Evaluation of Rate Constants for Consecutive and Competing First-
or Pseudo-First-Order Reactions, with Special Reference to Controlled-Potential Electrolysis
by Robert I. Gelb’ ant- Louis Meites Drpnrlmenf o j Chcmzstry, Pol&chnic
Insticute o j Brooklyn, Brooklyn, L V e York ~
(Received J u l y $1, 1863)
For a number of systems iiivolving consecutive and competing first- or pseudo-first-order chemical or elccti*olytic reactions, procedures are described for the evaluation of the rate constaiits. usiiig coulometric or other data on the extents of reaction a t different times. Scvcral of these systems arc exemplified by the controlled-potential electroieduction of a-furiltliosimc a t lwious pH values, and in cach such case the application of tht? kinetic ctqimt,ions is illustratcd with the aid of cxperimcntal data. Calculated plots of In i us. t arc givcii for most of the systems considered and should aid in their recognition when t’hey arc c?ricountcred urider novel circumstances.
Introduction I’ro(:cdiircs for t’lic evaluation of the rate constants of chcniical u i i d c1ecti~)Iytic st,cps occurring during c!oiit idlctl-potciit ial elcctrolyscs ha\-(? been described for a i i u n i l ) r r of WSPS. If oiily a siiiglc clcc!troii-ti.arisfcr st,cp can occur, and if its rate is cwit>imlledt)y thc rate of mass transfer of t tic iducit)lc siihstaiice :I to t,hc electrode-solution iiit,orfaw,tlic elcct,i’ol>,ticitate constant’fl is given by2
the rcducible substance A is regenerated by a chemical rcactiori TI3
+x
k
SA
+Y
occurring in the bulk of the solution as rapidly as . I is consumed electrolytically by the reaction
x
+ ne
B
23
the order and rate constant of the chemical step can be obtained from the e q ~ i a t i o n ~ - ~
in which the steady-state values of CAand CB are easily calculated from the values of the steady-state current, thc initial currelit, arid the initial concentration. The rate constants of electrolytic steps occurring ~~~
~
Science I’oundatiorl 17ndergradunte Research Participant, Surnlner, IBfi2: this paper is based in part or1 t h e thesis subrnit,ted by llohert I . Cklb t o t h e I$’ac-ultyof the Polytechnic Institute of I3rooklyn i l l p:irtiaI fulfillnierit of the requirements for the degree of B w h e l o r of Science in (’tiernistry. June. 1963. (2) J. J . Iiiigaiie, , I , Am. C h i m Soc.. 67, I9lG (1945). ( 3 ) L. XIeites aiid’S. A , Moros. Anal. (‘hem.. 31, 2 3 (1959). (4) L. AIeitea, Record Chem. Progr., 2 2 , 81 (1961). (5) S. .A. 3Ioros a n d L. RIeites, J . Elrctroanal. Chem., 5 , 103 (1963). ( 1 ) S:itional
EVALUATION OF RATECONSTANTS
63 1
simultaneously but a t considerably different rates can be obtained by graphical dissection of a plot of In i us. t.* Finally, a procedure has been describedl whereby the rate constant for the dimerization of the intermediate I to give the electrolytically inert dimer D can be evaluated in the scheme
A much mQreprecisely and (if a direct-reading current integrator is available) equally easily measurable timedependent variable during a controlled-potential electrolysis is the quantity of electricity &, (in mf.) that has been consumed since the start of the electrolysis. This is defined by the equation
B
A
+ nle +I
Qt =
k
21 +D
I
+ nze
B -3
B
The present paper describes procedures for the evaluation of electrolytic and chemical rate constants in cases in which two or more first- or pseudo-firs$order reactions, which may be either electrolytic or homogeneous, occur consecutively or simultaneously. Examples of several of these cases have been found in the controlled-potential electroreduction of a-furildioxime at various pH values,S and experimental data are used to illustrate the application of the procedures and the precisions of the values obtained by their use.
