1696
ANALYTICAL CHEMISTRY
tion of a very stable metal chelate with one of the oxidation states of a metal can cause a shaip change in the redox potential. This property of metal chelates has analytical applications. Potentiometric Titrations in Presence of Excess Chelating Agent. The oxidation potential of the cobalt(I1)-cobalt(II1) couple is so negative ( - 1.82 volts us. hydrogen) that even the stronger oxidants such as ceric sulfate cannot be used as a redox titrant ( E O = -1.61 volts us. hydrogen). The addition of ethylenediaminetetraacetic acid, hoTTever, shifts the potential about 1.2 volts [EOfoi cobalt(I1) ethylenediamine tetraacetate-cobalt(II1) ethylenediamine tetraacetate = -0.6 volt], so that a redox titration using ceric sulfate is feasible. Pfibil and conorkers ( 4 , 8) have devised procedures based on this principle for the determination of manganese and cobalt. Titration with Standard Chelating Agent. It is possible in some cases to titrate metal ions in buffered solutions with a standard chelating agent. I n this case the potential change at the end point is a result of the formation of a very stable virtually undissociated chelate with one of the oxidation states of the metal. PEibil ( 7 ) obtained excellent results in titrating iron(II1) with disodium dihydrogen ethylenediamine tetraacetate in an ammonium acetate buffer. The use of this method is dependent upon the availabilitj- of a suitable indicator electrodc and the rapid formation of a stable metal chelate. Polarographic Methods. Metal chelate formation generally results in a shift of the half-wave reduction potentials ( E l / ? )of the polarographic wave of the metal. This displacement in E l / , has been made the basis of a large number of amperometric titrations by many investigators. Literature references concerning this technique are too numerous to mention here. It is also possible to separate the reduction waves of several metals having comparable reduction potentials through preferential chelate formation. Because the spread in El/, values for the chelates is dependent a t least in part upon the relative affinity of the ligand for the metal ions, the proper choice of a chelating agent makes possible the resolution of the corresponding reduction waves. Numerous examples of this application may be found in recent literature. PEibil and coworkers ( 9 )have succeeded in resolving the waves of copper(II), lead(II), and thallium(1) by this method. Colorimetric Applications. The development of intense color
resulting from the interaction of a number of metal ions with chelating agents may be utilized for analytical applications. Spectrophotometric determinations of manganese, cobalt, and chromium have been reported by Pfibil and coworkers (6, 6). Sweetser and Bricker (20, 21) have recently reported simple and rapid procedures for the spectrophotometric titration of copper(II), nickel(II), iron(III), zinc(II), and cadmium(I1) ions using disodium dih] drogen ethylenediamine tetraacetate. The interest and the excellent progress in the application of chelating agents to specific analytical problems are evidenced by the fact that over 150 scientific papers have already been published on the uses of nitrilotriacetic acid and ethylenediaminetetraacetic acid alone in this important chemical field. The continuing investigations along these lines n ill certainly result in the development of nem- and useful analytical applications of this important class of chemical compounds. LITERATURE CITED
(1) Ackermann, H., Prue, J. E., and Schwarxenbach. G., A-ature, 163,723 (1949). (2) Bjerrum, J., thesis, Copenhagen, 1941. (3) Bjerrum, J., and Nielsen, N.,unpublished results. (4) Piibil, R., and HorLEek, J., Collectton Ctechoslou. Chem. C o w muns.. 14. 413 (1949). (5) Piibil, R., and Hornychorh, E., I b i d . . 15, 456 (1950). (6) Piibil, R., and Klubalovh, J., I b i d . , 15,42 (1950). (7) Piibil. R., Koudela, Z., and h l a t y s k a , B . , I b i d . , 16, 80 (1951). (8) Piibil, R., and gvestka, L., I b i d . , 15,32 (1950). (9) Piibil, R., and Zabranskg, Z., I b i d . , 16, 554 (1951). (10) Prue, J. E., and Schwarzenbach, G., Helv. Chim. Acta. 33, 985 (1950). (11) Schmarxenbach, G., A n a l . Chim. A c t a , 7, 141 (1952). (12) Schmarzenbach, G., Chimia, 3, 1 (1949). (13) Schwarzenbach, G., Hell;. C h i m . A c t a , 33, 974 (1950). (14) Schwarzenbach, G., and Biedermann, W., Ibid., 31, 331 (1948). ( 1 3 Ibid.. n. 456. il6j Ibid.: 459. and Bangerter, F., I b i d . , (17) Schwarzenbach, G., Biedermann, W., 29, 811 (1946). (18) Schwarxenbach, G., and Gyding, H., I b i d . , 32, 1108 (1949). (19) Ibid., p. 1314. (20) Sweetser, P. B., and Bricker, C. E., SAL. C H E M , 25, 253 (1953). (21) I b i d . , 26, 195 (1954).
