J. lnczedy Technical University Budapest, Hungary
Representation of Titration Errors in Logarithmic Diagrams
The titration error, arising from the fact that the end point of the titration does not coincide with the equivalence point of the titration, is important in practical analytical work. Many publications (1-6) and books (6-12) deal extensively and quantitatively, but separately, with errors in titrations of different kinds. Now an attempt has been made to give uniform treatment for the titration error in all kinds of titrations. The fundamental principle is the same for titrations of all kinds, i.e., the equivalence point and the end point are not the same. Both points can be expressed in terms of pH in acid-base, pM in complexometric, pM or pX in precipitations, and pe in the oxidationreduction titrations. The relative titration error in percent is expressed by the general formula
In the case of titration of strong base, [ H + ] K > 1, and the second term in brackets can be neglected. Plotting the calculated log A values at different endpoints (i.e., pH's) the diagram in Figure 1 is obtained. The error is equal to 0, (log A = -a), for any concentration a t pH 7.
C, denotes the total concentration of the titrant, Cd, the total concentration of the substance to be determined in mole/l referring to the end volume of the titrated solution. y and z are stoichiometric constants.
Figure 1 , Logarithmic diagram i f the titration ermr in the titrotion of a stmng bare with stmng acid at different concentrotionr.
LSD
I
I
I
I
'
"
'
.
/
2 3 4 5 6 7 8 E ) K ) l l p H
In the titration of weak bases eqn. (6) can be simplified
Acid-Base Titrations
Let us consider first the acid-base titrations in which a monoacidic base, B is titrated with strong acid, e.g., hydrochloric acid standard solution. B+HCI=BH++CI-
(2)
The total analytical concentration of the titrant and of the base at the end point can be expressed from the following charge balance and mass balance equations Ccl
+ [OH-]
=
Ce =
+ +
[H+l [BHf1 [Bl [BE+]
(3)
Since in the pH range of the equivalence point [ H + ] >> [OH-] and [ H + ] K>> 1. By plotting the logarithm of the calculated titration error values as a function of the pH of the end point, for the titration of a weak base ( K = lo9) in different concentrations, the diagram in ' ~ i ~ u 2' r ecan be obtained. The equivalence point depends on concentration, approaching pH 7 as the solution is diluted. The two branches of the curves in Figure 2 can be approximated by straight lines with equations
(4)
and inserted into eqn. (1)
The second term in the brackets in the right hand side is the mole fraction of the unprotonated base, which depends only on the pH of the solution and the protonation constant, K .
Inserting +B into eqn. (5), the following general formula -for titration of weak or strong bases-can be obtained Figure 2. Logorithmi~diagram of the titration ermr in the titrotion weak bare (K = 10'1 with strong acid, at different concentrations.
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log A' = 2
- log CB - pH + 2 - log K
Log A' = pH
(8) (9)
the accuracy of the detection of the end point is not better than &0.5 pM (this is usual), the criterion of the quantitative titration is
At the intersection of the two lines the pH pH*, = '/dlog K
- log CB)
is the equivalence point. Similarly equations can be deduced for titration of acids of different kinds with strong base standard solution.
In the complexometric titrations of metal ions using chelating agents as titrants in most cases 1:1complexes form according to equation M+Y=MY
(11)
(for simplificationthe charges are omitted). M denotes the metal ion and Y the complex forming multidentate ligand. For the complex formed the stability constant, K, is characteristic, hut for practical purposes the "apparent" or "conditional" constant, K'-introduced by Schwarzenhach (IS) and Ringbom (14)-is used. The K' value can be calculated using the side reaction functions, if the pH, side reaction equilibrium constants, etc. are known (14). The titration curve of a complexometric titration is constructed similarly to that of acid-base titrations, but instead of the pH, the minus logarithm of the apparent concentration of the titrated metal ion =
PM'
is used. For calculation of the titration error, we use the general formula eqn. (1).