Fundamental Considerations Previously described procedures for the evaluation of rate constants of electrolytic and chemical reactions occurring during con trolled-potential electrolyses have always been based on measurements of the current as the independent variable. MacSevin and Baker, for example, who performed direct measurements of current throughout such electrolyses, found that they led to current integrals that were in error by about =IZL’(%~; this appears to correspond to errors of the order of *t-IOOj, in the individual measurements. Such errors are probably attributable to random shortterni fluctuations of stirring efficiency and, with mercury working electrodes, of electrode area. Ficker and Meitese minimized the effects of these fluctuations by calculating currents from the equation
(3) where Q, is the total quantity of electricity accumulated in the interval 0 5 1 5 tl, and Z is the average current in the interval ti 5 t 5 tz. However, if this interval is too wide, the value of i obtained from eq. 3 will differ appreciably from the actual current a t 1 = (tl t 2 ) / 2 , tirhile if the interval is too narrow the experimental errors involved in the measurements will lead to a serious relative error in 3. These considerations limit the accuracy that can be obtained when rate constants are calculated from values of the current.
+
i dt
(4)
In principle, the n rate constants that characterize any particular system can be evaluated by solving the relevant differential equations (with due attention to their boundary conditions), making the appropriate substitutions into eq. 4 t o obtain an equation describing Q,, substituting the data obtained a t n or more different times into this equation, and solving the resulting set of simultaneous equations. For any system except the very simplest, however, the equations describing i and &, are so complex that such a procedure would be excessively laborious. 3 Ioreover, as will presently appear, the plots of In i vs. t are often much too intricately shaped to permit graphical evaluations of the rate constants. A procedure is therefore recommended in which a preliminary estimate of one rate constant (usually that for either the first or last electrolytic step) is first made and used, in conjunction with experimental values of Q t , to evaluate the rate constants of the other steps to obtain a set of preliminary estimates which is then further refined by it,erative calculations until satisfactory constancy is attained. Although the plots of In i v s . t are of little use in the evaluation of the rate constants, they assume more or less characteristic forms for different systems and for different ratios of the rate constants in a single system. A number of representative plots of this type have therefore been calculated and are presented below for their diagnostic value. It is assumed that the system under consideration involves no kinetic or induced current^.^ It is also assumed that the experimental values of Q I have been corrected for any contributions due to charging, continuous faradaic, and faradaic impurity currents3; such corrections are most easily made by performing point-by-point subtraction of the chronocoulogram obtained in an electrolysis of the supporting electrolyte alone under identical conditions. Finally, it may be mentioned that an identical treat~~~
(6) H . K. Ficker and I,. lleites, Anal. Chim. Acta, 26, 172 (1962). (7) L. hleites, J . Eledroanal. Chem , 5 , 270 (1963) ( 8 ) R. I . Gelb and L. Jleites, to be publizhed. (9) W. M.MacNevin and B. B. Bnker, Anal. Chem., 24, 986 (1952)
ROBERTI. GELBAND Lours MEITES
632
merit may be employed in other situations quite alien to controlled-potential electrolysis. The coulometric data serve merely to indicate the total amount of reduction that has taken place and could equally well be replaced by any other data that would serve the same purpose. Ii'or example, if the reductions are brought about by a chemical reducing agent (present in large excess to ensure that the reactions will be pseudofirst-order with respect to the substances being reduced), precisely the same ends will be attained by spectrophotometric measurements of the concentration of the oxidized form of the reducing agent used, by calorimetric measurements of the heat evolved, or by a wide variety of other techniques. In the cases considered here, the coulometric n-values serve as proportionality constants relating the extents of the various reactions to the measured values of Q1. Other proportionality constants will be needed if other techniques are employed.
from which the current-time curve is easily calculated if values of the various parameters are assumed. Plotting log i US. t , as is the custom, one obtains a curve that may be concave either upward (if p1 > pz) or downward (if PI < Pz), the latter possibility being illustrated by curve a of I'ig. l ; in either event it tends to become linear a t large values of t .