b.
RECEIVED for review July 29, 1954.
Accel>ted September 29, 1954.
7th Annual Summer Symposium-Developments in Titrimetry
Interpretation of Data Obtained in Nonaqueous Media ERNEST GRUNWALD Florida State University, Tallahassee, Fla.
In media of low and intermediate dielectric constant, activity coefficients of ionic species deviate greatly from unity. The deviations may cause inaccurate inflection points in potentiometric titrations and troublesome salt effects on color indicators. The deviations are best described by assuming the formation of ion pairs. The physical nature of the ion pairs, the magnitude of their dissociation constants, and the calculation of ion activity for partly associated electrolytes are discussed. The available generalizations are applied to one particular potentiometric titration, and methods are proposed for improving the accuracy of the inflection point. A quantitative treatment of medium effects on color indicators for acid-base titrations is developed. For optimum results, the choice of an indicator is governed by the pH at the equivalence point and the solvent sensitivity indexes of the indicator and substrate.
I
X T H E classical approach to the theory of titrations, the l a w
of the dilute solution ( 2 3 ) are thought to be an adequate first approximation. This approach is certainly correct for aqueous titrations and, when applied only to nonelectrolytes, for nonaqueous titrations. However, electrolytes deviate so greatly from the laws of the dilute solution in solvents of low dielectric constant that the interpretation of results on the basis of the dilute solution approximation may lead to considerable error. For the sake of illustration, the experimental activity coefficients, y, for hydrochloric acid in a number of solvents are shown in Table I. I t is seen that y values for the 0.01JI acid decrease markedly with the dielectric constant, D, from 0.905 in water (D = 78.5) to 0.138 in 82% dioxane (D = 10.6). The y values are compared with those predicted from the Debye-Huckel limiting law
- logy =
s d;
(1)
1697
V O L U M E 2 6 , N O . 11, N O V E M B E R 1 9 5 4 Table I.
Mean Ion Activity Coefficients for Hydrochloric Acid, 25.0" C." HC1
Solvent Via t er 4541, dioxane-55% mater Methanol Ethyl alcohol 82% dioxane-18T Water a
Limiting Law
70 error
0,905
S yded. 0.506 0.890
40.25 0.01 32.5 0.01 24.31 0.01
0.762 0.682 0.485
1.388 0.720 1.913 0.644 0.505 2.97
-4.7 -5.6 -4.1
10.56
0.349 0.1379
D .I1 78.48 0.01
yobad.
0.001 0.01
10.33
0.471 0,0927
-1.7
+35.0 -32.8
Baaed on E o >aluee and data cited by Marshall and Grunwald (24)
\There A' = 1.826 X l O 6 / ( D 7 ' ) 3 *, 2' is the absolute temperature, and I* is the ionic st,rength. I n solvents of high and intermediate tliclect,ric constant ( D > 40)) y is generally greater than predicted f r o m Equation 1. Hail-ever, a t lower dielectric constant, m:iy he consideru1)ly less than predicted from Equation 1 a t some concentrations and great,er a t others, as shown very clearly t)y t,hc data in 82% diosane. The deviations are due to shorttxnge ion-ion interactions which become increasingly important :is L) is lowered. The effert of these interactions on y can be calculated moPt accurately by presuming the esistence of quasiassoviation equilibria of the type of Equation 2 or 3 ( 3 )2 4 ) .
