1528
RICHARD E. COVERAND LOUISMEITES
Vol. 67
AUTOMATIC AND COULOMETRIC TITRATIONS I N STUDIES OF CHEMICAL KIXETICS. I. COMPLEX FORMATION BY RICHARD E. COVER'AND LOUISMEITES Department of Chemistry of the Polytechnic Institute of Brooklyn, Brooklyn, A'. Y . Received February 12, 196s When a pseudo-first-order process (such as ligand exchange) is displaced by producing or destroj'71ng a reactant at a constant rate, as by automatic or coulometric titration, the rate and equilibrium constants of that process may be deduced from the titration curve obtained by any of the common techniques of end-point detection. Three different chemical systems are considered.
Introduction In the calculation of titration curves it has always been assumed that chemical equilibrium is attained a t every point. When zero-current potentiometry is used for end-point location it has also traditionally been assumed that each of the couples involved is reversible and that electrochemical equilibrium is attained a t every point. Several recent developments have contributed to an enhanced understanding of potentiometric titration curves. These include the realization2 that an expression similar to the Kernst equation may be written even when the potential-determining couple is not ideally reversible, the d e m o n ~ t r a t i o nof ~,~ the relationship between the potentiometric titration curve and the voltammetric characteristics of the couples involved, and the recognition5 of the influence of the nature of the electrode surface on the potentiometric titration curve. The present paper deals with complications caused by the occurrence of slow chemical reactions in the solution phase, so that chemical equilibrium is not attained a t the instant of measurement. The procedure followed has been t o conceive of a number of hypothetical models for titration systems and to establish and integrate the corresponding kinetic equations. The integrated equations, relating the concentrations of various species to time, describe titration curves that are characteristic of such end-point detection techniques as amperometry and spectrophotometry. Through these equations, the zero-current potential may also be related to time. Analysis of the results gives rise to various procedures for the extraction of rate and equilibrium constants from titration-curve data. Throughout this work it has been assumed that the reaction is isothermal, that mixing is instantaneous, and that the volume change during titration is negligible. Generally these requirements can be closely approximated in practice. It has also been assumed that the rate of addition of titrant is constant. This is the case in coulometric titrations, in which the reagent is electrogenerated by the flow of' a constant current (that, contrary t o conventional practice, is not interrupted while measurements are made), as well as in automatic titrations, in which the reagent is continuously added by means of a constant-rate buret or (1) This paper is based on the dissertation submitted b y Richard E. Cover to the Faculty of the Polytechnic Institute of Brooklyn in partial fulfillment of the requirements for the degree of Doctor of Philosophy in J u m , 1962. (2) S . Glasstone, K. J. Laidler, and H. Eyrinx, "The Theory of Rate Processes," McGraw-Hill Book Co., Inc., New York, N. Y., 1041, pp. 375577. (3) J. Coursier, Anal. Chim. Acta, 7, 77 (1952). (4) I. M. Kolthoff, Anal. Chem., 26, 1685 (1954). (5) J. W. Ross a n d I. Shain, i b i d . , 28, 548 (1956).
