Theory of Cyclic Voltammetry for a Dimerization Reaction Initiated ElectrochemicaIly Michael L. Olmstead, Robert G. Hamilton, and Richard S. Nicholson Chemistry Department, Michigan State University, East Lansing, Mich. 48823 Theory of cyclic voltammetry is presented for an electrochemically initiated homogeneous dimerization reaction. The mathematical approach treats the linear partial differential equation by transformation to integral form and the nonlinear differential equation by finite difference. This approach should be of general interest, and therefore it is described in adequate detail. Results of the calculations both for planar and spherical electrodes are presented as diagnostic criteria analogous to those developed previously for a first-order reaction initiated electrolytically. Finally, a method is presented by which the dimerization rate constant can be evaluated from a single cyclic polarogram.
SECOND-ORDER COUPLED reactions occupy a significant role in electrochemistry. Nevertheless, only a very few general mathematical treatments have been published for diffusionlimited electrochemical techniques and electrolysis mechanisms involving second-order reactions. The reason for this is the difficulty of solving the partial differential diffusion equations, which are nonlinear for higher than first-order reactions. Thus, unless simplifying and restrictive approximations are introduced, in general these nonlinear partial differential equations can be solved only by numerical finite difference methods; the few general treatments published to date are on that basis (1-4). These publications deal with chronoamperometric ( I ) and chronopotentiometric (2) electrolysis and linear scan voltammetry (3) for the disproportionation reaction, and chronoamperometric electrolysis for an ece mechanism ( 4 ) . Among the various second-order coupled reactions, the case of electrochemically initiated dimerization
difference methods that some of the inherent advantages of the integral equation methods (simplicity and accuracy) are retained. This new approach apparently is similar to one used very recently by Mastragostino and co-workers (3). Because these authors give very few details of their method, and because the method we have used should be readily adaptable to other electrochemical techniques, a description of this method also is presented. THEORY
Model. We assume System I to be operative, where the electron transfer is reversible, and 0, R, and Z are soluble in the solution phase. The homogeneous dimerization reaction having rate constant, kp, is assumed to be irreversible and the product, Z, electroinactive. Finally, the working electrode is assumed to be spherical, because experimental applications frequently involve a slowly dropping DME, or hanging mercury drop electrode. Boundary Value Problem. Based on the model, Fick second-law diffusion equations for a stationary sphere may be applied to 0 and R. These differential equations conveniently are written in the following dimensionless form
where V2 is the Laplacian operator for a spherically symmetric coordinate system. Other terms in Equations 1 and 2 have the following definitions
Ofne-R (1)
(3)
is ofinterest, particularly because of the occurrence of radical coupling reactions (5). This mechanism is best studied using a two-step perturbation method (6). Of the two-step methods, cyclic voltammetry probably is the most generally useful (5). Thus, we have developed the theory of cyclic voltammetry for Mechanism I, and it is the purpose of this paper to communicate these results. The approach that we have used to solve the partial differential equations is a combination of the integral equation procedure used previously (7), and a finite difference method. This combined approach has the advantage over total finite
(4)
2R3
z
(1) G. L. Booman and D. T. Pence, ANAL.CHEAI., 37, 1366 (1965). (2) S. W. Feldberg and C. Auerbach, ihid., 36, 505 (1964). (3) M. Mastranostino, L. Nadio, and J . M. Saveant, Electrochim. Acta., 13, 721 ~ 9 6 8 ) . (4) M. D. Hawley and S. W. Feldberg, J . Phys. Chem., 70, 3459 (1966). (5) P. J. Elving, Pure Appl. Chem., 15, 297 (1967). (6) W. M. Schwarz and I. Shain, J . Phys. Chem., 69, 30 (1965). (7) M. L. Olmstead and R. S. Nicholson, J. Electrounal. Chem. Interfuciul Electrochem., 16, 145 (1968). 260
ANALYTICAL CHEMISTRY
y = at I-
There C represents concentration as a function of radial distance, r, from a sphere of radius, ro, and time, t . CO*is the initial concentration of 0, and the parameter a is defined as a = nFo/RT
(9)
where u is, dE/dt, and other terms have their usual meaning (8). ___.___--_ ~
(8) R . S. Nicholson and 1. Shain, ANAL.CHEM., 36,706 (1964).
