where the coefficients a, b, A and B are empirically determined positive values, and E is the electrode potential. Equation (1) holds throughout the latent period of transformation (O < t < A), where A is defined as the time corresponding to the maximum rate of change in electrode potential, [dE/dtl. The latency was interpreted to be directly related to the initial thickness of the AgCl film. It was concluded that I- is exchanged for Cl- and, if no current is drawn from the electrode, the I-Cl exchange is one-for-one. These relations suggest that the time integral of the I-Cl exchange rate is related in a simple way to the initial quantity of AgCl film per unit area. This paper establishes that the I-Cl exchange rate characterizes the transformation process in terms of (I-) and initial quantity of AgCl in the film. It is shown with the aid of a model that mass transfer limits the rate of I-Cl exchange.
Electrodes and solutions
The experiments illustrated in Fig. 4 (see below) made use of electrodes of the type shown in Fig. 1b. They were oscillated through about 90 deg at a rate of 31 c/min to achieve relatively constant stirring conditions. Experiments were performed in 100 ml of solution containing pI 3 and pCl 1, 2 or 3. The inflexion point of each transformation curve on the analogue records was used to determine A.
The over-all error of the experimental methods was judged to be less than 5 percent. The method of least squares was used to derive the coefficients of the equations fitted to the data.
A typical experiment showing the loss of I- from a gently stirred solution is represented in Fig. 2. Potentials of Ag/AgCl and Ag/AgI electrodes obtained simultaneously are plotted as functions of time. The measured surface area of the Ag/AgCl electrode was 0.135 cm2 and 3.40 x 10-6 mol AgCl was deposited on it. This represents an initial film density of 25.2 x 10-6 mol/cm2. The origin of the time axis corresponds to the time at which 5 x 10-6 mol KI was added to the solution. After the introduction of I- into the solution, El dropped rapidly to a minimum of about - 185 mV. After about 1000 s, the time rate of change of the potential reached the stable value illustrated. From the data points, it is apparent that the slope, [dE/dt], remains constant for a period, then abruptly diminishes. The intersection of the two straight lines shown in the graph is virtually identical to A, the time of maximum [dE(Cl,I)/dt].
The solution ionic strength and the calculated activity coefficient for I- do not change with time if I-Cl exchange is one-for-one. Therefore, since the solution volume is constant, the quantity of I- in solution at any time is given by
where 10 is the initial quantity of I- in solution and Eo is the extrapolated value at t = 0 taken from the data of Fig. 2. The I- lost from the solution is given by
where AE, = E1 - E0 and RT/F = 25-7 with units of mV. The results of applying this equation to the data of Fig. 2 are plotted in Fig. 3. The solid curve is a graph of (3) combined with the empirically derived relation between E, and time for (O < t < A),
At t = A, the calculated I is 3.37 micromol, compared with the initial electrode AgCl content of 3.40 micromol.
Figure 4 shows the relationship between latency and initial AgCl film content for 16 electrodes and surface areas of 0.087 ñ 0.003 cm2. Assuming dense packing, initial film AgCl density can be converted to thickness in /zm by multiplying by the conversion factor 0.258 micromole . cm2/micromol. The electrodes were of the type illustrated in Fig. lb, and the radius of rotation r was 1.01 ñ 0.04 cm. Although experimentally the initial AgCl film density, MO, was the independent variable, it is presented in Fig. 4 as the ordinate in order to facilitate discussion of the data.
It can be seen from the data that there is no resolvable dependence on (Cl-) for the relations between MO and A. The solid curve, a graph of the theoretically derived equation [see (I 7), in Discussion], was fitted to the data by the method of least squares. The coefficient of correlation between the data and the empirically derived function is 0.999.
The present experiments determine the rate of I-Cl exchange during the process of transformation. At t = A, the calculated quantity of I- lost from solution is virtually equal to the initial AgCl content of the electrode (Fig. 3). This finding verifies our previous assumption that Ag+ is conserved and the I-Cl exchange is one-for-one. Hence, the I-Cl exchange rate can be determined by measuring the time course of I- loss from solution; ie the exchange rate is characterized by the I- flux. For the constant-volume case, normalization with respect to electrode surface area a gives
where IJ I is the scalar I- flux in mol/cm2/s, I is the cumulative I- lost from solution in mol, and R is the quantity of Cl- lost from the electrode film in mol/cM2. It follows from material balance that
where MO and M are the initial and instantaneous electrode Cl- quantities respectively.
It can be shown from the data of Fig. 2 that the rate of I-Cl exchange is directly proportional to I for a given solution volume. Substituting the experimental result AEm't into the Nernst equation, we get
where 10 is the initial solution L. By defining m'FIRT @ m, and differentiating with respect to time, we have
Dividing both sides by electrode surface area a gives the flux,
The initial flux IJOI Ifor the experiment illustrated calculated by this method is 4.92 x 10-1.0 Mol/CM2/S.
The rate of loss of I- from the solution is continuous until the AgCl film is virtually entirely depleted (Figs. 2 and 3). The time of the abrupt change in the rate of I- loss is coincident with A, the time of maximum I dE/dt I for the transforming Ag/AgCl electrode. Therefore, we conclude that (9) holds throughout the latent period of transformation; ie that the I-Cl exchange process is continuous in the interval (O < t < A). This conclusion can be summarized by writing (5) in the integral form
If the relation between 9 and t is known, J(t) can be derived from (10). From the data of Figs. 2 and 3, we see that R -* MO when t = A. Experimentally, two known determinants of J, (I-) and the stirring rate, were fixed, and the relation between MO and A was plotted (Fig. 4). The deviation from linearity indicates that J is not constant with time. It will be shown that the growth of the AgI precipitate layer resulting from I-Cl exchange can account for the change of J with time within the framework of the Nernst diffusion-layer model.