equivalent device. The equations that follow are written for the latter case. They may be converted to the form appropriate for coulometric titrations by using the relation
i / F = NV'
= nvp
(1) where i is the constant electrolysis current (amp.), F is the faraday (96491.2 coulombs/equivalent), N is the oxidation-reduction normality of the titrant (equiv8' is the constant volume rate of addialent~/cm.~), n is the number of electrons tion of titrant (~m.~/sec.), involved in the oxidation or reduction of each molecule or ion of the species being titrated, v is the volume of the solution titrated ( ~ m . ~and ) , p is the constant stoichiometric rate of addition of titrant (m~le-cm.-~ sec.-l). All of the systems considered below are formulated in the following way. It is assumed that a species A is titrated with a species TI, and that the reaction between these may be written k-
A
+ T'ZA' +T A
(2)
kA
where A and T are the oxidized and A' and T' the reduced forms of the couples, and k~ and k - ~are secondorder or pseudo-second-order rate constants (mole-' sec.-'). It is further assumed that the forward reaction is very fast and that the equilibrium lies far to the right-in effect, that k-.4 = 0 and ICA = a. Then the rate equations for the system are when CA > 0, CT' = 0 and dCA/dt = - p when CT' > 0, CA = 0 and dCT'/dt
= p
(3)
(4)
where t is the elapsed titration time (sec.), CX is the and p is deconcentration of species X (moles/~m.~), fined by eq. l. If the A-A' couple is potential-determining, then the Sernst expression becomes simply (at 25 ")
where E is the zero-current potential a t time t , EO' is the formal potential of the A-A' couple, n A is the number of electrons involved in the rate-determining step of the electroreduction of A to A', and t* is the value of t a t the equivalence point. E and EO' are, of course, referred to the same reference electrode. This mode of formulation is simple and easily manipulated. The assumptions made will be closely satisfied by many chemical systems, particularly when the rate of titrant addition is low. Moreover, this formulation permits the potential-time function to be evaluated
COMPLEX FORMATION IN STUDIES OF CHEMICAL KISETICS
July, 1963
1529
even when both of the reacting couples are electroactive, since it eliminates the necessity of considering the titrant couple before the end point. An equation can be derived which, in principle, parmits the calculation of the zero-current potential from information on the composition of the solution even if the cathodic process is not the reverse of the anodic one. It isa 0.0591 CB ’ - log y n D - ,B’nE;’ CD
E=E‘f-
(6)
where y and p’ are effective transfer coefficients for the reduction of species D and the oxidation of species B’, respectively, n D and n B ’ are the numbers of electrons involved in the rate-determining steps of these processes, and Et is obtainable from free-energy considerations. The parameters y and 0’are assumed to be constant in the derivation of this expression; where it has been used in this work, it has been assumed that ( y n D - P’nB’) = 1. The procedure recommended below for the treatment of potentiometric data is a simple one that provides a sensitive diagnostic technique for the identification of the various kinetic complications that may occur. As was shown above, for the straightforward reduction of a single species the POtential before the equivalence point is described by eq. 5; consequently a plot of E or (E’ EO’) us. log t / ( t * - t ) should be linear and have a slope of - 0 . 0 5 9 1 / n ~ v. This is not so when a slow side reaction also occurs, Plotting experimental data in this way will then give non-linear curves whose comparison with the theoretical ones should provide much insight into the nature of the system. It is arbitrarily assumed here that the titrant is a reducing agent. Cases like those below in which the titrant is an oxidizing agent will be described by equations identical with those given here except for the sign of the change of potential with time.
Results and Discussion System 1.---This system may be represented by the scheme
-
T’
k-1
A +A’
B-
kl
That is, species A, which on reaction with the titrant T’ undergoes reduction to A’, is in slow equilibrium with another species, B, in the same oxidation state. The first-order or pseudo-first-order rate constants k, and IC-, are expressed in sec.-I. This system is described by the equations dCA/dt = --~ACACT‘ - k.-ICA
(8)
- ~ACACT’
(9)
k-icA
dCT’/dt
p
=
(7)
- kiCB
dCB/dt = dCA‘/dt
+ klCB
dCT/dt = ~ACACT’
(10)
It is convenient to define a time T such that CA--c 0 as t + r. Then, with the aid of eq. 3 and 4,the solutioiis of the rate equations become, for t 5 T (6) R. E. Cover, Ph.D. IXssiertation, Polytechnic Institute of Brooklyn, 1962.
+
where G1 = (kl IC-,) and Cx0is the value of CX a t t = 0. If species A and B are assumed to be in equilibrium a t t = 0, eq. 11 and 12 may be further simplified to
+
CA = 2- P [IclGl(t* - t ) - I C - ~ GI
- t ) + k-1
P
CB = - [k-lGl(t* Gi
Ic-le-‘’tl
(13)
- IC-~~-‘’~]
(14)
The exponential terms in eq. 11-14 may be made negligible by increasing the sample size or decreasing p. Then Ca and Cg will decrease linearly with time. If meither can be followed during the titration (by amperometry, spectrophotometry, or another technique), the rate constants cart be evaluated from data on the linear portions of the concentration-time curves. One has
12-1
-XB
:= (IB
+ t*XB)(KI + 1)
(16)
where I X and XX are the intercept and slope, respectively, of the linear portion of the plot of Cx us. t , mid K1 = kl/k-l. In applying eq. 15 and 16, the equivalence-point time t* would, of course, be calculated a priori from the concentrations of the solutions, the rate of addition of titrant, and the stoichiometry of the reaction. Equilibrium measurements are not necessary for the evaluation of K1, which is more conveniently obtained from these equations as
+
+
Ki = - SA(IB ~*XB)/XB(IA SA)
(17)
Obviously it is unnecessary to know the actual concentrations of A and .B (i.e.,the constant of proportionality relating these concentrations to the measured diffusion current, absorbance, etc.). Finally, the calculation can still be performed even if only one of these substances can be followed, since CA
+ CB
= p(t*
-
t)
(18)
which, of course, is valid only for t 5 7 . The value of r in any experiment may be estimated by considering its properties: (1) 0 5 r 5 t*, (2) CAk= 0 for 1 2 7 , and (3) only if t 5 T can CB vary linearly with time. When t 2 7 (so that CA = 0), theassumptions that A and B are in equilrbrium at t = 0 and that the exponential terms in eq. 13 and 14 are negligible a t t = T lead to the follo~vingsolutions of the rate equations.
