James N. Butler University of British Columbia Vancouver, British Columbia
Calculating 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
J. Chem. Educ. 1963.40:66. Downloaded from pubs.acs.org by UNIV OF SUNDERLAND on 10/30/18. For personal use only.
else the equations obtained
rigorously were cumbersome to evaluate. In this paper some explicit expressions for in acid-base and precipitation the titration error titrations will be presented and also a simple method for deciding qualitatively whether or not a given titration is feasible.1 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 difference in volume added between the equivalence point and the end point, divided by the volume added at the The titration equivalence point, is the titration error. error results from the failure to know the exact equivalence point, and is in addition to any errors resulting from volume measurements and weighing.
$
=
[Ag+] + P
CoVo
=
V + F„
CV V + Vo
=
(1)
=
K'0
(3)
This treatment of titration errors wilt be included in the author’s forthcoming book, “Ionic Equilibrium,” which will be published early in 1963 by Addison-Wesley Publishing Co., Reading, Mass.
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Journal of Chemical Education
(5)
=
Fo
V + F0
C
=
C0
K
+ C
1
When (6) is used to evaluate the volume factor in (5) obtain the equation for the titration error:
we
Titration
For the
error
reverse
=
$
-
1
titration
the signs of the terms
lA-g(7)
=
on
of' silver ion with chloride, the right hand side of (7) are
reversed.
At the equivalence point, $ =
=
1,
and (7) gives (8)
Kan
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.
Example. In the Mohr method for chloride determination, 50 ml of 0.1 M NaCl is titrated with 0.1 M AgNO:;. As an indicator, 1.0 ml of 0.1 M K2Cr04 is added. If the end point is taken when the first red precipitate of Ag2Cr04 forms, calculate the titration error. The red precipitate will form when the solubility product of AgsCr04 is exceeded, that is when
The fraction titrated is defined to be the ratio of the number of equivalents of titrant added to the number added at the equivalence point: 1
A so [Ag+]
[Ag+]
=
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. 1 in (4), we have: Setting $
(2)
where P is the number of moles of AgCl that precipitate per liter of solution. The only equilibrium that need be considered in most of the usual analytical methods (3) is the solubility product: [Ag+] [Cl-I
CuFq F0
l) V +
(4>
[Ag+]!
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 ITg2S04, PbS04, AgBr, Agl, AgSCN, etc. If V0 ml of chloride ion of molar concentration C0 is titrated with V ml of silver ion of molar concentration C, the mass balances (1,2) are
(4)
C0V0
Note that at the equivalence point, $ 1. Eliminating P between (I) and (2), using (3) to express [Cl-] in terms of [Ag+], and using (4) to eliminate CV, we obtain the equation of the titration curve (4):
Symmetrical Precipitation Titration
[CI-] + P
CV
=
[Ag+HCrOr]
> 1.1 X 10-12
Since [Cr04~] = 1.0 X 10-3, when roughly 50 ml of silver nitrate has been added, the end point will occur when
[Ag+]
=
3.3 X 10
3
0.1 C When this value is substituted in (7), together with Co 1 5.5 X 10-J. This titraand Ksa 1.8 X 10-!0 we get # tion error (0.06%) is half the usual buret reading error (0.1%). In practice, the solubility product at a finite ionic strength, will be larger than the value given, and an appreciable amount of the red precipitate must form before it is visible. Thus the value calculated represents the minimum possible titration error. =
=
—
=
-
A popular method for determining the end point a potentiometric precipitation titration is to choose the point where the slope of the titration curve is a maximum. For a symmetrical titration this is the best in
way of determining the 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 equivalence point Abo \ dpAg -0.434 CoC [Ag+] + Slope Co A C [Ag+] / =
The point of maximum slope occurs where the derivative of the slope is zero. By differentiating the above expression, we find that at the point of maximum slope [Ag+]»
A'.o
=
This is the
same as (8), which gave the value of [Ag+] at the equivalence point.
Unsymmetrical Precipitation Titrations
For titrations forming unsymmetrical salts, M3X, as silver oxalate or mercurous chloride, the titration curve equations are of slightly different form. For the titration of V0 ml of the divalent ion X of normality Ce (molar concentration 1/2 Co), with V ml of the monovalent ion M of molar concentration C, the such
mass
[X]+P
iy^V0
=
[M] + 2F
(9) (10)
=
The equilibrium is governed by the solubility product [M]>[X]
A8„
-
(11)
Eliminating P between (9) and (10), and using (11) and (4), we have the equation for the titration curve CqVq Vo
(*-D=
1M]
2[X]
-
[XIV.