t ;
.4
.50 .52
.3
a
.2
.40
h
h
Y
x .I .46
0 -I
-.I
I
0
2
log qJ
-.2
d&,,
I
I
50
0
I
I
150
100
I
I
Figure 2. Variation of with kinetic parameter for zero spherical parameter. A is dimerization, B is first-order reaction
-50 -100
n(E-Eo) mV Figure 1. Cyclic polarograms for iC/ = 0.005 ( A ) , 0.15 (B), 0.675 (C)
by Laplace transformation and combined with Equation 16 to give 1-s,”
The initial conditions are
X(X)K(Y- x ) d x
=
VO,
Y ) exp(b)Uy)
(18)
There the kernel function is given by (9)
y = o , z > o
u=o V(l
+ +z)-I
=
exp(--6)
The boundary value problem now has been reduced to Equations 2, 11, 14, and 18. This problem can only be solved conveniently by numerical finite difference methods, because Equation 2 is nonlinear. Details concerning incorporation of Equation 18 into the finite difference procedure are given in Appendix A, and a discussion of the accuracy of the method is given in Appendix B.
where exp(b) = CO*/CR* The boundary conditions are y
> 0, z u-0
V(1
+ 94-l
= [exdb)
+
RESULTS AND DISCUSSION 4W-l
There X is the switching time for the triangular wave, d & ( y ) is the current function, and Equation 17 is an abridged form of the Nernst equation (8). Based on the preceding formulation of the boundary value problem, apparently current-function curves will depend on two variables, IJ (Equation 7), a kinetic parameter, and 4 (Equation 8), a spheiical parameter. Whereas IJ may assume any finite value, the parameter, 4, seldom exceeds 0.1 for reasonable experimental conditions (8). Thus, discussions that follow are restricted to values of 4 in the range, 0 < 4
Figure 1 illustrates typical results of calculations for three values of the kinetic parameter (J.> and zero spherical parameter (9). These curves qualitatively are similar to those obtained when the succeeding chemical reaction is first-order (8). Thus, wherever practicable in discussions that follow, results for the present case are compared directly with published data for a succeeding first-order reaction (8, Case VI). In each instance the fact that the kinetic parameter for the present case includes the additional variable CO* makes unambigums distinction between the two mechanisms possible. For the first-order succeeding reaction three diagnostic criteria have been presented based on variations with scan rate of peak current, peak potential, and ratio of anodic to cathodic peak currents (8). The first two of these are discussed in the section on the single scan method, and the third is implicit in the discussion of the cyclic method. For spherical electrodes the current-function curves will be influenced by simultaneous changes with scan rate of the spherical and kinetic parameters. To present results of calculations which reflect this effect, it i s convenient to define a new variable directly related to experimentally measurable
Equation 1 and the initial and semiinfinite boundary conditions (Equations 10 and 13) can be reduced to integral form
(9) M. L. Olmstead and R. S. Nicholson, J . Electroanal. Chem. Interfizcial Electrochem., 14, 133 (1967).
y>o,z= 0
bU
bZ
- 4l.J = 4 V -
bV -
bZ
= 2/-x(y)
where
Y I exp(y
- 2ah)
y
> aX
< 0.1.