estimated with good accuracy from the interionic attraction theory, allowing for the finite size (5.0 A . ) of the ions ( 1 9 ) . It is seen that EOremains constant a t 1.823 =k 0.007, within the accuracy of the Beckman Model G meter, over a sevenfold variation in concentration. Further analysis shows that the error is not due to sodium error a t the glass electrode (2). As will be shown, the error arises from the interplay of ion association equilibria in the titration cell. I n titrations employing color indicators, the salt effects on the indicator equilibrium may be PO pronounced that the correct end-point color a t one concentration is incorrect a t another. The classical study of crystal violet in glacial acetic acid is a good example of the possible magnitude of this effect (5). THE NATURE O F IOY PAIRS
When the dielectric constant is greater than ea. 10, it is sufficiently accurate to consider only the formation of ion pairs, as indicated bj- Equation 2, and to neglect the formation of higher order aggregates. I n order to understand the factors determining the stability of these ion pairs, it is instructive to inquire into the physical basis of the ion pairing process. In the Bjerrum-Fuoss treatment (3, 9))solvation effects are neglected, and the potential energy of the ion pairs, E ( r ) , is given as a function of the interionic distance, T , according to Coulomb's law, by Equation 6.
E ( r ) = ZcZae*/DT
(6)
I n Equation 6, Zc, Z A are the valencies of the ions, and e is the unit of ionic charge. The treatment results in a distribution (9) such that the great majority of the ions are paired either a t rather short distances, or a t distances sufficiently long so that the ions may be thought of as free. ilt intermediate distances, the population is small, the population minimum occurring a t distance p defined by Equation 7 , where k is Boltzmann's constant. Thus, ion pairs are considered as associated when ry. r(min) = y = -ZcZae2/2DX2' (7) The Bjerrum-Fuoss treatment neglects the effect of solvation of the ions. I n polar solvents, the ions exist in a solvation shell or region of electrical saturation, the radius of which has been estimated from dielectric constant and preferential solvation data as ea. 5 A. for typical univalent ions (1, 7 ) . Therefore, when r < 10 A,, the solvation shells overlap, as shoTvn schematically in Figure 2 , where the dipole ari ows represent solvent molecules.
NaOH (millieguivalents)
Figure 1. Titration of Acetic-Hydrochloric i c i d \Iixture with Sodium H>droxide, 95q' Methanol, at 2.5'C.
C+A-
=
c++ '4-
C+mA-,, = inCi
+ nA-
Because of the ion association phenomena, the activity coefficients in mixtures of electrolytes may behave in such a way as t o cause serious error in potentiometric titrations. For example, Figure 1 shows the variation of -log aHaCl (a = activity), measured by means of the cell Glass electrode I solution X 1 AgC1--4g
(4)
when a mixture of hydrochloric and acetic acid in 95% methanol ( D = 36 0 4 ) is titrated with sodium hydroxide in 9570 methanol. a Because acetic acid is weak in this solvent ( K A = 1 X sharp change in the ordinate is expected a t the equivalence point, somewhat like t h a t expected vihen hydrochloric acid is titrated in water. in the presence of ammonium ion ( K A = 0.6 X A sharp change is indeed observed, as s h o m in Figure 1, but the volume of sodium hydroxide a t the inflection point differs by 6% from the equivalent amount. T h e error cannot be ascribed to inaccuracy of the cell, as shown by the data in Table I1 where solution X contains only hydrochloric acid. I n this case, E is given correctly b y the electromotive force relation
- log UHCLCI = Eo - log C H C C ~+ 2 s &/(l
Figure 2. IlIodel of SolventSeparated Ion Pair Dipole arrows repreaent solvent molecules
Table 11. Data for Hydrochloric Acid i n 95 Volume RIethanol, 25.0' C. HC1, 103.11 IP B -Ea"
E = Eo
+ 2.426 d i )
(5)
where E is e.m.f. (in pH-scale units), Eo is a constant, and y is
4
0,0981 0.1303 0.1571 0.1934 0.2356 0.4711 0.6851 From Equation 5 .
