Theoretical-Slope Method of End Point Detection J. R. Dean' and W. E. Harris University of Alberta, Edmonton, Alberta, Canada When reactions go to completion, the end point in an amperometric or photometric titration can be precisely and accurately located by the so-called tangent or extrapolation method. A method, called the theoretical-slope method, i s proposed for titrations involving a metal ion indicator in which the extrapolation method i s unsuitable because the ratio of conditional equilibrium constants is too low. The proposed theoretical-slope method was used to detect the end point in the titration of 5 X 10-5 mole of gadolinium, samarium, neodymium, or zinc by using cadmium as the amperometric indicator and EDTA as the titrant. An accuracy and relative precision of 1to 3% was obtained. In each of these systems, the experimentally determined ratio of the conditional equilibrium constant of the metal-titrant complex to the conditional equilibrium constant of the amperometric indicator-titrant complex was less than 10.
WHENREACTIONS GO TO COMPLETION, the end point in an amperometric or photometric titration can be precisely and accurately located by the tangent or extrapolation method (1-3). The method involves extending the two linear portions of the titration curve until they intersect. Several persons (4-7) have developed mathematical treatments for cases in which simple extrapolation of two straight lines to intersection is unsatisfactory. Most amperometric and photometric titrations can be carried out by measurement of the current or the light-absorbing properties of the titrant, of the substance to be titrated, or of the reaction product, In some cases where such measurements are not feasible, a metal-ion species exhibiting these measurable properties may be added to act as a so-called amperometric or photometric indicator (8, 9). When amperometric or photometric indicators are used and when favorable equilibria exist, reliable end points can be obtained by the straightforward extrapolation method. When equilibria conditions become unfavorable, end point location by extrapolation is difficult because of curvature in the end point region. A method called the theoretical slope method is proposed that enables end points to be found rapidly, accurately, and precisely under favorable conditions of equilibria. Experimental data are presented to illustrate the applicability of the method. 1
Present address, Chemcell Ltd., Box 99, Edmonton, Alberta
(1) . , J. T. Stock, "Amperometric Titrations," Interscience, New York, 1965, p'34. (2) . , D. A. Aiklus. G. Schmuckler. F. S. Sadek, and C. N. Reillev. -, ANAL.CHEM., 33,1664 (1961). (3) R. F. Goddu and D. N. Hume, ibid.,26, 1740 (1954). (4) E. Grunwald, ibid., 28, 1112 (1956). (5) V. A. Khadeev, Chem. Abstr., 52, 19670e (1958). (6) L. A. V. Bazay, ibid., 54, lSO(1960). (7) J. L. Latham and E. C . Lawley, Analyst (London), 92, 698 (1967). (8) J. T. Stock, "Amperometric Titrations," Interscience, New York, 1965, p 12. (9) G. Charlot and B. Tremillon, J. Electrmnal. Chem. 3, 1 (1962).
ob . '
'
0.4 '
'
'
. 0.8 '
i ' '
1.2 '
.yI
MILLIMOLES OF TlTRANT ADDED
Figure 1. Theoretical curves for the titration of 1.0 millimole of M"+ with 0.4 millimole of In#+ as the indicator ion K ' M T ~ + / K ' ~ , , , ~has T ' + values of: A , D ,101.6; E, 10
m;
B, 10a.O; C,102.0;
THEORY
A nonelectroactive metal ion Mn+ may be titrated amperometrically with a titrant T by using a second metal-ion species Zndz+ as the amperometric indicator. Consider the main reactions to be Mn++ T
-,
K.MT~+
MP+
(1)
and
Indz+
+T
KindT"'
i IndT*+
where KMTn' > K i f l d T " + (concentration stability constants). At the beginning of the titration, the titrant will react primarily with the metal ion Mn+. Ideally, when Mn+ has completely complexed with the titrant, the amperometric indicator Ind'f will begin to be complexed and the diffusion current caused by the indicator-ion reduction will decrease. The end point, corresponding to the amount of metal ion Mnf present, would be at the point where the amperometric indicator begins reacting. Metal ions, whether complexed or not, are unlikely to exist simply as aquated ions in an appreciable concentration. Because there are various other equilibria to be considered, the concentration stability constant is not the most informative one. Consideration of the competitive equilibria leads to conditional stability constants (10,11) and K'indTz+, which take into account the effect of hydrolysis, pH, and other side reactions. The conditional stability constants for Reactions 1 and 2 can be expressed as
'
(10) C. N. Rdlley, R. W. Schmid, and F. S. Sadek, J . Chem. Educ., 36,555 (1959). (1 1) H. A. Laitinen, "Chemical Analysis," McGraw-Hill, New York, 1960,p 229. VOL 40, NO. 8, JULY 1968
1213
"0 Figure 2.
