VICTOR W. BOLIE Iowa State College, Ames C H E M I C A L analysis by titration has been an established procedure for many years and shows no signs of diminishing in the future. Although the theory of titration is well known, the textbook treatments appear to handle the mat.hematics of the problem in an approxi- mate fashion. The purpose of this paper is to show the development of a titration euuation for a nrecinitation reaction. which is exact in the mathematical sense. The example used is the titration of a sodium chloride solution with a silver nitrate solution, although the theory would apply to other precipitation reactions. The equation is shown to produce the expected results a t three significant stages of titration. Following the development of the equation is a mathematical treatment involving automatic computation. A graph is presented, together with a discussion of the slope of the titration curve a t the equivalence point.
. .
Initially, the NaCl solution contains eocr= V[C1-In equivalents of chloride ion. Since it is in equilibrium with an infinitesimal amount of AgCl precipitate, the number of equivalents eoA,+ = V[Ag+Ia initially retained in solution is related to the initial chloride ion concentration by the formula [Ag+] [Cl-] = K.,. After adding a volume v of the AgNOa solution of normality n, the number elk+ of Ag+ equivalents precipitated is equal to the original number enA,+in the NaCl solution, minus the number eA.+ remaining in solution, plus the number nu of Ag+ equivalents from the AgNOa solution. e ' w = (eon@ - e m )
Consider the titration of a NaCl solution with a AgNO8 solution. Assume that an infinitesimal amount of AgCl precipitate exists in equilibrium with the NaCl solution initially. The following symbols may be used to develop a t.heoretica1formula for the titration curve.
= number of equivalents of C1- in preci itate [ ~ g + ]= equivalents/Iiter of Ag+ in the mixetsolution [Cl-I = equivalents/liter of C1-in the mixed solution
e'cl-
eeasi e"c1-
[Ag+l, [CI-1,
= = = =
value of e * e before any AgN08is added value of eel- before m y AgN08is added value of [Ag+l before any AgNOa!s added value of [CI-I before any AgNO, 1s added
VOLUME 35, NO. 9, SEPTEMBER, 1958
(1)
It is also clear that the number em- of equivalents of chloride ion remaining in sorution is equal to the initial number e0,,-, minus thenumber elcl- precipitated out of solution. eel- = eDcr-
DERIVATION OF THE EQUATION
+ nu
But, since etcr written,
=
- e'cl-
(2)
elA.+, the above equation may be
Substituting equation (1) into equation (3) gives, ect- = e'cl-
- e " ~ e- nu + e
~ e
(4)
Using the relationship between equivalents, concentration, and volume, together with the definition of the solubility product constant gives, (V
+ u) [CI-I
=
V[CI-10
- VIAg+lo - nv
+ ( V + v ) [Ag+l
Rearranging the terms gives the quadratic equation,
which predicts the fact that in practical cases, where n >> < , the final chloride ion concentration, existing after a large excess of titrating reagent is added, is very nearly equal t o the solubility product constant divided by the normality of the titrating solution
which has the solutions
METHODS OF SOLUTION OF THE EQUATION
The foregoing formula developed for the chloride ion concentration readily lends itself to numerical computation. For the purpose of calculation it is convenient to write equation (8) as,
Since the square root term in this expression is larger in magnitude than the first term on the right side of the equation, the solution corresponding to the negative sign is always negative and must be rejected. Thr result is the formula
rc1-1 = 2
dKn+
= ( - lO.000
and using the relation [Ag+], = K,,/[Cl-lo gives nu = V ([Cl-10 - IAgCIo)
which predicts the expectation that the equivalence point is attained by balancing out the original Ag+ deficiency. Finally, for cases in which the added titrating volume is in great excess, the conditions are such that ('"'-1" -
2)(")V + v
"8.
2lCI-lo
--
n 2
= --
+
= 2[-1
+ 4 1 + 4K,,/nZl
= ;[-I
+I +
n
=
%fl n
+
2,K., - 2 - 4 n 'n K8p= Kne J
