TITRATION ERROR IN POTENTIOMETRIC PRECIPITATION TITRATIONS JAMES 0. BIBBITS Aircraft Nuclear Propulsion Department, General Electric Company, Evendale, Ohio
K o m H o m and Furman1 have published mathematical formulations of the titration errors occurring in potentiomctric precipitation titrations together with supporting experimental data. However, their derivations, which are based on the use of equivalent concentrations, become unnecessarily complex when applied to the general case. In addition, the typographical errors which appear in the published version detract from the ut,ility of the material presented. It seems desirable to present derivations based on the use of molar concentrations, as it is this type of concentration that is involved in the Nernst equation. Although some of t,he material presented here duplicates that published by Kolthoff and Furman, it seems worthwhile to have the complete set of derivations appear in one place in the literature. I n many cases, the course of a precipitation titration may be followed potentiometrically and the end point determined by the rapid change in potential per unit volume of precipitant added. It is well known that when the composition of the precipitate is BA, the potentiometric end point and stoichiometric point are identical (assuming a reasonably low solubility, a lack of preferential ionic adsorption, etc.) whereas, when the composition of the precipitate is B2A, or in general, B,A,, the potentiometric end point is not identical with the stoichiomet,ricpoint. Let us consider the simplest case, where B+ is titrated with A- to form BA. Let us follow the course of the titration potentiometrically by measurement of the potential difference between a metallic B electrode and a suitable reference electrode. From the Nernst equation, the potential will change with changing concentration of B+ as follows: En = E'B
+ k In [Bfl
(1)
where 7c and EoBare constants, and [B+] = concentration of B+ in solution, Let us assnme for the sake of simplicity that the titration is carried out at const,ant temperature, that there is no volume change during t,he course of the titration, and that enough A- has been added so that a permanent precipitate has been formed. If we let c = original concentration of Bt, y = amount of A- added, z = molar solubilitv of BA. then
S=Z(C-~+Z)' where S is the solubility product of BA. KOLTHOFF, I. M., AND N. H. FURMAN, "Patentiometrie Titrations," 2nd ed., John W h y & Sons, Inc., New York, 1931, pp. 18-24,
VOLUME 35, NO. 4, APRIL, 1958
Equation (1) becomes EB
=
E'B
+ k In
(C
+ x)
-y
(2)
In actual laboratory titrations of this type, when E is plotted as the ordinate and y as the abscissa, a titration curve is obtained which has a point of inflection. The first derivative a t a point of inflection of this type is a maximum, and the second derivative is zero. Consequently, t o establish the titration error, that is, the difference between the stoichiometric point and the potentiometric end point, let us take the second derivative of E with respert t o y , set it equal to zero, and compare the result with the situation which is known to exist at the stoichiometric point. Differentiation of (2) with respect to y yields:
+
x). Since S , the solubility product, equals s(c - y 21). Further, dc/dy = 0. Apdx/dy = x/(c - y propriate substitution into equation (3, and rearrangement of terms yields:
+
d-. 8 ~dU
-
k e - 11
(-I)
+ 22
Differentiating equation (4) with respect to
!J
yield?,:
Substituting for dsldy and dcldy, equation (5) becomes. after simplification:
Since c - y = S/x - x, and c - ,1 equation (6) becomes
+ 2s = S/s +
2,
Since d2EB/dy2is equal to zero,
because kZO,and S/x point of the titration,
+ s #O.
Therefore, a t the end
S = x' (9) The equality shown in equation (9) also corresponds to the situation a t the stoichiometric point. Therefore, the potentiometric end point and stoichiometric end point are identical, and there is no titration error. Let us next consider the titration of B+ with A--
to form B2A, and again follow the course of the titration potentiometrically. A titration curve similar to that obtained with BA will be obtained, again showing a point of inflection. Again letting e = original concentration of B', v = amount of A-- added, & = mols~.solubility of B8A, then S = z(c - 2y 2dP, where S is the soluhility product of BzA.
+
In this case, the Nernst equation is given by Es = E m e+ k In (C - 2y 22) For simplicity, let F = c - 2y 2x. Then
+
+
If the original concentration of B + is c, then YoTitration error = 0.74 qx.100 e
I t is apparent from equation (23), that the titration error decreases as the value of S decreases, and as the concentration of B increases. Let us finally consider the case of the titration of B+" with A-", or nBh
(10)
(23)
+ mA-" = B.A,
(24)
If we let e =
original concentration of BtR,
y = amount of A-" added, and z = molar solubility of B,A,, then
Substitution into equation (11) of the following values for dF/dy and d2F/dyZ calculated with the aid of the relation S = x ( c - 2y ZX)~,
The Nernst equation is given by
yields:
Again let us obtain the second derivative of En with respect t o y and set it equal to zero. The first derivative of EB is:
+
where S is the solubility product of BA,,.
At the potentiometric end point, d2EB/dy2 = 0. Therefore -c+
2 y + 62 = 0
+
(15)
Since lc and (c - 2y 62) are not equal to zero, equation (15) after substitution and simplification leads to S = 642"
In like manner, the second derivative of E n with respect t o y can be shown to be
(16)
The solubility BzA a t the end point is 2
=
The concentration of calculated from 8
4
= [A--1
(17)
Bf a t the end point can be
= 6423 =
+S[B+]P -
4
Since k, n2, and m are not equal t o zero, and it is highly improbable that n or m will have values such that the denominator is zero,
(18)
or
Substituting for S in equation (29), and simplifying me obtain
[Bf] = 22/S
At the stoichiometric point, S = 4X"
Since the value of X at the stoichiometric point is
where X i s the true molar solubility of B2A and xis the molar solubility arrived a t by solution of the Nernst equation. The titration error, being the difference between the concentrations of B+ a t the potentiometric end voint and the stoichiometric point, is thus
Equation (21) can be simplified to Titration error = 0.74
i/S
(22)
it is immediately apparent that the stoichiometric point and potentiometric end point occur a t different values of y. The titration error for the general case could be formulated more definitely, as was done for B2A, but this seems unnecessary in view of the complexities involved. I n conclusion, it is pointed out that Kolthoff and Furman' have nresented exnerimental data with explanatory figures supporting the derivations given
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