(12)
Near the equivalence point, the volume factor can be approximated by (6), and the titration error is given by Titration
error
=
4>
—
1
Co
A C
(IM]
CoC
C0
AC
r
CoC
For the
-
If)
=
(bps ~21X1)
on
[M]3
Titration
’
Sl»p»
-
-
-»«
=
1
—
C0C
(m
-
IH*0
=
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):
+
(28) a
very large, we have the titration of a strong acid by a weak base. This titration is usually performed the other way around, so we shall give the expression (from (27) with opposite signs) for the titration of a weak base with a strong acid: error
=
$
—
1
=
(jH+]
sions 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+]2 77„ gives =
0.217
v
CoC Co
0.217
(29)
77a and 771, are very large, we have the titration strong acid with a strong base, whose titration is given by
Titration
error
=
$
—
1
=
—~
[H
1
]
^
(30)
As in the case of the precipitation titrations, the equilibrium constants to be used are concentration constants applying at the ionic strength of the solution at
the equivalence point.
If [H + ] at the equivalence point is of the same order of magnitude as either A„ or Aw/Ab, the equivalence point will occur in the buffer region of either the acid or the base.
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Journal of Chemical Education
CoC Co
+ c
(32)
[H+]'
CoC Co
+
C
(33)
Aw
=
0.21
v
/
A* 'A
CoC
VCo
(34)
+ C Aw )
For the titration of a weak base with a strong acid, differentiation of (29) and application of the equivalence 1 gives point condition $ =
“ 2175TTcnFp
1 in (29). If where [H + ]' is obtained by setting $ the equivalence point is more acid than pH 6, the sharpness index is given by =
0.217
v
CoC
( Co +
(36)
c
For the titration of a weak acid with a weak base, differentiation of (27) under the assumption that both [H+] and [Oil-] are small at the equivalence point gives
For rough calculations, the
3
10®
=
If both error
2.17 X
where [H+]' is the value of [H+] obtained by setting $ 1 in (28). If the equivalence point is more basic than pH 8, the sharpness index is given by
'
Aw
a
1
+ C A,V‘A
where the value of 77„. at 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 $ = 1 gives
Aw \ [H+l/
Ab[H+J
of
(31)
d‘t>
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 expres-
A';, is
Titration
IdpH
=
=
[H A
If
0.2
-
Q
In converting from pH to [H+] remember that -log [H+] 7w+ pH
InCo
22
where 7 is ionic strength.
v
[H +
_
T
+
])
Aw
log
v
=
°’21'
Co
(3'}
+ C [Bp
where [B\ [HA ['is the concentration of free base or acid at the equivalence point. Using the mass and C0V0, charge balances (18) through (23), with CV =
—
and neglecting [H + ] and
[HA], we
can
obtain ”
“
0-217
[OH-] compared to [B] and
jl +
(%f-b)V,[
(38)
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 [H2A] and ]
[A-] at the equivalence point, the sharpness index
is
given approximately by v
=
0.217
(§£)'A
(39)
where ZCi is the ionization constant of II2A and ZC,2 is the ionization constant of HA-. Equation (39) breaks down when Kal/ZL2 is smaller than 103, but this is not
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 at the equivalence point can usually be estimated to within one unit. From (31), this means that if the sharpness index is larger than will be less than 0.1%, and the 103, the titration error accuracy of the titration will be governed by the bureta
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 10:!. Strong acidstrong base titrations, where C and Co are larger than 10-3 molar, also fall in this class, provided C02 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 at the equivalence point can be 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.
reading
Literature Cited (1) Butler, J. N., J. Chem. Educ., 38, 141 (1961). (2) King, E. J., “Qualitative Analysis and Electrolytic Solutions,” Harcourt-Braee, New York, 1959, chap. 4 ff. (3) Butler, J. N., J. Chem. Educ., 38, 460 (1961). (4) Bolie, V. W., J. Chem. Educ., 35, 449(1958). (5) Ricci, J. E., “Hydrogen Ion Concentration,” Princeton University Press, Princeton, N. J., 1952. (6) Smith, T. B., “Analytical Processes, A Physico-Chemical Interpretation,” 2nd ed., Edward Arnold and Co., 1940. (7) Davies, C. W., J. Chem. Soc., 1938, 2093.
Volume 40. Number 2, February 1963
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