G 238 5.988 5 832 5.042 5.470 4.887 4.577
Data taken from ( 2 ) .
1.810 1.819 1 816 1.828 1,825 1.833 1 832
%
ANALYTICAL CHEMISTRY
1698 Because of the electrical saturation effects, the effective dielectric constant is then much less than the macroscopic value, and E ( r ) is correspondingly lowered (6). However, the overlapping of the solvation shells occurs at the expense of solvation energy, since solvent molecules are set free. The two effects work in opposition and the net effect is difficult to predict. I t very short distances, when the ions are in direct contact without intervening solvent molecules, we expect bonding phenomena to become important and hence a low potential energy.
11
n \PI 3
I
I
I
I
6
IO
l
I
20
60
l
r (A')
o
T HO C H a ] +
(8)
tion energy for its reaction with azide ion, which must picrre the solvation shell, is 4.0 kcal. greater t,han that for the parallcl reaction with water which is already part of the solvation shell. The activation energies would have been of the opposite order in the absence of the solvation effects. h potential maximum somewhere to the right of point a in Figure 3 is clearly indicahed. Even more definitive is the evidence from kinetic studies of the solvolysis of the optically active compound 9 in glacial acetic acid (52). The kinetic data for racemization and products formation
C H , O~ H C H , - C H C H , . O S O ~ C ~ H ~ (9) require a scheme involving two distinct ion pairs, one intimate and one solvent-separated, as shown in Equation 10 where compound 9 is represented by R-X, X = OSO&eH,.
R-X
0.623 0.519 0.419 0.350 0.228 0.1703 0.1360
a,. 5 - ciyi =
There is significant evidence to the effect that the removal of the last layer of intervening solvent molecules must cost considerable energy. Therefore, the potential energy function for actual electrolytes should resemble qualitatively the function shown in Figure 3 for a univalent elrctrolyte a t 25" C. in a solvent of D = 20. For r > 10 .\., E ( r ) is given to a good approxiniation by Equation 6, which in the Iiwsent case has been extrapolated into the region where electrical saturation and desolvation effccts are important (dotted portion of Figure 3 ) . The desolvation effects cause a sharp rise in E( r ) near the distance s, and the shortrange bonding interactions cause a second minimum a t T = a. Ions are treated as apsociated if r < q . The distance s is t,hcln the equilibrium distance of the solvated or solvent-Fi:p:irat,ed ion pairs, and a is the equilibriu~ndistance of t,he intiinate or touching ion pairs. Thus ion pairs are thought t o consist of two distinvt and chemically different species. The evidence for the existence of two distinct ion pair species comes chiefly from research into the mechanisms of organic reactions. For example, the reactions of di-p-tolylmethylcahonium ion have recently been studied in 90% acetone (2f). Thc nctiva,
0.624 0.521 0.417 0.349 0.228 0.1723 0,1379 b A' = 2.59 X 10-4,
e R'X-$
[R+IX-] e R f
L
products
+ X-
k!
0.776 0.688 0.594 0.527 0.395 0.327 0.284
y
I n order to obtain the activity coefficient for an ionic species when there is ion-pair formation, two approaches suggest themselves: One approach is to treat Equation 2 as a real chemical equilibrium, involving more than one chemical species, analogous to, say, the ionization of a weak acid in water; the other is to treat Equation 2 as a quasi-equilibrium in which a single species is distributed among several distinguishable microscopic environments. The two approaches lead to the bame final result: The activity of a given ionic species is equal to that of the free-ion portion of the species:
D . 20 T . 298" C.
H
1.5 3 B 10 30 BO 100 a See ( 2 4 ) .
EFFECT OF ION ASSOCIATION O N
Figure 3. Plot of E ( r ) cs. r for Representative 1 to 1 Electrolyte
p
Table 111. Data for Hydrochloric Acid in 82% Dioxane, 25.0" C. 106C. .\I Yobsd a Y O S I O ~ .b acalcd.