Experimental The apparatus, reagents, and techniques used to obtain the experimental results cited below will be described in a later paper.* Results and Discussion nie
nle
System I . A -+-B +C
For the reaction scheme 81
+ nle B B + n2e +C
(5b)
dCA/dt = -PICA
(6)
A
--f
(W
82
one has
whence, since CA = COAat t
=
0,
CA = COAe-@lt
(7) 0
and
2
1
3
Time, ksec.
dCn/dt
=
PiCA - PZCR
(8)
where the p's are the electrolytic rate constants, described by equations of the form of eq. IC. Combining eq. 7 and 8, we obtain dCll/dt
=
PlCoAe-81t- P z C ~
Figure 1. Plots of log i us. t for system I : ( a ) (solid line), calculated from eq. 11, using P I = 2.20 X 10-3 sec.-l and 82 = 3 . 4 i X 10-3 sec.-L; ( b ) (open circles), data obtained in the electroreduction of a-furildioxime: see text for experimental conditions.
(9)
Integrating this and imposing the boundary condition Cn = 0 a t t = 0 gives
Integrating eq 11yields Q , = (nl
+ n z ) V C o -~ niVCo"-811 + (12)
The total current a t any instant is given by The .Journal of Physical Chemistry
whence, of course, a t very long times
EVALUATION OF RATECONSTANTS
&t
-
&m
=
(nl
633
+ n2)VCoA
(13)
'This sytem is exemplified by the electroreduction of a-furildioxime a t -0.19 v. os. s.c.e. in 6 F perchloric acid; a t this potential, which is on the plateau of the fiwt wave of a doublet, the value of Qm obtained exn2) = 6.010 f perimentally corresponds to (nl O.CiO2. The experimental plot of In i us. 1 is shown by curve b of Fig. 1 for an electrolysis in which VCOAwas 0.24585 mmoles. Extrapolating to 1 = 0 yields io := 2.20 wf./sec. (approximately 220 ma.). If another portion of the same solution is electrolyzed a t -0.40 v. us. s.c e. (which is on the plateau of the second wave) under identical conditions, precisely the same value of Qm is obtained, but the plot of In i us. t is perfectly linear and has an intercept io = 3.30 pf./sec. I n the light of eq. l a , this ratio of initial currents may be equated t o a ratio of n-values; in other cases, it should generally be possible to arrive a t estimates of nl and n2 from chemical considerations or auxiliary electromewic data. Combining eq. l a with the estimate T L ~ = 4 and the experimental values of io a t -0.19 v. and' VCOAyielded a preliminary estimate of P1 = 2.24 X 10-3 see.-'. One may now write eq. 12 for the experimental values of Q t a t two finite timea that are as widely separated as possible, substitute the preliminary estimate of p1into the equation written for the longer time and solve the result for pz, substitute this into the equatioii written for the shorter time and solve the result for p1, and so on. Large numbers of values of Q t are not needed, and a direct measurle of the reliability of the results is provided by selectjing different pairs of points. I n a typical case, four pairs of points in the range 0.2 5 QL/Qm 5 0.95 gave p1 = (2.20 f 0.02) X set.-' and pz = (3.47 i 0.05) X set.-', where the deviations are mean deviations. From these figures it may be inferred that the attainable precision can be expected to be of the order of f 2 % or better. This compares favorably with the precision that has heretofore been obtained iin evaluating rate constants for single-step processes yielding linear plot,3 of In i us. t. Cases may occasionally be encountered in which a priori information regarding the values of n1 and n2 is unavailable. A useful mode of attack is then to assume that p1and p2 are equal; as A and B will usually have similar sizes and structures, this can hardly be in grosii error. On this basis eq. 12 becomes
+
Q,
=
(ni $- ~ ~ ) V C O-AVC0Ae-"(nl
+ pnzt) (14)
where p is the common value of P1 and 02. eq. lS, this may be rearranged to give In
(QJIjCOA)
=
-Pt
In view of
+ In (nl + Ond)
where Q R is the quantity of electricity that remains to be accumulated after the first t seconds of electrolysis and is defined by the equation QR
Q m
-
(16)
Qt
Equation 15 shows that a plot of log QR/VCOAus. t will have a zero intercept equal to log nl; since the plot will not be linear, the data used in constructing it must be obtained a t relatively short times. Then n2 can be deduced from a n experimental value of Qm with the aid of eq. 13, and finally two estimates of p1are available: one from the initial current via eq. la, the other from the initial slope of the plot of eq. 15. The evaluation of p1 and PZ can then be carried out as described above. me
System I I .