CM= [M']
log A' = 2
- log CM- pM' - log K' + 2
log A" = pM'
+ [MY]
+ [MY1 = [M'l + [M'l[Y'1K1
pMbQ = '/%(logK'
(18) (19)
(13) (14)
the formula can be transformed
- log Csr)
(20)
Equation (20) gives the equivalence point. Precipitation Titrations
For precipitation titrations the deduced error formulas are very similar to that of complexometric titrations. In the simplest case, where a salt of MX compositionis formed a n d X is to be determined
K, is the precipitation formation constant, the reciprocal value of the solubility product; Cx is the hypothetical total concentration of the titrated ion, calculated from the total amount of the ion and from the end volume of the solution. If the volume change during the titration was negligible, the original concentration can be used. Oxidation-Reduction Titrations
Assuming first that a reductant is titrated with an oxidizing agent, the reaction of the oxidation-reduction titration can be described with the following general equation y Ox, z Red8 y Red, z OX^ (22)
=
+
Using the followingmass balance equations CY = IY'I
since in this case A is lower than 1%. The asymptotes of the two branches of the curves in Figure 3 can he described by the following equations
At intersection
Complexometric Tihations
-log [M'l
> 10'
K' (10)
+
The oxidation-reduction systems involved and the corresponding equilibrium constants are the following ones ~.
K6v =
Expressing [Y'] from eqn. (14) and inserting it into eqn. (15), after some arrangement we obtain
(titrant) [Oxtl [el" oxd ye e Reds
+
*
Since in the range near to the equivalence point
[ W
>> 1,we can simplify
Using the last formula, rapid calculation can be made to obtain the titration error, if [M'] a t the end point approximates the indicator transition point and CX and K' are known. Calculating the percentage error a t any pM' and plotting the log A against pM', we obtained a diagram very similar to that obtained from acid-base titrations. See Figure 3. M, and It follows from formula (17), if CM = 770
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Figure 3. Logarilhmic diagram of the titrotion error in chelotornetric titralions of h o different cases llhe values of the stability constant of the corn. plex formed ore of the order of 10' and 1OL'1 and a t two different concentrotions.
[Redl], [Redb], [Ox,], [Oxdl are concentrations, while [el is the electronactivity, introduced first by Jorgensen (16) and used by several authors (16-18). The negative logarithm of the electronactivity is used in the description of oxidation-reduction equilibria similarly to the pH in the description of acid-base systems. During an oxidimetric titration pe changes very similarly to the way pH did in the acid-base, or pXI in the complexometric,or pX in precipitation titrations. Between the oxidation-reduction potential and the pe there is a close relation pe=-
E 2.3 RT/F =
0.059
K r p e o= ~ 6.1; log KA" = 3.08; log KrSr = 13.0; log Kc.' -~ = 25.0. The equations of the straight lines, which correspond to the branches of the curves are log A' = log K J 2 - ype (31) log A' = zpe + 2 - log K,' (32) The pe of the intersection
+
gives the equivalence point. The criterion of the quantitative titration can he deduced if we consider that the substance to be determined must be oxidized (or reduced) during the titration to an extent of 99.9%. This means that
(at 25C')
E denotes the oxidation-reduction potential in volts.
or using the logarithmic form of eqn. (24)
The standard oxidation-reduction potential of a system is expressed by the K' constant
pee,
> -1 log Ka' Y
+ Y1 3
(34)
-
Using eqn. (33) also we obtain y Log Kt' - 2 log Kdr > 3(2 Y) (35 ) If both oxidation-reduction systems in the titration involve two electrons, the difference between the log K' values must be greater than 6. By the use of eqn. (34) andof the conditional equilibrium constants, it is also very convenient to calculate the suitable conditions for t,itrations in those cases where the reduced or oxidized form of the systems react in side reaction with complex forming agents or with protons. The conditional oxidation-reduction constant, K' can be deduced and used similarly to those used in complex chemistry
+
For calculation of the titration error, using eqn. (I), the following mass balance equations are necessary Ct = [Oxtl [Redtl (25) Ca = [Om] [Reddl (26)
+
+
Assuming that the substance to he determined was completely in reduced form before the titration, which is a fundamental criterion of the quantitative determination, the following equationis also valid (27) y[Redrl = z(Oxs1 Using eqns. (25),(26), and (27) the titration error is
Introducing eqns. (23) and (24) we have
[Red'] and [Ox'] are analyhal concentrations without any respect to side reactions. The connection between the conditional and real constant is the following one
Since in the range of the equivalence point Kdl[e]V< 1, the second term in the denominator can be neglected and the formula simplified
where
A%
1
100
1 [-Kr[elz -~d[el~]
(30)
On the basis of this equation it can be concluded that the error is independent of the concentration of the reactants at first approximation and depends only on the K' values and on the electronactivity at the end point. Actually the error is not independent of concentration, since the KTconcentration constant values are dependent on the ionic strength of the solution. If the titrant is a reductant and an oxidizing agent is to be determined, the same formula can be used; only the signs of the two terms in brackets must be changed. Using the deduced formula, we calculated the titration errors at diierent pe values in the titration of hexacyanoferrate (111) ions with ascorbic acid a t pH 6, and in the titration of iron (11) ions with cerium (IV) sulfate standard solution (see Fig. 4). While the latter is a "symmetric," the former is an "unsymmetric" titration. The constant values used are: log
and ao,cm are the side reaction functions an.dw = 1 [Alp, [A]%+ . . . ao,Cs) = 1 [Blyl [Bl2y2+ . . . [A] and [B] are concentrations of the species reacting with the reduced and oxidized form, respectively. orn.d(*,
+
+
+
+
Figure 4.