VOL. 41, NO. 2, FEBRUARY 1969
261
Table I. Current Function for Planar Electrode ( E - E')nr niV 140 120 100 80 60 40 20 10 0
- 10 - 20
- 30 -40 - 50
- 60
- 80
-100 - 102 -100 -YO - 60 -40 -20 0
10 20 30 40 50 60
70 80
100 120 140
0.005
0.0625
0.150
0.004
0.004
0.009
0.009 0.020 0.042
0.020 0.042 0.085 0.161 0.270 0.329 0.381 0.419 0.441 0.447 0.438 0.421 0.399 0.353 0.312 0.309 0.292 0.230 0.163 0.070 0.052 -0.173 -0.216 -0.243 -0.253 -0,249 -0.236 -0.218 -0.198 -0.179 -0.145 -0 118 -0,099
0.085
0.161 0.271 0.330 0.383 0.422 0,444 0.449 0.440 0.422 0.400 0.353 0.312 0.308 0.292 0.233 0.171 0.086 0.024 -0.134 -0.174 -0.199 -0.210 -0.208 -0.198 -0.183 -0.166 -0.150 -0.121 -0.098 -0.081
0.004 0.009
0.020 0.042 0.085 0.162 0.273 0.333 0.386 0.425 0.447 0.451 0.442 0.423 0.400 0.352 0.311 0.307 0.292 0.236 0.179 0.103 0.005 -0.094 -0.131 -0.155 -0.167 -0.167 -0.160 -0.149 -0.135 -0.122 -0.098 -0,079 -0.065
0.350
0.004
0.009 0.020 0.042 0.085 0.163 0.277 0.338 0.393 0.433 0.454 0.457 0.445 0.424 0.400 0.350 0.308 0.305 0.291 0.240 0.190 0.125 0.042 -0.043 -0.077 -0.101 -0.113 -0.117 -0.114 -0.107 -0.098 -0.088 -0.071 -0.057 -0.046
quantities. Thus, by elimination of scan rate between the spherical and kinetic parameters, a new variable, p, is obtained.
Calculations performed for various values of 4J and $, with p held constant, reflect the manner in which current function varies with scan rate for a spherical electrode (9). Single Scan Method. Table 1 contains values of the current function for severai values of the kinetic parameter and zero spherical parameter. These data plus results of calculations performed for values of not included in Table I were used to develop diagnostic criteria. Figure 2 illustrates variation of with $ for a plane electrode (this plot is equivalent to a plot of i,/du-os. u). For purposes of comparison, results for the first-order case also are included in Figure 2. For the present case d ; x p varies between the limit corresponding to kinetically uncomplicated electron transfer at high scan rate (small $), and the limit corresponding to kinetic control at low scan rate (large $). For these two limiting cases values of d ; x p are represented in Figure 2 by dashed lines. For spherical electrodes and scan rates where the spherical parameter becomes significant, values of ~ T X , ,in Figure 2 are subject to increases. A quantitative meawre of - this effect can be obtained from the data in Table I1 where l / ' i r x p is given for several values of p as $is increased.
+
dqP
262
ANALYTICAL CHEMISTRY
kzCo*/a
0.675 0.004 0.009
0.020 0.042 0.086 0.166 0.283 0.347 0,403 0.443 0.462 0.462 0.448 0.424 0.398 0.346 0.305 0.302 0.290 0.243 0.198 0.142 0.071 -0.004
-0.036 -0.059 -0.073 -0.079 -0.080
-0.076 -0.070 -0.064 -0.051 -0.041 -0.037
~
1.5 0.004 0.009 0.020 0.042 0.088 0 171 0.297 0.365 0.423 0.462 0.476 0.469 0.449 0.421 0.392 0.340 0.299 0.296 0.286 0.245 0.207 0.160
3.5
10
35
0.004 0.009
0.004
0.004 0.010 0.023 0.057 0.137 0.295 0,467 0.509 0.515 0.494 0.461 0.425
0.020 0.043 0.092
0.185 0.327 0.399 0.456 0.486 0.488 0.470 0.442 0.410
0.009 0.021 0.046 0.103 0.219 0.385 0.455 0.497 0.504
0.487 0.456 0.422
0 . 100
-0.035 -0.006 -0.017 -0.033 -0.042 -0,046 -0.046 -0.044 -0.040
-0.032 -0.026 -0.020
Figure 3 illustrates variations of peak potential, (E, Ea)n,with $ for a plane electrode. For small $ the peak potential is nearly constant and equal to -28.5 mV (8). For increasingly large values of the kinetic parameter the peak shifts anodically and approaches a limit given by (IO, I I ) n(~, E") =
RT - [In$ 3F
- 3.121
for $ greater than about 10. Figure 3 also includes variation of peak potential for the first-order succeeding reaction. The limit for large $ in this case is given by (8) RT n(Ep - E") = - [In$ 2F
- 1.561
Based on Equations 50 and 51, and Figure 3, differences in the slope of peak potential plots should be useful for determining reaction order when $ is greater than about three. In addition, changes in concentration will shift peak potential plots along the scan rate axis for the dimerization reaction, whereas for a first-order reaction curves will not be affected by changes in concentration. Variation of peak potential with scan rate is affected by sphericity, but to a lesser extent than peak current, because (10) J. M. Saveant and E. Vianello, Compt. Rend., 256, 2597 (1963). (11) R. S . Nicholson, ANAL.CHEM., 37,667 (1965).