A --+B
A convenient procedure for the evaluation of in this simple case, where eq. l a and l b have always been employed, may be deduced from eq. 12 by setting n2equal to 0 and simplifying; this gives Qt
=
~IVCOA- nlVCoAe-Bt= Q m (l e-8t)
(17)
which is most easily used if it is written in the form
where Q R is defined by eq. 16. I n a typical case, calculations based on eq. 18 and using 10 points over the range 0.3 5 Qc/Qm 5 0.95 gave, in mean, P = (2.16 f 0.01) X loe3 set.-'. The precision of this value is much better than that of the one obtained from the equation
p = - In (&/id t z - tl (which is equivalent to eq. 3) because of the previously mentioned errors that aflict measurements of current. The procedure is not only far easier than the customary graphical one, but it yields a result that is probably better (because it does not involve the uncertainty that always characterizes a graphical construction) and that is accompanied by a valuable indication of the reliability. I n using eq. 18, it is only necessary to avoid times so long that the experimental errors in Q R (including the associated errors in any corrections for the continuous faradaic background current) become appreciable. I n general, the precision can probably be expected to deteriorate as QR/Qm decreases below about 0.1. me
(15)
=
1238
System I I I . A +B --+C +D Volume 68, Yurnber S
March, I S @
634
ROBERT I.GELBASD LOUS MEITES .~
For the reaction scheme
A
+ ,pie
Pl
+B
(1 9 4
k,
B-C
(19b) Bs
C
+ n3e --+D
(19c) where eq. 19b represents a first- or pseudo-first-order chemical reaction occurring in the bulk of the solution, while eq. 19a and 19c represent electron-transfer steps whose rates are controlled by the rates of mass transfer of A and C, respectiveIy, to the electrode-solution interface, one has
which is analogous to eq. 9 but involves the rate constant k z of the chemical step described by eq. 19b. Integrating eq. 20 and imposing the boundary condition CB = 0 a t t = 0 gives
This in turn may be combined with the equation
0
and the boundary condition Cc
=
0 at t
=
2
1
0 to give
+
3
Plt.
Figure 2 . Calculated plots of log i us. t for system 111: ( a ) P1:kZ:P3 = 1 : 3 . 2 : ( b ) PI:k2:@3 = 1 : l . l : O . S ; ( C ) Pi:k*:P3= 1:0.1:0.5.
The total current at any instant is given by
Like eq. 11-which is easily obtained from eq. 23 by letting IC2 increase without limit and replacing p3 and n3 by p2 and n2,respectively-this permits the calculation of plots of In i us. t if values of the parameters are assumed. The three characteristic shapes are shown in Fig. 2 . If both k2 and B3 are larger than pi, the curve is concave domnward; if either k 2 or P3 is larger than, while the other is smaller than, PI, the curve has a shoulder; if both IC2 and 6 3 are smaller than 61, the curve is everywhere concave upward. These are illustrated by curves a, b, and c, respectively. T h e Journal of Physical Chemistry
whence one can obtain the expected description of Qm
Qm = (nl
+ n3)VC0~
Equations 24 and 25 together give QR
=
nlVCoAe-P1t-
[
Pik2P3n3VCo~ k2 - Pi
e-"' 2 P3
- k2)
-
e -81 1 Pl(P3
-
P1)
+
EVALTATIOS OF RATECO~YSTASTS
635
These equations are sufficiently complicated to make a detailed description of their application seem warranted. R e have found this system to be exemplified by the reduction of a-furildioxime at potentials on the plateau of the first wave a t pH values between 1 and 2 . Here the experimental values of Q e correspond to (nl n3) = 6.00 A 0.02. It was assumed that nl = 4, not only by analogy with the conclusion reached. as described above, for the reduction in 6 F perchloric acid under otherwise identical conditions, but also because this assumption yields a preliminary estimate of pl that is not very different from the value obtained in 6 F perchloric acid, and wide varistions of ,B1 between these media seemed unlikely. This assumption, together with eq. l a , gave the preliminary estimate PI = 2.0 X see.-'. Other estimates of p1 were obtainable from the values of Q 1at shorl times: if so little C has yet accumulated that the current is almost entirely due to reduction of A, one may employ eq. 17 for this purpose. Estimates of p3 may be obtained from equations that provide crude approximations to the values of i and Q a t long times (where only C and D are present a t appreciable concentrations). Assuming that the starting material mas C instead of A, one has