LogariIhmic diogmm of t h e titrotion error in the titrotion of ond in the titrotion of iron[ll] ions with serium[lV]-wlfote standard solution.
hexocyanoferrote[lll]ions with ascorbic acid
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0's and 7's are complex products or protonization constant products. In all calculations concentration constants can be used if they are corrected to the corresponding ionic strength. If the criterion of the quantitative determination is not fulfilled, by suitable pH change or by the use of a complexing agent, which shifts the values of the conditional constants, the titration may be realized. Literature Cited B n ~ m nJ. , N.. J. C n m . Enac., 40, 66 (1963). C ~ m l s z l o ~ n ~H. ~ sL., o ~J.. CXEM.EDDC.,40, 63 (1963). B u ~ c l i n n J., , w o FERNANDO, Q., J. C H ~ MEDUC., . 43, 546 (1966). F s s a ~ ~ oQ., o , AND BUTCHBB.J., J. CXEY.EDDC..44, 166 (1967) ( 5 ) E n o ~ r L.. , m o SVEXLA, G.. Anal. Chim. A d a , 40, 473 (1968). (1) (2) (3) (4)
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(6) K o m x o r ~ I. . M., A N D M B N Z ~ H.. L , "Die Massanaiyse," Springer Verlag, Berlin. 1931. (7) KOLTOPF. I. M.. AND STENOER,V. A,. "Volumetric Analyeis," Interscience (division of John Wiiev & Sons, Ino.). New York,1942, Vol. I. (8) E n n e r , L.. "Beveretes a kemiai analisisbe." T~nkonyvkiado,Budapest. 1943. Vol. 11. (9) L A ~ E N E N H.. A,, "Chemical Analysis." MoGraw-Hill Book Co.. New York, 1960. (10) R m a n o u , A,, "Complexstion in Analytiaal Chemistry," John Wiley & Sona, Ino.. New York. 1963. (11) BnvLEa, J. N.. ''Ionic Eouilibrium," Addiaon-Wesley Publishing Co., Reading. Massaolrusetts, 1964. (12) SEEL. F., "Grundi&gen der analytisehen Chemie und der Chemie in wssarigen Syatemen," Verihg Chemie, Weinheim. 1955. (13) SCX"YARZENB*CB. "Die kompiexometrisehe Titration, Ii. Enke Verlag, Stuttgart. 1957. (14) R ~ ~ a n oA,, x , J. C x m . Eonc.. 35, 282 (1958). (15) J ~ R R E N B EH., N , "Redox malinger," Gieilerup. Co~enhagen,1945. (16) F A U L ~L.. O , "General Chemistry," W. A. Freeman and Co., Sao Francisco, 1954. (17) SILLBN, L. G., "Grsphic Presenbtiotion of Equilibrium Data." Intersoienee (division of John Wilev &Sons, Inc.), Naw York. 1959. (18) J o n m s s o ~S.. , Elerncnfo, 49, 1 (1966).
