Location of the End Point on Certain Graphical Titration Curves ALOIS LANGER
AND
D. P. STEVENSON, Westinghouse Research Laboratories, East Pittsburgh, Penna.
and we have
The equations for certain types of titration curves are derived and possible methods for graphically finding the end point described. The practical aspects resulting from the analysis are discussed.
i
kaAo
=
+
(4)
kbB,
When B, 2 S/Ao
we have ('40
- P) (Bz - P)
=
s
(5)
and
I
N TITRATIOSS such as polarometric (amperometric)
( I , 6), conductometric, radiometric (4, and others, in which the readings of the end-point indicator are proportional to the concentration of ions or molecules in the solution, graphical methods are usually most convenient for determining the equivalence or end point. Various constructions have been suggested for finding the equivalence point from the titration curve. Some of these methods, that of Majer (6),for example, are difficult to apply because they require the accurate knowledge of certain constants which are in most cases unknown. Others, such as the tangent method ( I ) , are uncertain to apply when the solubility of the product is high or the solutions to be titrated are dilute and therefore the shape of the titration curve starts to deviate strongly from straight lines. In this paper an attempt is made to derive the equations for the ideal titration curves under idealized conditions. An analysis of the curve obtained for the end point in a precipitation reaction indicated the correlation to Majer's solubility method, the approximation of the tangent method, and led to a new graphical construction for determining the end point which should suffer from none of the aforementioned difficulties if the specified conditions can be realized.
- P) + k b ( B z - P)
i = ka (Ao
(6)
Solving Equation 5 for p and substituting in Equation 6, we obtain for i as a function of B, z. = - k. - kb (A0 - Bz) + kb d ( A o - B,)Z 45 (7)
+
+
2
This is the analytic expression for the titration curve (for examples, see Figures 1 and 2). By definition, the equivalence point is given by B, = A0 or i
= (ka
+
kb)
lop3 would make determination from such reactions unsatisfactory, unless K itself were accurately known, so that
Figure 6 represents the titration of potassium chromate in a mixture of acetic acid and hydrochloric acid, at zero potential with 0.01 M lead nitrate. I t is obvious that such a curve can hardly be used to find the end point by the tangent method, since such a procedure would give us values completely misleading, whereas a construction derived from theorem (I) and described in Figure 3, or the construction mentioned previously, still gives reasonable values. These examples lead to the conclusion that the titration curves under certain specified conditions follow closely the theoretically derived shape and that the end point can be found by determining the intersection of the asymptotes. It can be seen in Figure 5 that the sharpness of the transition
ANALYTICAL EDITION
October 15, 1942
0
mI.
2 /E
3
4
5
6
?
FIGURE 6. TITRATION CURVES Titration of 25 ml. of 0 . 0 0 1 M potassium chromate diluted with 70 ml. of 0 . 2 Macetic arid and 5 ml. o f 0 . 2 M hydrochloric acid. Titratedwith 0 . 0 1 M lead nitrate at zero potential.
113
as low and the concentration of the solution as high as possible. The first condition is given by the precipitate itself and the supporting electrolyte. Only the supporting electrolyte can be varied to give the lowest solubility, as, for example, by keeping the pH of the solution in a limited range, as was found in the titration of zinc with o-oxychinoline, by cooling the solution as in the titration of potassium with dipicrylamine, by adding alcohol as in the titration of molybdate with lead nitrate, or by other means. In case of low solubility the volume correction factor is also of a simpler form. If the amount of the given substance is small and the solubility unfavorable, it is necessary to increase the concentration by reducing the volume. Experiments showed that the mercury dropping electrode is very convenient for microtitrations even in volumes less than 0.5 ml. From other investigations it can be said that the given analysis of the end point on certain titration curves has some practical application, besides a theoretical interest. The described method of finding the end point can be used with advantage for solubility determinations from titration curves ; however, only curves can be practically evaluated which depart considerably from straight lines.
Acknowledgment from the initial linear portion of the titration curve t o the hyperbolic portion is not realized experimentally, probably because of peculittr solubility relations caused by the known dependence of solubility on particle size. Probably this relation makes it also necessary that after each addition of the reagent sufficient time (from 1to 5 minutes, depending on the precipitate and nature of the mother liquid) has to be given for the precipitate to mature enough so that a definite solubility equilibrium is established in the solution before the indicator reading is taken again. If the titration is performed too quickly, the curve sometimes departs markedly from that obtained when a longer time is taken after the addition of the reagent. The points on such a rapidly taken curve are often scattered and because of the solubility change during the titration, it deviates from an ideal curve and gives no exact result for the end point with any method. I n order to obtain a predictable titration curve which can be solved for the end point, besides the particle size-solubility consideration, the solubility must not change because of the varying composition of the solution during the titration. For example, in the titration of copper with benzoinoxime in a strong ammonia solution (S), the solubility was changing because some ammonia was expelled by each mixing with the nitrogen during the titration. In some other cases, the soluhility can be affected during the titration by the addition of an alcoholic solution of the reagent, a reagent dissolved in a strong acid or alkali, and so on. The volume change during the titration has to be either negligible or corrected. The correction factor is a complicated function. The indicated simple graphical constructions are limited to a reB“- = AB solid only. However, the stateaction A”+ ment that the intersection of the asymptotes to any titration curve in a precipitation reaction gives the correct end point is general. In many constructions the possible graphical errors must also be considered. The theoretical analysis has shown that the shape of the titration curve depends on the ratio S/Ao and that the tangents can replace the asymptotes as a good approximation if this ratio becomes smaller than approximately 10-3. Obviously, for practical analytical purposes such conditions will be sought that the tangent method can be used for finding the end point. The indicated analysis shows that there are two ways to obtain such conditions, keeping the solubility of precipitate
+
The authors wish to thank E. U. Condon for helpful discussion of several aspects in this work.
Literature Cited J. Am. Chem.
(1) Kolthoff, I. M., and Pan, Y . D.,
Soc., 61, 3402
(1939). (2) (3) (4) (5) (6)
Langer, A,, IND.ENQ.CHEM.,ANAL.ED., 12, 511 (1940). Ibid., 14, 283 (1942). Langer, A., J . Phys. Chem., 45, 639 (1941). Majer, V., Z.Elektrochem., 42, 123 (1936). Schoenflies, A., “Einfuhrung in die analytische Geometrie”, p. 130, Berlin, Julius Springer, 1925.
Alkalimetric Standardization of Iodine Solutions F. L. HAHN’, Casilla 221, Quito, Ecuador
T H E reaction: SO3--
+ H20 +
I1
= SO1--
+ 2HI
may be used to check a volumetric iodine solution by means of a known solution of sodium hydroxide. A solution of pure sodium or potassium sulfite is prepared, which is slightly alkaline to thymolphthalein. With this solution a measured volume of the iodine solution is decolorized; an excess of l or 2 drops of sulfite does not matter. Then the acid is titrated with sodium hydroxide and phenolphthalein. Because of the very small sulfite content, the solution is not buffered and the change is very. sharp a t the end point; no difference was observed in the volume of acid required in several trials. To test the procedure, an approximately 0.1 N solution of oxalic acid was titrated with the sodium hydroxide solution; on the other hand, the same oxalic acid solution was titrated with permanganate, the permanganate was used to check, iodometrically, a thiosulfate solution, and the latter was used to titrate the iodine. RESULTS. 10 ml. of iodine = 9.70 ml. of sodium hydroxide directly, and 9.62, 9.69, and 9.67 ml. average = 9.66 ml. by the indirect way. 1 Present address, Instituto de Quimioa Agricola, Guatemala City, Guatemala.