I
-30F-
0.2 -I
I
0
2
-2
-1
I
0 log( k2Cfgr)
109 $
Figure 3. Variation of peak potential with kinetic parameter and zero spherical parameter. A is dimerization, B is firstorder reaction
Figure 4. Variation of ia/iowith kinetic parameter and zero spherical parameter. A is dimerization (a7 = 4), B is firstorder reaction
effects of electrode geometry are manifested primarily as variations of current rather than position of the wave on the potential axis (9). Data which illustrate this fact are included in Table 11. Cyclic Triangular Wave Method. The data of Figure 1 and Table I show that, as anticipated, the anodic portion of the polarogram is most sensitive to the rate of dimerization, and hence best suited for measurement of kz. Therefore an effort was made to correlate the ratio of anodic to cathodic peak currents, ia/ic,to the rate constant kz. A number of cyclic polarograms were calculated for several values of $ with $I equal zero, and three values of the switching potential, (Ex - E")n. For each case the ratio ia/ic[the base line for calculation of ia was the extension of the cathodic wave (S)] was evaluated. For each of the switching potentials the ratios ia/icthen were plotted against the quantity log(+ar) [ = log(k&*r)lo, where UT is
portional to U T . Thus, it is possible to incorporate effects of switching potential in a new variable, w, defined by
and r is the time from E" to EA. Figure 4 shows this plot for = 4. Plots for UT equal 3.5 and 5.0 are nearly the same except for small shifts along the log(kzCo*r) coordinate pro-
+
log w = log ( k z C o * ~ ) 0.034(UT
- 4)
(24)
Values of ia/icas a function of w are given in Table 111. A plot of the data of Table I11 can be employed to evaluate w graphically from experimental ratios of ia/ie. For a plot of w us. r, the slope is kzCo* exp[0.078(ur - 4)], and consequently if switching potential is known, kz can be calculated. Thus, for linear diffusion, rate constants can be measured with the aid of a single theoretical working curve. For a first-order succeeding reaction one of us recently described a semiempirical method by which r and ia/iccan be determined from a single experimental polarogram (12). Similar correlations are possible for the present case. Thus, Eo can be determined (within 3/n mV) as the potential at 85 of the cathodic peak for 5 1.5, and r calculated by r = (EA - E")/u. The ratio ia/iccan be obtained from a single cyclic polarogram with the relationship
+
UT
(12) R. S. Nicholson, ANAL.CHEM., 38, 1406 (1966).
Table 11. Peak Current Function for a Spherical Electrode 0.0
6.25
p X 1040 25.0
56.25
100.0
0.451 (28.8) 0.453 0.454 0.457 0.459 0.463 0.468 0.474 0.481 0.489 0.499 0.511 0.526 0.544 0.577 (23.2)
0.452 (28.8) 0.455 0.456 0.459 0.463 0.468 0.474 0.481 0.490 0.500 0.512 0.526 0.545 0.568 (26.9)
kiCo*la
0.448 (29.0) 0.449 0.449 0.450 0.452 0.454 0.456 0.459 0.463 0.468 0.473 0.480 0.488 0.499 (22.8) 0.516 0.542 0.600 (5.3) Values in parentheses are peak potentials, (Eo- EJn, mV. 0.005 0.010 0.015 0.025 0.040 0.0625 0. 100 0.150 0.225 0.325 0.475 0.675 1.000 1.500 2.700 6.000 24.00
a
0.447 (28.5) 0.447 0.447 0.448 0.448 0.449 0.450 0.452 0.454 0.457 0.461 0.465 0.470(23.4) 0.476 0.486 0.499 (11.6) 0.514
0.449 (28.7) 0.451 0.452 0.453 0.456 0.458 0.462 0.466 0.472 0.478 0.486 0.495 0.507 (25.7) 0.521 0.546 0.587 (17.0)
VOL. 41, NO. 2, FEBRUARY 1969
263
Table III. Ratio of Anodic to Cathodic Peak Current for Plane and Spherical Electrodes P x 104 0.0 6.25 25.0 56.25
100.0
W”
0.02
0.990 0.979 0.969 0.950 0.923 0.887 0.838 0.785 0.726 0.670 0.613 0.563 0.513 0.469
0.04
0.06 0.10
0.16 0.25 0.40 0.60 0.90 1.30 1.90 2.70 4.00 6.00 a For finite p,
w
E
k2Co*7 and
UT
0.990 0.981 0.970 0.951 0.925 0.890 0.843 0.793 0.737 0.685 0.632 0.588 0.545 0.509 =
0.990 0.980 0.970 0.951 0.926 0.893 0.847 0.799 0.747 0.698 0.650 0.611 0.574 0.545
y a7
P
Y
r
3.5 4.0 4.5