+
i &t
=
=
nlvc0
P ~ ~ ~ v C O ~ - (27a) ~ ~ ~
+ n3vC0(1--
Qp, = 723VC0e-P31
1
(27b) (27c)
where VCois taken as the number of millimoles of A originally present. In the present case these yielded p3 = 1.3 X l o w 3sec.-l; as expected, this is not grossly different from the preliminary estimate of 61. '4 priori, one might assume p3 = p1 for n-ant of any knowledge to the contrary. We have found that the final value of p3 is usually intermediate between these two preliminary estimates, and it therefore seems appropriate to use their mean (which in this case is sec.-l) in the subsequent clalculations. 1.7 X One may now write eq. 26 for three different times: tl near the start of the electrolysis, ta near its end, and tz intermediate between these. Substituting the preliminary estimates of 01 and 63 into the equation for Q, a t t = tl yields an equation with IC2 as the only unknown. Its solution can be used, together with the original estimate of p1 and the equation for Q R a t t -= tal to obtain a better value of p3, and so on. This i6 continued until additional steps produce no significant change in the value of ICz. The nature of the convergence is illustrated in Table I by the summary of the calculations based on the above data. All values are given in ksec.-l. The first line gives the preliminary
estimates of 01 and p3 and the value of k2 calculated from these; in each subsequent line it is the italicized value that has been calculated from the values of the other parameters.
Table I Pa
P1
First approximation Second approximation Third approximation Fourth approximation Fifth approximation Sixth approximation Seventh approximation
2.0 2.0 2.0 2.0
ki
1.7 1.88 1.88 1.92 1.91 1.92 1.92
1.99
1.99 1.99
10 10 6.67 6.67 6.67 6.67 6.0
Further refinement is unwarranted, for there is no reason to believe the mass-transfer coefficients p1 and p3 to be constant to better than the 0.5% indicated by the last four lines. The values obtained may be tested using them to calculate values of Q 1 or Q, a t times not previously employed in the calculations, and comparing these with the experimental values. The rate constants shown in the last line of the above summary yielded, for instance, Q t = 0.3799 mf. a t t = 1500 sec., where the measured value was 0.3789 mf. The mean difference between the calculated and measured values was of the order of 1%,which is comparable with the probable uncertainty in measuring Q t . In connection with eq. 26, it was mentioned that the assumption p1 = p 3 may often be warranted. This leads to the equations
i
=
~
~
n
~
+~
~
~
~
e
-
~
1
~
and
while Q m is still, of course, given by eq. 25. In this case, after obtaining a preliminary estimate of p1 from the extrapolated value of io, one may select a time so long that the terms in e-"' in eq. 29 are negligible, so that the equation for Q R becomes
Neglecting
122
in the denominator (as in the similar Volume 68, +Turnher 3
M a r c h . 1964
ROBERTI. GELBAND LOUISMEITES
636
procedure recommended for handling several related equations in other systems) yields the preliminary estimate
which may be substituted into the coefficient in eq. 30 to obtain the better estimate