RICHARD E. COVERAXD LOUISMEITES
1530
9101. 67
101
IC
(56 1.c
0.I
i t
QO II t.0.1
I
I
I
00
-0.2
o'o(E-d,vo~;? of ( E - Eo') us. log t / ( t * -
Fig. 1.-Plots t ) for system 1. The A-.4' couple is assumed to be potential-determining; n.4 = 1, k-I = look, 1.67 sec.-I, t* = ( a ) 600, ( b ) 6000, and (c) 60,000 sec. Curve ( d ) represents the simple equilibrium case in which the side reaction B A does not occur.
Icl(t* - t )
CT
= C A ' = p (t*
k-1 - -~ exp kl(t* klG1
[
-
Gi
(19)
k-l]
- t) -
"-3 -
Gi
"-Ill
-GI
(21) It is clear from eq. 19 that a plot of 111 CB us. t in this interval will be linear and will have a slope equal to -k l . Consequently, if Cg is followed throughout the titration, ICl can be evaluated. Then
The value of ICl may also be determined by following CAI or CT in this interval, since, from eq. 20, a plot of either In ( p i * - CA') or In (pt* - CT) us. t n-ill be linear aiid will have a slope equal to -ICl. The rate constants can also be evaluated from potentiometric data if the potential in the interval t 5 T is controlled by the A-A' couple and if the value of CA is known a t some time during this interval. If, in addition, the exponential term in eq. 11 is negligible for an appreciable portion of this interval, then since CA'
= pt
(23)
I
I
-0.2
-0.1
(E - E)', v o L T s .
Fig. 2.-Plots of ( E - Eo') us. log t / ( t * - t ) for system 1. The B-A' couple is assumed to be potential-deterrnining; n.t = I, k-I = look, = 1.67 sec.-I, t* = (a) 60, ( b ) 690, and (c) 6000 sec.
throughout this interval, one can use the value of CA' together wiih the measured potential and the Nernst equation to calculate the change in C.4 relative to some known value of CA during the titration. At any pair of times t1 and fz one has
Ez kl(t* - t )
I
0.0
tz - E1 4-0.0591 -logn9
tl
0.0591
= ___
nA
Chz log -(24) CAI
This permits the evaluation of the function R,, which is at varous times during the defined as the ratio CA/CA,, titration. When the exponential term in eq. 11becomes negligible, Rt becomes a linear function of time. Data obtained in the time interval in which this is so may be used for the evaluation of the rate constants in the following manner. Let RM and RN be two values of Rt a t the corresponding times thf and t N on the linear portion of a plot of Rt us. t, aiid let R,' = RMMRN. Then it can be shown6 that where
If CAk0(the value of CA a t t = 0) is known, and if the system was in equilibrium at t = 0, then kl is given by
COMPLEX FORMATION IN STUDIES OF CHEMICAL KINET~CS
July, 1963
i58i
and
If the value of CA is known a t t M but not a t t = 0, one ~ Ca,/ may evaluate CA at t:q (from the relation C A = RI') and then calculate kl from CAM - C A = ~ p(tx - t~)(&i - h)/Qi (29) and k-, from eq. 28. In a titration of this system, the end point precedes the equivalence point if Ca is follom-ed or if the A-A' couple controls the potential before the end point. This is because CA --$.0 as t -+ 7 , while T is given by
Since klGl > 0, clearly T < t*. The absolute error, t* - T , is constant and equal to l/KIGl, but the relative error, ( T / t * ) - 1, approaches zero with increasing t*. The end point aho precedes the equivaleiice point if it is the variation of CA', CB, CT, or C(T'that is followed. Increasing t*, either by increasing the sample size or by decreasing p, tends to decrease the error in all of these cases. Figures 1 and 2 show plots of the potential function ( E - EO') us. log t,/(t* - t ) and illustrate the behavior that will be observed in this system on varying t*. For a simple reversible system, of course, such a plot should be a straight line with a slope of -0.0591/n v. and an intercept [at t / ( t * - 1 ) = 11 of 0. If the A-A' couple is potential-determining, however, the curve is not linear (Fig. 1) and the end point precedes the equivalence point. If the B-A' couple is potentialdetermining (Fig. 2 ) , a noli-linear curve is again obtained but the curvature is in the opposite direction and the end point follows the equivalence point. In either case the curves approach linearity as t"' increases and the end point error decreases. For the case in which the A-A ' couple is potential-determining the limiting expression as t* + is