1.018 0.9678 0.8563
-0.6539 -0.5370 -0.3322
0.3729 0.3337 0.2610
> 0, z
-
z*
Vz,,v) (1
Definitions of the experimental quantities (iCp)o,(iuP)o, and (is& are included in Figure 1. The constants P, y, and are given in Table IV for several values of UT. For spherical electrodes no convenient method could be found for incorporating simultaneously effects of sphericity and switching potential. Thus, values of i,/ic were determined for fixed U T and four values of p. These data, which are rigorously applicable for UT = 4, are included in Table 111. The data of Table I11 demonstrate that sphericity causes an increase. of the ratio iu/ic. This result could be anticipated based on the following reasoning. For spherical electrodes the diffusion process for substance R is divergent, and the concentration of R in the vicinity of a spherical electrode at the same time during the cathodic scan is less than for a plane electrode of the same area. This effect decreases the rate of the succeeding reaction relative to a plane electrode, and results in increased availability of R for oxidation on the anodic scan. Thus ia/ic for a spherical electrode should be larger than for a plane electrode. The results presented above make cyclic voltammetry a convenient technique for the quantitative study of the dimerization mechanism. Moreover, the numerical analysis method described clearly is applicable to other cases of coupled second-order reactions, as well as to other electrochemical techniques. Applications of this type currently are under investigation, and results will be communicated in the future.
r
APPENDIX A
Reduction of Semiinfinite Boundary Conditions. Boundary condition, Equation 14, is not in a form suitable for use in the general finite difference solution of Equation 2, and therefore it is necessary to replace it by the approximate form ANALYTICAL CHEMISTRY
0.990 0.980 0.971 0.953 0,929 0.898 0.855 0.812 0.766 0.723 0.683 0.652 0.625 0.607
4.
Table IV. Constant Coefficients for Equation 25
264
0.991 0.980 0.970 0.952 0.927 0.895 0.851 0.806 0.756 0.711 0.667 0.632 0.600 0.578
+
$JG>-~ =
[exp(b)
+ Gy1-l
(‘4-1)
There the constant, zs, is a value of the dimensionless z coordinate such that for z 2 zs changes in the function, V(z,y), are negligible (b2V/bz2N 0). This statement is equivalent t o the assumption that the semiinfinite condition on species R may be replaced by a finite diffusion condition which does not significantly change the problem. A similar assumption was employed by Booman and Pence ( I ) . The magnitude of zs can be estimated from well known solutions for potentiostatic electrolysis at a spherical electrode. In this case Delahay (13) gives the following equation for concentration profiles.
Thus, for large arguments of the error function complement (ca. 2, or greater) concentration profiles become nearly invariant. Therefore, for some total electrolysis time, ts, the diffusion layer thickness, rs - ro,is approximately given by
Hence, in terms of the dimensionless space and time coordinates the following holds
(‘4-4) Equation A-4 is an adequate approximation for calculation of zs because variations in concentration during potentiostatic electrolysis extend further from the electrode than for the analogous situation in cyclic voltammetry. Derivation of Difference Equations. The finite difference technique used to solve Equation 2 is similar to one used by Booman and Pence ( I ) . In this method, the z domain is divided into N equal intervals of width, Az, and the y domain is divided into M equal intervals of width, 6. The maximum value of y is thus y, = M6. The dimensionless time variable is serialized by the integer, m, such that y = m6 and m takes on values, rn = 1, 2, . . . , M during the calculations. (13) P. Delahay, “New Instrumental Methods in Electrochemistry,” p 60, Interscience, New York, N. Y.,1954.