25 50 75. INDICATOR REACTED, (%)
100 INDICATOR REACTED, (ye)
Correction graph for the application of the theoretical slope method of end point detection Each line corresponds to a different value of log ( K ' M T ~ + / K ' I , , ~ Tas"indicated +) A. 10g(K'MTn+/K'lndTZ+) V d U H from 3.6 to 1.0 B. ~ O g ( K ' M T n + / K ' i n d T Z t ) Values from 1.0 to 0.0
and therefore
(4) where CM,+is the total concentration of the metal ion not complexed with the titrant, C l n d z + is the total concentration of the amperometric indicator not complexed with titrant, CT is the total concentration of uncomplexed titrant, C M T ~ +is the total concentration of metal ion complexed with titrant, and C I n d T I + is the total concentration of indicator ion complexed with titrant. The curvature in the end point region is dependent on the Figure 1 is a plot of several theoretivalue of K'MTn+/K'lndT.+. cal titration curves corresponding to differing ratios of K I M T . + to K ' I n d T = + . In curve A of Figure 1, the titration curve in region 1 has a slope of zero and an intercept of 0.4 (amount of indicator ion initially present). In region 2, the titration curve A has a slope of minus one. The other titration curves approach curve A at the beginning and at the end of the titration-that is, they approach a slope of zero in region 1 and a slope of minus one in region 2. If the extrapolation method of end point detection were applied to curves D and E, erroneous values would be obtained. The straight line in region 1 theoretically would have a slope of zero and an intercept corresponding to the amount of indicator ion added. If the value of the conditional stabilIndT"+were suffiity constant for the reaction hdz+f T ciently large, an accurate and precise end point would be defined by a line in region 2 of slope minus one drawn through a point approaching 100% reaction of the indicator ion. If a line of slope minus one is drawn through a point in region 2 that corresponds to less than 100% reaction of the indicator ion, an end point of lower value than the equivalence point will be obtained. The proposed theoretical slope method includes a correction factor for the end point. This correction factor is a function of the percentage of indicator ion that has reacted at the par-
+
1214 *
ANALYTICAL CHEMISTRY
ticular point chosen in region 2 and also a function of the ratio of conditional stability constants. At any time during the titration cM"+
Clnd'+ cMTn+
+
+
c1
(5)
= CZ
(6)
= CIndTz+
f C l n d T * + f CT
=
x
(7)
where C1 is the concentration of Mn+ initially present, Cz is the concentration of I d + initially present, and X is the overall concentration of titrant. From Equations 3 to 7, Equation 8 can be obtained. (K'lndTs+
- K'MTn+)(Clnd*+)2 + [ ( X ' K ' l n d T * + CT'K'lndTz+ - Cz'K'Indp+ f Ci.KJMTn+ X * K ' M p +f C T . K ' M p + f 2(Cz*K',p+)]* Clndz+
f X . K ' M T ~ +' CCi*K'Mp+.C2 ~ - CT*K'MT.+*Cz- K'MT"+*CZ'C~ = 0 (8)
Substitution of values of K ' M T ~K'I%dT= +, t, CZ, and C1 in Equation 8 allows one to calculate the value of Cl,&+ as a function of concentration of titrant added by making successive approximations for CT. A computer program can be written that will perform this calculation and also calculate titration curves for various values of C1, CZ,K ' M T+, ~and K ' m d T z + (12). With this program, various end points can be determined by finding the intersection of a line in region 1 of slope zero and an intercept corresponding to the initial concentration of indicator ion, with lines of slope minus one through various points in region 2. The points chosen in region 2 have been expressed as percentage of indicator ion that has reacted. A plot of percentage of end point error against percentage of
(12) J. R.Dean, Thesis, University of Alberta, Edmonton, Alberta, Canada, 1968
