Quintus Fernando and John Butcher
of Arizona Tucson, 85721
University
calculation of Titration Error in Precipitation Titrations A graphical method
O n e of the first instances in which a student encounters simultaneous equilibria is either in a discussion of fractional or selective precipitation or in the familiar laboratory exercise for the determination of chloride by the Mohr method. Frequently, however, the student gains only a superficial knowledge of the simple concepts involved and is thus unprepared to deal with the more complicated situations which are presented later. We have found that it is well worth the time to emphasize the early study of simultaneous equilibria. A productive topic in this respect is the calculation of theoretical titration errors in precipitation titrations. The Mohr titration is ideal for this purpose and is preserved as a laboratory exercise in our undergraduate chemistry curriculum since, if for no other reason, it does offer a great deal of insight into calculations involving simultaneous equilibria. The calculation of titration errors in precipitation titrations has been described previou~ly.~-~ In this article, a graphical method is introduced which greatly clarifies the nature of the calculation and illustrates the significance of the various equilibria.
titration errors, it is convenient to define the end point as the point at which the solution becomes saturated with AgzCrOh. Consider the tit,ration of an anion, Z-, with silver nitrate to form an insoluble salt, AgZ. Before and a t the end point, the titration solution is described by two equations: the solubility product: and the electroneutrality equation: [Ag+l
+ Cz-
=
[Z-]
+ Ck+
(2)
Here, C s and CA.+ are the formal concentrations of Z- and Ag+, respectively, in the titration solution; these quantities include the Ag+ and Z- present as the solid AgZ. The stoichiometric fraction of the anion titrated, X, is given by:
and the fractional titration error is:
The Mohr Titration
The Mohr method is a precipitation titration in which silver ion is the titrant. The indicator is chromate ion, and the end point is the point a t which the red oranCre of Ag,Cr04 is observed. For the calculation of
' SMITH,T. B., "Analytical Processes,'' Edward Arnold, London, 1940, p. 148. COETZEE, J. F., in KOLTHOFF, I. M., AND ELVING,P. J., ''Treat,ise on Andytical Chemistry," Part 1, Vol. I, Interscience (division of John Wiley & Sons, Inc.), New York, 1959, p.,?92. BUTLER, J. N., J. CHEM.EDUC. 40, 66 (1963); also Ionic Equilibrium," Addison-Wesley Publishing Ca., Reading, Mass., 1964, p. 187.
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/
Journal of Chemical Edvcotion
When defined in this manner, the titration error is negative before the equivalence point is reached and positive after the equivalence point. Substitution of eqn. (2) into the numerator of eqn. (4) gives:
where [Z-] is, of course, calculated from eqn. (1). Note that neither the silver nor the anion concentration at the equivalence pnint appears in either eqn. (5) or in its derivation.
Given, then, the chromate concentration a t the end point,, the silver ion concentration at that point and, hence, the t.itrat,ion error are readily calculated. The same derivation applies, and the same method of calculation is used, whether the end point occurs before, at, or after the equivalence point. The variation of both terms in eqn. (5) as well as of [CrOs] are readily shown graphically as functions4 of pAg when dilution by the titrant is negligible. For example, consider Figure 1,which represents the solution conditions during a titration of 0.050 F chloride with silver, the indicat,or being 0.005 M c h r ~ m a t e . ~The signs shown on the graph are those of the corresponding terms in eqn. (5). Whether one plots -log [Ag+] and - log [Cl-1, as in the figure, or the negative logarithms of [Ag+]/Ccr and [Cl-]/Car is a matter of personal preference. Both graphs are equally informative, although, in the first case, it is necessary to remember to divide by Ccr to obtain the titration error. The [Cl-] curve in Figure 1 is horizontal until the K., of AgCl is satisfied (i.e., pC1 pAg = 10.00); thereafter, the [Cl-] curve is a straight line plot of the solubilit,yproduct equation:
end point pAg (and thus the titration error) is less sensitive to the concentration of the indicator ion than it would be if that ion were univalent. Another conclusion which can be drawn from eqn. (5) arises from the following considerations: The pAg at the end point is determined solely by [CrOa=]. Further, the largest possible error is obtained when the second term in the error equation is negligibly small (i.e., K., is small). I n that case, and when [Ag+],.d is the usual M, the titration error is:
.*