and so on. The result of this series of successive approximations can then be substituted into eq. 29, written for the value of Q t a t some relatively short time, and the equation thus obtained may be similarly solved for p1 to obtain an estimate that is better than the preliminary one. I n a typical example, using data obtained in the reduction of a-furildioxime in a buffer of pH 9.7, the preliminary estimate of p1 was 3.3 X sec.-l. When this was substituted into the equation for Q R a t t = 3100 sec., solution by successive approximations gave k2 = 3.52 X sec.-l. Substituting this into the equation for Q R a t t = 500 sec. gave p1 = 3.5 X sec.-l, and using this value in the equation for Q R a t t = 3100 sec. gave kz = 3.32 X lo-* sec.-l. Averaging the two values of k2 and substituting their mean into the equation for Q R a t t = 500 sec. gave p1 = 3.42 X set.-', and this in turn gave k2 = 3.41 X lop4 sec.-l, which was identical within o.3yOwith the value used in the last calculation of pl. It may be noted that only two points are needed in the calculations. It is, of course, desirable to repeat the calculations with other pairs of points, and these should be so chosen that the mean of the two times in each pair, t, is different for different pairs. A drift of k2 with varying 1 will then indicate that p1 and p 3 cannot justifiably be assumed to be equal, and in this case eq. 26 should be used as described above. I n the example cited here, the values of kz and p1 obtained from different pairs of points had mean deviations of only about f1-2y0. nae
nle
System I V . A
-+
E
+
iBs
1260 -+ F
(31e)
where eq. 31b and 31c represent first- or pseudo-firstorder chemical reactions occurring in the bulk of the solution while eq. 31a, 31c, and 31e represent electrolytic steps whose rates are controlled by the rates of mass transfer of A, C, and E, respectively, to the electrode-solution interface, the concentrations of A, B, and C are described by eq. 7, 21, and 22. I n addition, one has dCD/dt
=
PZCc - k i C ~
dCE/'dt
=
k 4 C ~- PBCE
and
i
PlnlvCA
=
+
f
03%T/'CC
(32)
@6%vCE
It is only necessary to combine the above equations with the boundary conditions CD = CE = 0 at t = 0 to obtain an explicit equation for the current. We have found no example of this system, and as its analysis would differ from that of system 111 merely in being I
I
I
1
I
n~e
B -+ C -+- D -+ E --+F
For the scheme
+ rile -+ B PI
il
(31%)
I
I
I
I
I
I
k*
B-C
0
4
2 012.
Figure 3. Calculated plots of log i us. t : ( a ) system Iv, P1:k2:P3:k4:P5 = 1:2:0.8:0.1:0.5, n1 = n3 = n5; ( b ) system V, P ~ : k z : k =~ 5 : l : l .
The Journal of Physical Chemistry
6
EVALUATION OF RATECOXSTANTS
637
somewhat more complex, a detailed description of the results and their possible applications in the interpretation of experimental data seems unwarranted a t this time. Calculated plots of In i us. t, of which the one shown as curve a in Fig. 3 is typical, are, as would be expected, recognizably more complex than those for system 111, as is seen by comparing this curve with curves b and c in Fig. 2 . me
System V . B
A +C
For the scheme
+ nle +C 81
A
Coa/CoB = k q / k 4
(334
a
+b
B W A
=
ab
1
dCA/dt
=
-PICA
+ k 2 C ~- k-2C.4
Pi =
+ k2 + k-z
(42)
P1k2
(43)
(34)
and
Equations 38-40 and 42-44 may be combined to yield dCB/dt
=
k-2CA - k,CB
(35)
Solving eq. 34 for CB, differentiating, and substituting the expressions for CBand dCB/dt into eq. 35 yields
(45) This provides a value of PI; k2 and evaluated from eq. 42 and 43. me
where c1 and c2 are arbitrary constants and a and b are the roots of the auxiliary equation of eq. 36, given by (P1
can then be
n2e
System V I . A +C ; B +D
whence
+
IC2
-I-
k-2)
--
If two unrelated species A and B are reduced simultaneously in accordance with the equations
+ me +C B + n2e --+ D 81
A
4 ( P 1 + k , + kP2)2 - 4j1&
(61 f
k2
4- k-2) -1 d ( P i f
k2
i
=
@l?%1vCoAe-81' f
(47)
P2n21."oBe-8zt
whence (37c)
Qt
=
i = PlnlVC.4 -- PlnlV(cle-"
TLiVC'a $. nzVCo~-
+ n2VCoBe-82t) (48) nlvcOA + n2VCOe (49)
(nlT'CoAe-81'
The current a t any instant is given by
+ c2e-bf)
Qm =
(38) and
while Qm is given by