indicator ion that has reacted is shown in Figure 2 for various values of K ' M T+~ / K ' z ~+.~ T = Each curve in Figure 2 is independent of the ratio of the concentration of Mn+ and Indz+ present. Each has a maximum ordinate value of 100 at zero per cent reaction of the indicator and a minimum ordinate value of zero at 100% reaction of the indicator. The shapes of the curves in Figure 2A are such that for large small end point errors or correction ratios of K'MTn+/K'rndT.+, factors will be obtained for all points in region 2. For example, for a ratio of conditional stability constants of 103.0 a correction factor of less than 1 % will be necessary if the point chosen in region 2 corresponds to greater than 10% reaction of the indicator ion. Because the uncorrected end point is not significantly different from the equivalence point, no correction need be applied. For smaller ratios of conditional stability constants, it becomes more and more important to obtain readings corresponding to nearly complete reaction of the indicator ion. For example, for a ratio of 102.0the error is unacceptably large when 25% of the indicator ion has reacted and a reading after 5 0 z reaction or greater is preferred. If the ratio of conditional stability constants is small, as in this case, a correction may be needed to obtain a valid end point. To obtain this correction from Figure 2 , one must know the approximate ratio of the conditional stability constants and choose a point in region 2 that will not require an excessively large correction. The value of K'.wTn+/K'lnd~i+ can be obtained (12) by adding a known amount of standard titrant to a known amount of standard Mn+ and In@+ solution adjusted to the desired pH and ionic strength. The amount of I n P remaining is rneasured, and the ratio of conditional stability constants calculated in the manner illustrated by Schwarzenbach, Gutt, and Anderegg (13). EXPERIMENTAL
Samarium, gadolinium, neodymium, and zinc were titrated amperometrically (dropping mercury electrode) with cadmium as the amperometric indicator and EDTA as the titrant. Schmid and Reilley (14), and Tanaka, Oiwa, and Kodana (15) investigated amperometric titration of cadmium with EDTA. Tanaka, Oiwa, and Kodana obtained satisfactory agreement between experimental end points and equivalence points when the cadmium solution contained 0.001 % gelatin. Polarographic measurements were made with the aid of a Metrohm Polarecord E261. All solutions were prepared with deionized water. Stock 10-2M solutions of samarium nitrate, gadolinium nitrate, and neodymium nitrate were prepared from Alfa Inorganic reagents (99.9% pure). The solutions were standardized by precipitation as the hydroxides, ignition, and weighing as the oxides. Stock solutions (lO-*M) of zinc nitrate and cadmium nitrate were prepared from Fisher Certified reagents and standardized by precipitation and weighing as the ammonium phosphate salt. EDTA solutions were prepared from Fisher certified disodium ethylenediaminetetrqacetate and standardized against magnesium iodate (16). Procedure. Introduce an aliquot of metal-ion solution of unknown concentration into a titration vessel. Pipet an aliquot of standard cadmium ion solution in an amount
(13) G. Schwarzenbach, R. Gutt, and G. Anderegg, Helu. Chim. Acta, 37,937 (1954). (14) R. W. Schmid and C. N. Reilley, J . Amer. Chern. SOC.,80,2101 (1958). (15) N. Tanaka, I. T. Oiwa, and M. Kodana, ANAL.CHEM.,28, 1555(1956). (16) F. Lindstrom and B. G. Stephens, ibid., 34,993 (1962).
Table I. Summary of Results of Amperometric Titrations of Gadolinium, Neodymium, Samarium, and Zinc Using Cadmium as the Amperometric Indicator and EDTA as the Titrant p, 0.1; pH
Amount Substance taken, titrated moles X lo4 Neodymium 0.847 0.847
4.2, O . O O l ~ o gelatin Amount of cadmium indicator Amount of substance found, present, moles X lo4 moles X IO4 1.971 0.836 f 0.011 0.986 0.84
0.508 1.693
0.971 0.986
0.51 1.70 zk 0.02
Gadolinium
1.670 1.670 0.835 0.835 0.501 0.501
1.971 0.986 1.971 0.986 1.971 0.986
1.64 1.64 =t0.02 0.82 =t0.02 0.84 0.52 0.52 f 0.01
Samarium
1.709 1 ,709 0.855 0.513
2.172 1.086 2.172 1.086
1.71 =k 0.030 1.730 i 0.010 0.860 f 0.030 0.522 rk 0.010
Zinc
1.630 0.845
2.172 2.172
1.69 0.86
=k
0.070
f 0.04
approximately equal to the metal ion added. Adjust the pH and ionic strength to the same value at which K'.v~n+/ was determined (pH = 4.2, p = 0.1). Add sufficient K'znd~=+ gelatin to give a final solution of 0.001 Dilute to approximately 50 ml. Set the potential at -0.7 V against SCE. Record the initial reading (A) of diffusion current. Add an amount of standard EDTA, so that the diffusion current reading ( B ) is between 5 and 15 % of the original value. Measure this diffusion current. Add EDTA in excess of the amount needed to react completely with the metal ion and the cadmium ion. Measure the residual current. Subtract the residual current reading from the two others, and correct for dilution if necessary. Calculate the results as follows: Calculate the end point, using the formula
z.
