J. Phys. Chem. 1982, 8 6 , 3492
3492
Reply to the Comment on "Photosensitlzatlon of Titanium( I V ) Oxide wlth Tris( 2,2'-bipyridine)ruthenlum( I I ) Chloride. Surface States of Titanium( I V ) Oxide"
Sir: The apparent failure to satisfy the second boundary condition arises from a printing error; the inadequate expression for the decay i,, with time was due to the approximation used. The differential equation governing a solution-sensitized semiconductor electrode according to the relation i
2
3
4
5
6
7
0
9
IO
hu
A e B in solution
-
Flgm 1. Plots of the anodlc photocwent evolution vs. time, according
to eq 8 and 9, for different values of p .
where z = k f t . The time derivative of the anodic photocurrent is 6iap
- = FAkkfbo erfc ( P z ) ' / ~exp[(p - l ) z ]
-
6t
< 1,erfc (Pz)lI2 and exp[(p - l)z] both vanish at t When p > 1, exp[(p - l)z] tends to infinity for large z but, since erfc ( P Z ) ' / ~N exp(-pz)/(rpz)'/2 for large z m.
Si, - N FAkkfboexp(-z)/(rpz)'12 at
for sufficiently large values of z and for arbitrary p . Hence 6ia,/6t is only vanishing when z tends to infinity. The plateau value of the anodic photocurrent is obtained from eq 9: iap(z m) = FAkbo/(p112+ 1)
-
In Figure 1,we represent i, as a function of z for various values of p . This dlows a comparison of the predictions of eq 11in ref 1and those of this work (see eq 8 and 9). Consequently, we conclude, in agreement with the assumptions made in eq 2, i.e., a constant concentration of A and a constant rate of production of B, that the fastrise-time anodic photocurrent reaches a plateau when t 1 D / k 2 . This conclusion is in contradiction with the one of previous work where, according to eq 11 of ref 1, the current falls off in a pure diffusion mode for times t 1 D f k2, contrary to what one would expect from the assumptions underlying the differential equation (eq 2). Acknowledgment. We thank Professor J. Nasielski for his interest in this work. R. D. thanks the "Institut pour la recherche scientifique dans 1'Industrie et 1'Agriculture" for a fellowship. Universit6 Libre de Bruxelles
B B+ + e- at the electrode where k is an electrochemical rate constant [cm/s], has the form
(10)
For p
Facun6 des Sciences Service de Chimie Organique et
ki
k
J. Brocas R. Dewm A. Klrsch-De Memaeker'
Organique Physique
B- 1050 BruxeUes, &/gium Received: February 16, 1982; In Final Form: June 18, 1982
0022-3654/82/2086-3492$01.25/0
where bo = @JaaA/kf. In our original treatment, as Brocas et al. surmized, we oversimplified the treatment of the generating and fluorescent terms. The equation may be solved directly; taking Laplace transforms, and using the transformed boundary conditions 6(x) 0 x: w
- -
D ( a b / a x ) , = k6(o) we find
kb,kf exp(-xQ) b- = - - bokf s(kf + S ) s(kf + S)(k + DQ) Q = [ ( k f+ s ) / D ] " ~ By using standard operational formulae and the convolution theorem, we can invert this to yield the solution of Brocas et al. From their integrated form cap
=
FAkbo P-1
(erfc (pz)lI2[exp(p -
1)2]
-1
+ p'12 erf z ' / ~ )
and bearing in mind that p