+
Qm = n i V ( C o ~ COB) =
(46b)
the current a t any instant is given by
-t- k-2)' - 4Pika
2
I n addition, a t t
(464
81
2
(37b) b=
(41)
and (by rearranging eq. 41)
one has
a =
K
where K is the equilibrium constant of the reaction described by eq. 33; being a formal or conditional equilibrium constant, this will naturally vary with the composition of the supporting electrolyte. A typical plot of In i us. t calculated from these equations is shown as curve b in Fig. 3. It consists essentially of two linear segments, which can be dissected either algebraically, using the procedure recommended below for system VI, or graphically, by the procedure of Ficker and Meites,6 to yield values of the parameters a and 6. One .may then write
kz
k-
=
(39)
0 eq. 37 becomes COA
= c1
+
e2
(40)
and finally, if A and B are assumed to be in equilibrium a t t = 0
Q R == nlVCoAe-81'
+ n2VCOne--82z
(50)
It will be assunied that &> P2, so that A is taken to be the species whose concentration undergoes the more rapid decay. There is a trivial case in which Pi = pz; in this event a plot of log i us. t is strictly linear and there is no variation in the values of p obtained by Volume 68. Number 3
March, 1364
638
ROBERTI. GELBAR'D LOUISMEITES
applying eq. 18 to the experimental values of Q R a t different times. At a sufficiently long time one may obtain an experimental value of Q R that is described by the equation QR
=
n9VCoBe-oz1
(51)
so that (since nz may be obtained from the value of Q m in a reduction of B alone, while the values of V and COB for the rhixture will be known) one may estimate PZ from the equation
The resulting estimate of pz may be substituted into eq. 50, together with the experimental value of Q R a t some relatively short time, to give
and so on. As in the case of the simple system I1 this procedure should yield values of the rate constants that are appreciably more precise than those obtainable by traditional procedures, including the graphical one of Ficker and ,Ileites.6 The number of approximations required will be small unless arid 02 are very nearly equal or unless the initial concentrations of 4 and B are very widely disparate. me
System V I I . C +A
A
nae
B +D
(66b)
It is clear that system V above represents the still more special case in which P3 = 0. Experimental data conforming to the present case might be most easily analyzed by assuming initially that p1 = p3 and deducing their common value by a procedure entirely similar to that used in evaluating p1 in system V [ c j . eq. 45 and its accompanying discussion]. This mill in turn provide estimates of k z and IC-,. From these, the estimate of p1 can be refined by calculations based on the value of Q R at some relatively short time; the value of Q R at some relatively long time can then be used to obtain a better estimate of p3, the values of p1 and p3 can be used to refine the preliminary estimates of lcz and k-2, and so on in obvious fashion. be very large, one will have Should both k , and b >> a, and the exponential term in b will then vanish a t all t > 0. Equation 55 then becomes
d(ln i) - a = kt
Plk2
k,
+ 63k-2
+
k-2
(57)
or, in view of eq. 41
There are several important special cases of the general scheme 41
h
+ nle -+ C
(%a)
kz
B Z A OS
B
+ n3e --+D
(54c)
in which t!vo species A and B are initially in equi librium and can both undergo reduction a t the poten tial employed. One such case is that in which the equilibrium between A and B is labile and in which the reductions of A and B involve equal numbers of electrons (so that 12 = nl = 1 2 3 ) . For this case one may derive the equation
where el and cz a re dimensionless constant's and T h e Jotimal of Physical Chemistry
The value of a can be obtained from experimental data by a treatment exactly similar to that of the simple system TI. There are three particular cases of interest : (a) if K >> 1, a = P I ; (b) if K
