CHRONOPOTENTIOMETRY
Oct., 1957
at 860°K., the temperature at which the time to ignition is ten seconds, is 0.6 X assuming the ore-ignition reaction Mg
+ BaO, +MgO + BaO
(2)
The activity can also be calculated from the particle size of the powdered material. In this case, it is expressed as the ratio of the number of moles of reactant on the surface to the number of moles in the bulk of the particle, a relationship which has previously been applied to propagatively reacting system^.^ A comparison of the values for the activity obtained by these two independent means should provide a basis for predicting the limiting reactant in the pre-ignition reaction. The values obtained for the activity compare favorably for the magnesium-barium peroxide composition. The activity calculated for the 12.6 p barium peroxide particle from the moles-surface to moles-bulk ratio is 13.7 X lo", and for the 112 p particle of magnesium the ratio is 1.0 X 10". Both of these values nominally agree to within an order of magnitude with the experimental value of 0.6 X lov5. The agreement is good for this type of treatment even though the calculation alone is not sufficient to enable us to establish definitely magnesium or barium peroxide as the limiting re-
1405
actant. However, in conjunction with the following considerations the values indicate fairly clearly that it is the magnesium. Qualitatively, on a basis of particle size magnesium would seem to be the limiting reactant. It is a much larger particle than the barium peroxide and consequently has a smaller surface area per given weight of material. For reaction to take place, magnesium atoms must come into contact with barium peroxide molecules and this can only occur at the surfaces of the particles. It would appear then, that the rate of reaction is limited by the particle having the smallest surface area, vix., the magnesium particle. From a quantitative stand-point, these moles-surface to moles-bulk ratios were calculated assuming spherical particles. Atomized magnesium is specially manufactured to ensure this but barium peroxide powder is not. Since the barium peroxide is an irregular particle, it is likely that its surface area is greater than that calculated from the average particle diameter of 12.6 p obtained experimentally. If this is the case then the value of the activity calculated above from the moles-surface to moles-bulk ratio for barium peroxide will be correspondingly greater, lending support to the choice of magnesium as the limiting reactant.
CHRONOPOTENTIOMETRY-STEPWISE POTENTIALTIME CURVES FOR AN ARBITRARY NUMBER OF REDUCIBLE SPECIES BY TOMIHITO KAMBARA* AND ISAMU TACHI Department of Agricultural Chemistru, Kuoto University, Kyoto, Japan Rdcezved April 18 1967
Equations are derived for potential-time curves and transition times for reduction of an arbitrary number of substancns in chronopotentiometry. A correlation between the general equations for cbhronopotentiometry and polarography is established. These results agree with equations previously derived by other authors for the partidular case of two substances. It is also shown how the general equations of chronopotentiometry can be deduced from the simplified model of the diffusion layer of von Stackelberg.
Theoretical analysis of voltammetry at constant current (chronopotentiometry) was developed by Weber, Sand and Karaoglanoff some fifty years ago for the reduction or oxidation of a single substance. Further work was done quite recently by Gierst and Juliard and particularly by Delahay and co-workers. 1, A theoretical analysis of the method in the case of two reducible substances was made by Berzins and D e l a h a ~ . ~The . ~ relationships obtained by these authors can be generalized for the case of more than two substances as was pointed out by Reilley, Everett and Johns.5 A general formula for transition for several substances is derived in this paper first by a rigorous method and then by * Department of Chemistry, Faculty of Science and Engineering, Ritsumeikan-University, Kyoto, Japan. (1) For a general review, see P. Delahsy, "New Instrumental Methods in Electrochemistry," Interscience Publishers, Inc., New York, N. Y., 1954,p. 179-216. (2) P. Delahay and G. Mamantov. A n a l . Chem., 27, 478 (1955). (3) T. Berzins and P. Delahay, J . Am. Chem. SOC., 76, 4205 (1953). (4)See also P. Delahay and C. C. Mattax, ibid., 76, 874 (1954). (5) C. N. Reilley, G. W. Everett and R. H. Johns, Anal. Chem., 27, 483 (1955).
the von Stackelberg simplified model for the diffusion layer.6 Potential-Time Curves for an Arbitrary Number of Reducible Substances.-We consider the reduction (oxidation) of an arbitrary number of substances under conditions prevailing in chronopotentiometry : mass transfer solely controlled by linear diffusion. The concentrations of the reducible (0) and reduced (R) species are solutions of Fick's equation for the following initial and boundary conditions. Fort = 0 coi (2,0) = c o i ( m , t ) = CRi (2,0) = C R j ( 0 0 , t ) =
*cOi *CRi
1
(1)
where the "Care the bulk concentrations. For t 0 and x = 0
Z-
E
io
9
niFDoiM!o,/dx
= 1
( G ) M. von Stackelberg, 2. Elsklrochem., 45, 465 (1939).
