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Diffusion Barrier Model for the Cyanide Ion Selective Electrode SIR: A recent publication (I) has reported measurements of selectivity ratios for a cyanide ion selective membrane electrode. The electrode whose response was investigated was an Orion solid state electrode which employs a mixed silver sulfide-silver iodide membrane. The following operational model has been suggested for this electrode (2). The mixed silver sulfide-silver iodide membrane responds to iodide ion in a sample solution through the solubility equilibrium of silver iodide. AgI S Ag+
+ I-
I
I
12- Solution
j
I I
Diffusion Barrier
I Bulk of
I Membrane I
(1)
The activity of silver ion near the membrane surface is governed by the activity of iodide through the solubility product expression. The membrane potential is in turn generated by the silver ion activity. The final result is that the membrane potential is given by E = constant
- RT -. In uI F
(2)
where u Iis the activity of iodide ion in the sample. When the membrane is brought in contact with a sample containing cyanide ion, it is postulated that the following reaction proceeds at the membrane surface: AgI
+ 2CN- e Ag(CN)Z- + I-
K
-
lo4
(3)
The equilibrium constant for Reaction 3 is large enough to ensure that the surface concentration of cyanide is rather low and that a concentration of iodide equal to half the original cyanide concentration will be generated. The liberated iodide governs the membrane potential through Equation 2. However, as Reaction 3 proceeds, solid AgI is leached from the membrane producing soluble Ag(CN)2- and I-. Normally the sample solution is gently stirred causing the replenishment of CN- at the membrane surface and the removal of Ag(CN)i- and I- to the bulk of solution. As time passes, a porous layer of the residual solid silver sulfide forms on the membrane. The region of the membrane containing solid silver iodide is now separated from the sample solution by a porous silver sulfide diffusion barrier. Cyanide ion is brought to the outer edge of the diffusion barrier by stirring of the solution. It must diffuse through the barrier to the region of membrane which contains solid AgI. There Reaction 3 generates Ag(CN)1 and I- which diffuse through the barrier to the outer boundary of the barrier where they are carried away by convection. The thickness of the diffusion barrier varies with time. This variation is caused both by the continued removal of solid AgI at the boundary between the bulk membrane and the diffusion barrier and by the tendency of the solid silver sulfide to break off at the boundary between the barrier and the solution (2). (1) B. Fleet and H. von Storp, ANAL.CHEM., 43, 1575 (1971). (2) J. W. Ross, in “Proceedings of Symposium on Ion Selective Electrodes,” R. A. Durst, Ed., Nat. Bur. Stand. Spec. Publ., 314,1969, p 84.
The response of the electrode must be interpreted in terms of the mass transfer of species through the barrier and the equilibrium considerations outlined above. The model of the barrier which will be employed is the same as that of Fleet and von Storp ( I ) , oiz., a slab of uniform properties and of thickness d separating the bulk of the membrane and the sample solution (Figure 1). Boundary Value Problem. The Fick diffusion equations for CN- and I- in the region 0 2 x 5 d a r e
bCCN = DCNazcch2 at
(4)
bX
DCNand D I are the diffusion coefficients of I- and CN- in the diffusion barrier. Initially it is assumed that the diffusion barrier contains no CN- or I- (Equation 6). The experiment is initiated by bringing a sample solution in contact with the diffusion barrier (x = 0) as in Figure 5 of reference 2. The concentrations of CN- and I- in the sample are C& and CI*, respectively. Stirring ensures the invariance with time of CCNand Cr at x = 0 (Equation 7). The CN- which diffuses through the barrier and reaches the bulk of the memANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972
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875
brane is consumed through Reaction 3 whose large equilibrium constant causes CCNto be negligible at x = d, assuming Reaction 3 proceeds rapidly with respect to diffusion (Equation 8). Equation 9 states that the inward flux of CNequals twice the outward flux of I-. The equation embodies the stoichiometry of Reaction 3 and the assumption that neither CN- or I- accumulates at x = d. Diffusion Equation 4 has been solved for the initial and boundary conditions given by Equations 6-8 (3). The solution is:
Equation 10 permits the computation of CCNat any point in the diffusion barrier at any time. At sufficientlylong times, the exponential term vanishes giving a steddy state form of Equation 10. The remaining series may be shown to be the Fourier series of the function 1 - x/d. Thus Equation 10 becomes as t + ~0
-CC-N
X
CCN*
Equation 11 may also be obtained by imposing the steady state on Equation 4, i.e., by setting bCc~/bt= 0. The resulting differential equation along with conditions 7 and 8, gives Equation 11 directly. Similarly Equation 5 may be solved for the steady state condition utilizing Equations 7, 9, and 11. The result is
where CI* is the concentration of I- in the sample solution. Figure 1 gives steady state concentration profiles for CCN*= 10 CI*and DIIDCN= 0.8. As mentioned earlier, the electrode response is thought to be governed by a1 at x = d. We may write ax = ~ I C I where 71 is the activity coefficient for iodide. The value of CIat x = d from Equation 12 may be placed in 2 and y I may be extracted from the logarithm term and included in the constant. This requires that yI be constant as it would tend to be if the ionic strength of the samples were maintained at a constant value ( I ) . The result is E = constant’
- RTIn F
[
cCN*
+2D1 DCN
cI*
]
(13)
the basis of the stoichiometry of Reaction 1 . A selectivity ratio of 2.0 is to be expected for this model only if DI = DCN. It should be noted that the thickness of the diffusion barrier does not appear in the expression for the selectivity ratio. Discussion. An estimation of the magnitude of the selectivity ratio requires knowledge of the ratio of the diffusion coefficients of I- and CN- in the diffusion barrier. We may assume that the ratio of diffusion coefficients in the barrier is equal to the ratio of diffusion coefficients measured in free solution. The diffusion coefficient of I- in 0.125M sulfuric acid at 25 “C was found to be 2.0 X lod6 cm2/sec (4). This is in agreement with the value computed from the limiting ionic equivalent conductance of I-. The diffusion coefficient for CN- in 0.25M sodium chloride, 0.05Msodium hydroxide may be computed from polarographic data (5). It is about 2.5 X lod5 cm2/sec. These values give a predicted selectivity ratio of 1.6. Unfortunately, diffusion coefficients are not available for the medium employed by Fleet and von Storp (0.1M NaOH), so a ratio pertaining to the actual experimental conditions cannot be computed. The experimental value of the selectivity ratio was about 1.3 ( I ) . The discrepancy between this value and the value of 2.0 based solely on the stoichiometry of Reaction 1 has prompted the suggestion that reactions other than 1 play a role in the electrode response ( I ) . The calculations presented in this communication demonstrate that a more complete development of the diffusion barrier model produces a predicted selectivity ratio which is much closer to the experimental value than was previously thought to be the case. Solution of the boundary value problem to obtain CI as a function of both time and distance (analogous to Equation 10 for CCN)would permit the prediction of the time dependence of the electrode potential after a step-functional change in CCK*(cf. Figure 5, Reference I ) . However, it is doubtful that the potential would be a sufficiently sensitive function of the model parameters (DcN,DI, and d) to provide a meaningful evaluation of the model from studies of potential-time curves. DENNISH. EVANS Department of Chemistry University of Wisconsin Madison, Wis. 53706
where constant’
=
RT YIDCN constant - - In F 201
In Equation 13, the selectivity ratio is predicted to be 2D1/ DCNrather than the value of 2.0 which was anticipated ( I ) on
RECEIVED for review November 12, 1971. Accepted January 11, 1972. This work was supported by Grant GP-19579 from the National Science Foundation. (4) A. L. Beilby and A. L. Crittenden, J . Phys. Chem., 64, 177
(3) H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids,” 2nd ed., Oxford, London, 1959, pp 102-105.
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ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972
(1960). ( 5 ) W. H. Jura, ANAL,CHEM., 26, 1121 (1954).