+
0.0591 1 K1 E = EO' - log ___ %A Ki
System 2.--This scheme
0.0591
- __ nA
t
log t*
-t
(31) system may be represented by the T'
A
k-
-----)
A'
2
€3' k2
That is, species A is reduced to A' by the titrant T', and A' is in slow equilibrium with another species, B', which is in the same (reduced) oxidation state. The first-order or pseudo-first-order rate constants lc2 and 16-2 are expressed in see.-'. This system is described by the equations
- kAcAc'S' (32) dCA'/dt = ~ACACT'kzcn' - 1 ~ 2 C k ' (33) dCB/dt .= k - 2 c ~ ' - k#n' (34) dCA/dt
=
+
With the aid of eq. 3 and 4, the solutions of the rate equations for the interval t 5 t* are found to be
CB'
=
where Gz = k2 tions are CA' =
5
P
-
P = -
+ k-2e-Q"t]
(37)
+ Lz. For the interval t 2 t* the solu-
Gz2 CB'
[Ii-2 G2 t - k-,
{kzGd* - k-2 exp[Gz(t* - t)]) (38)
1
{lc-zG2t* - k-2 exp[G2(* - t ) ]
(22%
(39)
assuming that the expoiiential terms in eq. 36 and 37 are negligible at t = t * 7 a condition that will be satisfied if t* is sufficiently large or if p is sufficiently small. When, in the interval t 5 t * , the exponential terms in eq. 36 and 37 are negligible, CA' and CB' will increase linearly with time. If either can be followed during the titration, the rate coilstants can be evaluated from data on the linear portions of the concentration-time curves. One has
+
ka = ASA'/IA'(K~ 1) ==
(40)
+ 1)
-SB'/IB'(K~
(41) where I X and SXwere defined above (cj'. eq. 15 and 16) ~. to eq. 17 and Kr = l ~ ~ / k - Analogously 12-2
Kz
=
-~A'IB'/#B'IA'
(42)
If only one of these two concentrations can be followed, the other caii be calculated from the relation f CB' = pt
(43) The rate constants can thus be evaluated in a manner similar to that outlined for system 1 in the discussion of eq. 15-18. They may also be evaluated from potentiometric data if the A-A' couple is potential-determining in the interval t 5 t * , if either K2or the value of CA' a t some instant during the titration is known, and if the exponential term in eq. 36 is negligible for an appreciable time in this interval. Then the value of CAgiven by eq. 35 can be used, together ~ 7 i t hthe measured potential and the Neriist equation, to calculate the change in C.4' relative to some arbitrarily chosen value of Ca' during the titration. The function Rt', defined as the ratio CA'/CA~',can then be evaluated a t various times during the titration (cj. eq. 24 and the accompanying discussion). When the exponential term in eq. 36 becomes negligible, Rt' becomes a linear function of time. As in the discussion of eq. 25-29, let RRI' and RN' be two values of Rt' on the linear portion of a plot of ICt' us. t, and let R2' = R ~ ; ~ ' / R NThen '. CA'
kzG2 = k-zQ2
(44)
where Q 2 = (1 - BZ')/(BZ'td - h') (46) If K , is known, li2 and k-? caii be calculated from the
equations
Kn
= (Qz
- lcz)/kz
(46)
RICHARD E. C O V E R
1532
ASD
LOUISRTEITES
Vol. 67
10
11
(T. t
1.1
0
/
0.0
I
t 0.2
- E'),VOLTS.0.0 Eo') us. log t / ( t * - t ) for +O.l (E
Fig. 3.-Plots of ( E system 2. The A-&4'couple is assumed to be potential-determining; n A = 1, k-2 = l00kz = 1.67 see.-', t* = ( a ) 60, ( b ) 600, and (c) 6000 sec. Curve ( d ) represents the simple equilibrium case in which the side reaction ii' B' does not occur.