(2)
>
TOMIHITO KAMBARA AND ISAMU TACHI
1406
where io is the current density, I the current, q the electrode area, ni the number of electrons involved in the i-th electrode reaction, Doi the diffusion coefficient of the i-th reducible substance, and x the distance from the electrode. Furthermore, the condition of continuity of flux must be obeyed a t the electrode, Le., for t > 0 and
x=o
Do hCoi/& = -DRibcRi/hX
+ Zo,(t)
m
Y
T, =
Ti
i=1
ri being the transition time for the i-th system. There follows from (13) the equation
(3)
By solving the boundary value problem, and introducing the concentrations in the Nernst equation (exchange currents for electrode reactions large enough) one obtains by Laplace transformation the interfacial concentrations OC given by ocoi = *co,
Vol. 61
I
with
which is the general formula suggested by Reilley, et U Z . , ~ on the basis of the equation derived by Berzins and Delahay3 for the particular case of two substances. The potential-time curve for the p-th system is, on the basis of eq. 12 where '/2Ep is the polarographic half-wave potential. The potential '/%Ep is observed a t time'
Ki = (fOi/fRi)(DR,/Doi)'/'exP (si) (6) T i = ( n i F / R T ) ( E - '12Ei) (7) 'hEi = 4Ei ( R T / n F ) In (foi/fRi)(DRi/Doi)'/z(8)
+
where OEi is the standard potential for the i-th couple foi f~~are the corresponding activity coefficients. There follows from (2)
On the basis of the foregoing considerations one can develop a graphical procedure for the analysis of potential-time curves. The method together with experimental results will be published elsewhere. Transition Time for Varying Current.-If the current varies during electrolysis, one can derive by the method followed above an equation corresponding to eq. 12
(9)
or, when = 0 (no reduced species initially present a t any appreciable concentration) 2iot1/2 TV2
5
niFDoilh*Co,
i=l
2
[l
+ tanh( - F)]
(10)
This is the general equation for potential-time curves in the case of an arbitrary number of reducible (or oxidizable) substances. The right-hand side of eq. 10 is proportional t'o the mean current intensity observed for stepwise composite polarographic waves with a solution having the same composition as in corresponding chronopotentiometric measurements. According to the polarographic theory each reversible reduction wave is represented by the relationships Id,i
= KniDoi'/2*Coi
K
= 607mz/81~'/6
where F ( t ) is the current-time relationship. For instance, for a current (kL1/z)increasing linearly with t ' / z one obtains
where I i is the current along the i-th wave, I d , i the diffusion current for the i-th species, and K the familiar Ilkovic factor. Equation 10 for potentialtime curves in chronopotentiometry can now be written in the form
Transition Time for the p-th System.-The p-th system is jnvolved in the electrode reaction after complete depletion of the (p - 1)-th reducible species at the electrode surface. One then deduces from (12)
kt
It
Zi
;;i7P=C; 2=1
(19)
This equation which was derived by Sendas b.y a different method shows that transition times are proportional to concentration when the current increases with the square root of time. If the current is reversed ut the transition time 71 as in the method used by Delahay, et a1.,3,4one obtains9 the equation 2io[2(t T#/2 - 11/21 = (20) K'/2 K
-
nr,
which agrees with the result derived by Berzins and Delahay3 by another method.
Application of von Stackelberg's Model to Chronopotentiometry.-The important results of chronopotentiometry which have been established by rigorous analysis' can also be deduced from a simplified model of the diffusion layer according to a method developed by von Stackelberg6 for polarographic currents. From Sand's equation' for the transition time for a single substance, one deduces for the diferential diffusion layer thickness (7) The approach in the foregoing treatment is similar to t h a t used in the interpretation of the Heyrovski-Forejt method of oscillographic polarography: T. Kambara, Leubold Polarographische Be?., 2, 59 (1954). See also H. Matsuda, 2. Elektrochem., 60,617 (1956). ( 8 ) M. Senda, Rev. P o l a r o u m p h y ( J a p a n ) , 4, 89 (1956). (9) One has F ( t ) / q = iosdt) 2ioSTl(t) where Sa@)is defined by (Laplace transform) L-1 0
< t < k, and & ( t )
-
["I
-
= 1fork
< t.
= sr(t). One has S b ( t ) = 0 for
CHRONOPOTENTIOMETR~
Oct., 1957'
(21)
The concentrations are
1407
where 7 2 is the transition time for 0 2 . These results agree with those derived by Berzins and Delahays by more rigorous method.
*C
.and
According t o von Stackelberg6 the integral dif'fusion layer thickness A (Fig. 1) corresponds t o the total amount of electrolyzed substance, ie., for constant current electrolysis (von Stackelberg considered constant potential electrolysis) 601
=
1
2
-7r ao,
1
2 Ao1 (*cot
-
'C
I1 Fig. 1.-Diffusion layer according to von Stackelberg. The foregoing analysis can be extended to the case of more than two substances. As an application, we shall derive the equation of the potential-time curve for the p-th wave for an irreversible process. This case was not considered in the first part of the paper. Thus
I
J
0~01)
The results embodied in eq. 21 and 22 can now be readily derived from the above model. It follaws from Fig. 1 that the areas of the triangles OAB and 0A"B' are proportional to the quantity of electricity consumed at time t . Hence, one has the results
The contribution of p-th system to the current density is which agree with eq. 21 a s d 22. I n the case of two substmces one haB %heequations 4i0t p = nl*ColGO, n,(*Co, - a C ~ I ) 6 ~ z ( 2 7 )
+
i, = npFDOo*cop - OCO,
(35)
60
Furthermore, the current for an irreversible electrode process is (the backward reaction is neglected) -a,n,*F(E - 'E,) i, = npFoCOpOkpexp (36) RT where %, is the rate constant a t the standard potential OE,, cyp the transfer coefficient, and np* is the number of electrons involved in the rate-determining step. Hence, the potential-time curve obeys the equation I
which hold for the time intierval T~ < t
< TI + 72.
Hence
(29) Since the left-hand side of eq. 28 w n t a i n s only quantities for substance 01, and the right-hand side quantities for substance 0 2 , both sides must be c q u d to a constant independent of the experimental con
