] and [OH-] are small at the equivalence point gives
wbere [B]' = [ H A ] ' is the concentration of free base or acid a t the equivalence point. Using the mass and charge balances (18) through (23), with C V = CoTro, 68
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Journal o f Chemical Education
and neglecting [H+] and [OH-] compared to [B] and [HA],we can obtain
For the intermediate equivalence point in the titration of a diprotic acid with a strong base, it can be shown by similar methods that under the assumption that [H+] and [OH-] are small compared to [HzA] and [A=] a t the equivalence point, the sharpness index is given approximately by
where Ii,, is the ionization constant of HzA and K.z is the ionization constant of HA-. Equation (39) breaks down when K.,/KsZ is smaller than lo3, hut this is not a situation of practical importance for titrations. Other equivalence points in polyprotic acid-base titrations can be treated analogously. These various expressions for the sharpness index can be easily evaluated when the equilibrium constants are known, and the value of the sharpness index can be used to give a qualitative or rough quantitative estimate of the error in a given titration. With an indicator, using no special methods such as color matching, the pH a t the equivalence point can usually he estimated to within one unit. From (31), this means that if the sharpness index is larger than lo3, the titration error will be less than 0.1%, and the accuracy of the titration will be governed by the buret-
reading errors. An example of such a titration is the titration of acetic acid with 0.1 molar strong base, for which the sharpness index is 2 X lo3. Strong acidstrong base titrations, where C and Co are larger than molar, also fall in this class, provided CO* is excluded from the solutions. Using a pH meter to determine the end point by finding the point of maximum slope in the titration curve, the pH a t the equivalence point can he estimated to within 0.1 units. From (31), this means that if the sharpness index is smaller than 10, even the most precise measurements of the end point pH will result in a titration error greater than 1%. An example of such a titration is the titration of HCN with 0.1 molar strong base, for which the sharpness index is 10. It is hoped that the use of the simple equations derived in this paper will clear up some of the mystery surrounding the calculation of titration error and will make it easier for analytical chemists to decide whether or not a given titration is feasible. Literature Cited
(1) BUTLER, J. N., J . CHEM. EDUC.,38,141 (1961). (2) KINR, E. J., 'LQ~alitative Analysis and ~lectrol$tic Solutions," HareourbBrace, New York, 1959, chap. 4 ff. (3) BUTLER, J. N., J. CHEWEDUC..38,460 (1961). (4) BOLIE,V. W., J. CHEM.EDUC., 35,449 (1958). (5) Rwcr, J. E., "Hydrogen Ian Concentration," Princeton University Press, Princeton, N. J . , 1952. (6) S m m , T. B., "Analytied Processes, A Physico-Chemical Interpretation," 2nd ed., Edward Arnold and Co., 1940. (7) D ~ v m sC. , W., J . Chem. Soe., 1938,2093.
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