-+
and = kz2/(Q2
- kz)
(47) If, on the other hand, K Zis not known but the value of CA' is known a t some time 0, k2 may be calculated from the expression IC-q
and k-2 may then be calculated from eq. 47. Figure 3 shows plots of the potential function (E EO') us. log t l ( t * - t ) and illustrates the behavior that will be observed in this system on varying t*; it is assumed that the A-A' couple is potential-determining. The shapes of these curves are characteristic of the system, as is also the case for those shown in Fig. 1 and 2. They approach linearity as t* increases, and the limiting expression as t" --t 00 is
E = Eo'
IC2 0.0591 t - 0.0591 ___ log ___ - -log nA 1 $. Kz nil t* - t
(49) The use of amperometry, spectrophotometry, or a similar technique for locating the end point in a titration of this system may give rise to very large errors, as is shown by Fig, 4, which mas calculated with the same values of the different parameters as n-ere used to obtain Fig. 3. Similar curves for the other systems considered here and in a subsequent paper may be found in the junior author's dissertation.6
Fig. 4-Plots of C ~ ~ / C Az's. + " t / t * for system 2 . All parameters are identical with those for the similarly labeled curve8 in Fig. 3.
System 3.--This scheme
__
system may be represented by the T'
k-1
B
kl
A
---j
A-
2
BI
A' kz
That is, species A is in slow equilibrium with another species, B, in the same oxidation state; on reaction with the titrant, T', species A is reduced to A', which in turn is in slow equilibrium with another species, B', in the same (reduced) oxidation state. The titration of osmium(1V) with chromium(I1) in hydrochloric acid solutions has been founds to exemplify this system under certain conditions, and will be described in detail in a later paper. System 3 clearly combines the processes represented by systems 1 and 2; as would be expected, many of the equations valid for the latter systems are valid for system 3 as well. This system is described by rate equations identical with eq. 7, 8, 9, 33, and 34. If a time r is again defined such that CA* 0 as t * r, these rate equations can be integrated as before. For t 5 7,the variations of CAand Cg are described by eq. 11-14, derived above for system 1; and the variations of CAI and Cgl are described by eq. 36 and 37, derived above for system 2. For t 2 7 ,the variations of CA, Cg, CT, and CT' are described by eq. 19-21 derived above for system 1; the variations of CA' and Cgl in this interval, however, obey equations that are unique to system 3. These are rather complex and illsuited to the evaluation of the rate constants, and therefore are not presented here; they may be found in the junior author's dissertation as eq. 213 and 214.
COMPLEX FORMATION IN STUDIES OF CHEMICAL KINETIC$
July, 1963
1.533
(78 1.0
-
0.1
-
0.0 I t 0.1
I
I
0.0
(E-€*'),
-0.1
0.0 I
I
-0.2
VOLTS.
Fig. 5.-Plots of ( E - EO') us. log t / ( t * - t ) for system 3. The A-A' couple is assumod to be potential-determining; n A = 1, IC-, = kM2 = lookl = 1OOkz = 1.67 see.-', t* = (a) 600, (b) 6000, and (c) 60,000 sec.
If CA, CB, C A I , CB', or CT is followed during the titration by amperometry, spectrophotometry, or a similar technique, the rate constants kl, k-l, kz, and k-2 can be evaluated by the procedures described above. They can also be evaluated from potentiometric data if the A-A' couple is poltential-determining in the interval t T and if the exponential terms in eq. 11and 36 are negligible for an appreciable time in this interval. Under these conditions the Nernst equation becomes