and is independent of K.,. This is the maximum M. possible overtitration when [CrOa=] = 5 X On the other hand, when K., is large, the first term is negligible and the error becomes large and negative: E
=
- 10"' X K,, X 100 Cz -
+
-log [Cl-I = 10.00 - pAg
with a slope of -1. The [Cr04=]curve is likewise horizontal nnt,il the K,, of silver chromate is reached; t,hen, chromat,e obeys the solubility product equation: -lag [CrO&-I = 11.70 - 2pAg
The end point occurs a t the break in this curve, i.e., at the point a t which Ag,CrO, just begins to precipitate. Thus for the example considered in the figure, pAg at the end point is 4.70; pC1 is 5.30; and the titration error is: PA9
The graph ran also be used to illustrate several other points regarding the RiIohr titration. For example, the equivalence point obviously falls a t the intersection of the silver arid chloride curves, i.e., at the point of zero error. Changing [CrO&-] results in an upward or downward shift of the horizontal segment of the [CrOa=]rurve. Thus, to obtain E = 0, the break in t.he chromate curve must occur at pAg = 5 and [CrO4-1 must be 10-'.'A/T. At higher values of [Cr04=],the titrat~ionerror is negative since IC1-lend
0,
> lAgt1en,
nt
I n Figure 1, changes in Cor likewise affect only the horizontal portion of the [CI-] curve; an effect on the titration error arises, however, from the Ccr term in the denominator of eqn. (5). The -2 slope of the chromate curve reveals that the
' firr.rsN, L. G., in KOLTHOFF,I. M., AND ELVING,P. J., "Treatise "on Analytical Chemistry," Part 1, Val. I, Interscience, (division of John Wiley & Sons, Inc.), New Yark, 1959, p. 302. The scales on the axesin Figure 1are inverted so t h a t thefraction t,it,rated increases toward the right side and concentrations of the various ions increase toward t,he top of the graph. Throughout this discussion cert,ain liberties have been taken with the values of the equilibrium const,ants in order to simplify the mrmerical ealcr~lations. SWIFT,E. H., ef al., .4nal. Chem., 22, 306 (1950).
Figure I. Grophicol calcvlafion of the titrotion error-the Ccr = O.05OF; CE,O,- = O.OOSOF.
Mohr titrotion:
The Volhard Titrotion
The classical procedure of Volhard involves the addition of an excess of silver to a solution of halide, followed by the back titration of the excess silver ion with thiocyanate. The indicator is Fe3+ and the end point is taken a t the first permanent red coloration due to the formation of FeSCN++. I n order to describe the Volhard titration of, e.g., chloride, the following equilibria are considered: Few
+ SCN- e FeSCN++
AgCI 10-2.2" the equation
is:
I n the region where [SCN-] is near 10-2.20,the plot curves as shown in Figure 2. Taking the end point at [FeSCN++] = 10-5.20,the titration error is:
This represents an undertitration of chloride; is., a11 excess of thiocyanate has been added. If, as recommended by S ~ i f tthe , ~ iron concentration is raised to 0.20 F, the dashed curve in Figure 2 applies, and thc titration error is:
ThusX = 1 when CncN-= CAp+- CC,-and X = 0 when CBON= CA.+. The titration error is given by:
Substitution of eqn. (8) into the numerator of eqn. (10) gives :
Experiments have shown that the red color of FeSCN++ is just perceptible when its concentration is about 6 X Thus it can he assumed that the end point of the Volhard titration is reached when [FeSCN++] reaches that value. If CFez+and Ccr are known, the concentrations of the other species a t the end point and, hence, the titration error can be calculated. Alternatively, the graphical method can be used to emphasize the significance of the various equilibria. Figure 2 shows a graph for this calculation when Cor = 0.10 F and CFsa+ = 0.010 F. Either pAg, pC1, or pSCN could be chosen as the master variable; pSCN has been used in this case since it allows the most direct calculation of the other variables. The [Ag+] line in the figure is the plot of the AgSCN solubility product expression: The expression for [Cl-] is obtained from the ratio of
This corresuonds to an error of -0.5%. - when Ccr = 0.010 F. When the solid AgCl is not allowed to equilibratc with SCN-, either by filtering the solution or by the use of nitrobenzene, onlv the titration of the excess An+ with SCN- need bk considered. I n that case, the titration error is given by:
-
E
=
IAg+] - [SCN-I - [FeSCNr'] Ccr
Figure 2 still applies if the [Cl-] curve is rcmovcd, aud the titration error for the case represented in that figurc (CF,a+ = 0.010) is then E =
10b.M -
10-6.40 10-1.00
-
10-5SC
100
while, when Gea+ = 0.20;
Another method of interest is the Liebig titration of cyanide. Although the description of this method requires the consideration of acid-base and complex equilibria in addition to precipitation, the titration error calculation can be handled in the same manner as has been described.