The I-Cl exchange rate can be defined as a diffusion flux, in terms of the one-dimensional Fick's law,
where D is the diffusion coefficient and d(I-)/dx is the I- concentration gradient.
Calculations using this equation are simplified by use of the Nernst model, which postulates a constant concentration gradient in the diffusion layer adjacent to the electrode surface (Fig. 5). The initial thickness of the diffusion layer is represented as xo, and the concentration in the solution outside this region is constant. Figure 5 is a representation of
the Ag/AgCl/solution system at t = 0 and at some later time (O 1 < t < A). It is assumed that xo is unaffected by the nature of the solid/solution interface; that is to say, it does not matter whether the solid is AgCl or the AgI- precipitate layer. The increasing thickness of the AgI-precipitate layer effectively increases the thickness of the diffusion layer. Thus, from Fig. Sb, the effective thickness of the diffusion layer at any time is given as xo + x(t). It is assumed that (I-) at x @ 0 is negligible relative to (I-) in the bulk solution; hence, the concentration difference across the diffusion layer is given by the solution concentration and is constant. With these simplifying assumptions, Fick's law can be written
If the flux does not change much with time, it can be assumed that the thickness of the AgI layer increases linearly with a velocity v,
In the region vt < xo, (12) can be approximated by the use of the binomial expansion to give
where Jo = -D(I-)Ixo and is the flux at t @ 0, and k is vlxo. Combining (10) and (14), we get
Integrating, we have
Applying this result derived from the model to the experimentally determined variables, we let t = A and SI = MO - M', since 9 - MO when t = A. Here M' represents the amount of AgCl remaining in the electrode film when t @ A. Then, (16) can be written in terms of MO and A,
This quadratic form was fitted to the data and the result is shown in Fig. 4. The experimentally derived values for the variables in the model are given in Table 1.
Cross-term diffusion effects and the effect of (I-) and (Cl-) upon the I-Cl exchange-diffusion coefficients were not considered in the calculations of Table 1. Instead, the value of the KCl-KI exchange-diffusion coefficient for concentrations of 0.01 M was used. The accuracy of our experimental methods does not warrant more precise calculation, since there was no resolvable dependence of I-Cl exchange rate upon (Cl-) over the range of 0.001 to 0.1 M.
The usefulness of our model can be tested by considering the consequences of some calculations using the values of Table 1. The calculated thickness of the diffusion layer, xo, cannot be interpreted literally because of the limitations of the Nernst diffusion model.3 However, the value obtained is of the same order of magnitude as those calculated with the same model for other systems.4 Therefore, the data appear to be consistent with the hypothesis that mass transfer, rather than the dissolution of AgCl, limits the rate of I-Cl exchange. This can be summarized by
where m is the rate constant defined by (8), and k, and k-I are the specific rate constants for the dissolution and formation of AgCl; in these terms, m << kl. Compared to the diffusion layer in free solution, the physical structure of the AgI-precipitate layer approaches more nearly an ideal unstirred layer; ie it resembles a porous diaphragm. Consequently, the calculated thickness of this layer (I3) can be interpreted more literally. The fraction of the volume of the total layer occupied by crystalline AgI is approximated by the ratio of the rates of increase in the volume of AgI and the volume of the AgI-precipitate layer. This value is about 0.6 for the data of Table 1. Since tortuosity of the channels was not considered in the calculation, this value is an underestimate of the packing density, but is reasonable in view of the loose, spongey character of the AgI layer.
In terms of the Nernst model, the calculated effective diffusion layer thickness xo for the experiment of Figs. 2 and 3 is 2.03 x 10-2 cm. This value is about 6 times the calculated xo for the experiment of Fig. 4. Referring to (12), for an xo large relative to x(t), the flux is nearly independent of x(t), and is linearly related to (I-). Consequently, any contribution to the total flux due to the growth of the AgI layer was not resolvable as such in Fig. 2.
Both types of experiments yield approximately 10-8 mol/cm2 for the quantity of AgCl remaining on the electrode when transformation takes place. However, in view of the accuracy of the methods it is sufficient to point out that the calculated quantity of AgCl remaining after transformation is not distinguishable from zero.
It is concluded that the I-Cl exchange proceeds continuously during the latent period of transformation, which terminates (t = A) when the exchange is virtually complete. Accordingly, the experimentally measurable parameter A, together with the initial electrode AgCl content, MO, can be used to determine the average I-Cl exchange rate, (J),
The instantaneous exchange rate was shown to be directly related to solution (I-). The experimental results were interpreted in terms of the Nernst diffusion model.
It is concluded that the I-Cl exchange rate is limited by the exchange transfer of I- and Cl-.
E(Cl, I) (mV), potential of an Ag/AgCl electrode in a solution
containing both Cl- and I-
EI (mV), potential of an Ag/AgI electrode
A(s), time of maximum dE(Cl, I)/dt; terminus of latent period of transformation
i(mol/cm2/s), I-Cl exchange rate; flux
MO(mol/cm2), initial density of AgCl film on Ag/AgCl electrode
R(Mol/CM2), cumulative quantity of AgCl replaced by AgI
1O(mol), initial quantity of I- in solution
I(mol), cumulative quantity of I- lost from solution
Potential is referred to the saturated calomel half-cell (sce).
Acknowledgements: The authors express their gratitude to Prof. J. G. Albright for invaluable discussion in the preparation of this manuscript.
The work was performed under the auspices of the United States Atomic Energy Commission.