(10)
(11)
Ciffiy;'
11-here a i is the degree of dissociat~ionand yi' is the activity coefficient, of the free-ion portion. Furthermore, for a single electrolyte CY is related to c by LY'(Y')'C/(~
- a)
=
K
(12)
where K is essentially an equilibrium constant and therefore :I func~tionof solvent and 2' (4,8). For mixtures of electrolytes, one obt,ains several simu1t:tneoua equilibrium expressions of the form of Equation 12. For example, in the case of two electroIj.tes sharing a common cation, C + A - and C + B-, the act'ivity of rach ion is given 1))- Equation 11, subject to the rest,raints:
In Equation 13, C C A and CCB are the formal concentrations of the two c,lectrolytes, a~ = C A (free)/ccA, CYB = cB(free)/ccB, a n d y' is the mean ion activity coefficient of the free ions. Prediction of y values r i a Equations 11 and 12 has proved satisfactory ( 2 4 ) . The ion-pair dissociation constant, K , which allows for the theoretically difficult short-range interionic effects, can be measured directly by means of conductivity data. The frce-ion activity coefficients, y', can be obtained with good acruracy from the interionic attraction theory, since the calculation involves only r values greater than q. The exact form of the expression for y ' depends on the relative magnitudes of s and q. However, all expressions for y' are insensitive to the magnitude of s, so that y' is, to a good approximation, independent of the electrolyte and therefore a colligative function of the free-ion strength. Details of the conductometric prediction of 21 values have been described in a recent article from this laboratory (24). The success of the method may be illustrated by some data for hydrochloric acid in 82% dioxane ( D = 10.6) which are shown in Table 111. iMAGNITUDE O F K VALUES
Two facts stand out: I n a given solvent, K values vary greatly for different electrolytes, and for a given electrolyte, K values vary greatly with the solvent. The former effect is illustrated in Table IV. The K values shown for liquid ammonia are based
1699
V O L U M E 2 6 , NO. 11, N O V E M B E R 1 9 5 4 on conductivity data (15, 2 2 ) and are somewhat tentative, as the method of calculat8ionis not the most accurate method now available ( 2 4 ) . I n 95% methanol, the I< value for hydrochloric acid is a conductometric estimate, and is consistent n-ith the electroniotive force data shown in Table I1 ( 2 ) . The K valuer for the other salts are potentioinctric eetinmtes based on data fur cell 4 ( 2 ) . These values, too, are tentative pending conductomet,ric verification. 9otwl)ly the value for sodium chloride is smaller than might htrvc l x ? c ~expected ~i from conductivity d:it:i in other solvent,s and: if c-ovect,, strongly suggests intinintc ion-pair formation. There is 110 doubt, however, that I< v a l w s for different elcct,rol\-tes mnj- vary greatly, in sonic‘ by its much as two ordei,r of inngnitude. The tendency of the various ions toward ion-pair forination seems to follow a definite p:it,tcarn,which is fairly independent of the solvent. The cations um~illyobey the Fequenre: K + > Sa’ > H30+ = S H l f = S R i - , nnd the anions: RCOy- > c‘1- > I3r- > S O 3 - = I - = CIO1-. ( R = alkyl radical or H.) I n the prrdictioii or the solvent8effect, 13jc~1~ruin’~ tllcory S < ~ I ’ V ( ~ S :is a valuable guide. hlthough the theory nclglects sol.vat,ion :rnd short-range bonding intcractions, it fits the actual behavior of a number of electrol:-tes in polar solvents with fair accurrtcy (10, 24). hccording to the theory, K is given by Equation 14
K = 1000/32~~Yq3Q(b)
(1-4)
\\.here S is Avogadro’s number, b = 2q,!s, arid Q ( h ) is :i function the values of which have been t,abulated (8, 20). In Table \-> K has been computed for various value3 of 1) a t 25.0°, using s = 0.0 -4. It is seen that the effect of Polvcrit change is large, a fourfold variation of D resulting in a thousandfold variation of I