End point (millimoles) = VTMT- V I M I
+
where
Correction factor (10) 100
VI is the volume of the aliquot of cadmiumion solution added in milliliters, M I is the molarity of the standard cadmium solution, VT is the volume in milliliters of the EDTA added at the current reading B, and MT is the molarity of the EDTA. Calculate the percentage cadmium ion that has reacted, using the formula Percentage of indicator [VIM, ion reacted = = 100
-
VOL 40,
-
(:)
VIMI.IOO]
VIM,
(;)
100
NO. 8, JULY 1968
1215
p:
8
2
p:
w
c s l 4 2 n
0 '
1.0
12
'
2.0
3.0
1.0
2.0
3.0
5
L
n
E
O1
I
+ '
J
C
2 LOG( K;~+"
4
K; ndT+ X)
Figure 4. End point error as a function of conditional stability constant ratio for end points obtained by extrapolation
10
0
3
0
01 0
1
1
1.0
2.0
.
I
1
3.0
0 '
1.0
2.0
3.0
VOLUME OF EDTA ADDED, rnl
Figure 3. Titration curves of gadolinium and neodymium with 0.1018M EDTA using cadmium as the amperometric indicator A . Titration of 1.670 X loe4 mole of gadolinium and 1.971 mole of cadmium 8. Titration of 1.670 X mole of gadolinium and 0.986 mole of cadmium C. Titration of 1.693 X loF4mole of neodymium and 1.986 lo-' mole of cadmium D. Titration of 0.847 X mole of neodymium and 1.971 10-4 mole of cadmium
X X X X
The titrations were carried out in 50-ml volumes. Arrows indicate the equivalence points.
To obtain the end point in millimoles take this value of the percentage of indicator ion reacted, use the correction factor from Figure 2, and apply Equations 9 and 10. RESULTS AND DISCUSSION Results (12) of the determination of the values of the ratios of conditional equilibrium constants K ' G d y - / K ' c ~-,~ K,.,,dy s -1 K'Cdyn-, K'Smy-/K'Cdyi-, and K ' ~ ~ ~ z - / K ' c dfor ~ i -various amounts of gadolinium, neodymium, samarium, and zinc, as well as cadmium, indicate the following average values at pH = 4.2, = 0.1, and temperature = 21.0 "C: K'Gdy-/K'Cdgi-
= foo'68*o'09
K ) , . , -, ~/ ~K ' ~ = ~ ~100.14*0.o9 ~ K'smv
-/K'Cdys-
K'Zngi-IK'cdyi -
= 10°.58*0.09 = 100.25*0.08
The values of K'Smv-/K'Cdyland K'#dy-/K'Cdyl- agree with those obtained elsewhere (13) under similar conditions. Figure 3 shows typical amperometric titration curves obtained for the titrations of neodymium and gadolinium. The titration curves of neodymium with cadmium as the indicator ion have a ratio of conditional equilibrium constants of about and are therefore nearly straight lines. They represent one of the most difficult titration curves in which to find the end point. Results of these titrations and titrations of other amounts of samarium, gadolinium, neodymium,
1216
0
ANALYTICAL CHEMISTRY
and zinc are summarized in Table I. The data show an accuracy and relative precision of 1 to 3 %. The limiting precision of data such as shown in Table I depends on the precision with which diffusion currents can be measured and the precision with which the ratio of conditional stability constants is known. In addition to the experimental evaluation of the theoretical-slope method shown in Table I, statistical evaluation of the effect of the ratio of K ' M T n + / K ' z n d T 2 + on the extrapolation method of end point detection was carried out (12) on computer calculated curves (Figure 4). Figure 4, a plot of accuracy against log( K ' , M T n T / K ' z n d T . +), indicates that for accurate results by the extrapolation method, K ' M p + / K ' r n d T z L should be greater than about l o 2 or lo3. In addition it was found (12) that as K t M T . + / K ' l n d T z - approaches lo3. the precision reaches its best value, whereas for lower ratios the precision is poorer. When the ratio approaches one, the theoretical-slope method is recommended because of its greater statistical accuracy than the extrapolation procedure. When the ratio is of the order of l o 3 or greater, extrapolation gives end points of comparable accuracy. Even here, however, the theoretical-slope method has the attractive feature of potentially being more rapid. On the basis of statistical and experimental evidence, the theoretical-slope method has been shown to be a rapid and precise method of end point detection for titration systems where the ratio of conditional stability constants is less than 100. It can be used as an alternative to the extrapolation method when the ratios are greater than one. Although experimental data illustrated in this paper have been obtained for amperometric titrations employing amperometric indicators, the theoretical-slope method also could be applied with success to photometric titrations employing photometric indicators (12). ACKNOWLEDGMENT The helpful discussions with, and suggestions of, J. Breckenridge are gratefully acknowledged.
RECEIVED for review March 11, 1968. Accepted April 6, 1968. Financial support to J.R.D. by the National Research Council of Canada is acknowledged.