For each value of m, V(z,ma) is defined at the midpoint of each Az interval in the z domain and in addition at the points z = 0 and z = z8. Values of V(z,ma) at these points designate the array V = (VI, VZ,. V , + 2). Elements of this array are given by
Other approximations for V2(z,y ) also are possible. For example, if V(z,y ) at the nth point in the z domain is given by
.
Vl = V(0, m6) V , = V([n-;]Az,m6>; VN+2 =
(A-5a)
1 1
(A-28)
The coefficient matrix for Equation A-26 is tridiagonal and the V array is conveniently evaluated by the method of Gaussian elimination (16). The V array is then used to determine new values of B, by Equation A-11 or A-13, and the V array is re(16) M. C. Pease, "Methods of Matrix Algebra," p 32, Academic Press, New York, N. Y . , 1965. 266
ANALYTICAL CHEMISTRY
APPENDIX B
Some preliminary calculations were performed to estimate accuracy, and to determine optimum values for the numerical parameters 6, Az, and E. Because optimum values for 6 and Az will depend on values of A and $, the following five pairs of these parameters were used in the preliminary calculations: (1) $ = 0, 4 = 0; (2) = 1, 4 = 0 ; (3) $ = 0, 4 = 0.1; (4) # = 1, 4 = 0.1; and ( 5 ) $ > 1.5, 6 = 0. Effect of 6 and Az. The magnitude of 6 was dictated in part by the method used to approximate Equation 18. This method has been used successfully to solve similar integral equations at the 0.5 % error level when the integration interval (6) was less than 0.3 (7, 9). Thus, to ensure adequate approximations in Equation 18, initially values of 6 = 0.05, 0.1, and 0.15 were employed. Each numerical difference equation is solved for intervals of width 6 and 262. These intervals are optimum if Equation A-4 is approximately satisfied. Thus, the magnitude of Az is given by Az = d 6 . During initial calculations values of Az = 0.1,0.25, and 0.4 were employed. For the first four $-4 pairs given above, calculations of d & ( y ) were performed for a single linear scan with b = 7 and y , = 9. The peak value of the current function, and the peak potential, ( E , - Ea)n, were tabulated for each 6-Az pair. For each $+ pair the maximum variation between any two d&,and (E, - Eo)n values was found to be 0.06% and 0.15 mV, respectively. For pairs where $ = 0, the current function always was in excellent agreement with published values of d ? r x ( y ) both for planar and spherical electrodes
d&,,
(8).
+
For the last $-+ pair, values used for were 3.5, 10, and 35. In this case, space and time coordinate intervals selected were 6 = 0.025, 0.05, and 0.1 and Az = 0.075, 0.1, and0.25. For $ = 3.5, all combinations of the above intervals produced values of ' d & ( y ) which agreed internally to three signifiFor $ = 10, the maximum cant figures (0.04% of variation of was 0.08% and for $ = 35, 0.19%. Effect of E. The value of e simply determines the degree of self-consistency acquired in the iterations. Because it is desirable to have these results as accurate as possible, E of 10-18, which is the smallest value compatible with the number of single precision significant figures in the computer employed (CDC 3600), was selected. Additional Factors. In addition to the parameters mentioned above, there are other factors which influence results of the numerical calculations. Most notable of these are the approximations involved in treatment of the diffusion process
d&,
d&,).
with a finite bound (z,) and in the B, terms. Validity of the former approximation (Equation A-4) was proved by using values of zslarger than those calculated from Equation A-4 and noting no change in calculated current functions. Booman and Pence ( I ) discuss various approximations of the B, terms, and justify use of either Equation A-11 or A-13. All preliminary calculations were performed in duplicate using both of these approximations. Comparison of results showed variations that were negligible with respect to those resulting from changes of 6 and Az. Because Equation A-1 1 required
fewer iterations, however, Equation A-1 1 was employed for all of the results presented in the text.
RECEIVED for review August 1, 1968. Accepted October 23, 1968. Research supported by the National Science Foundation and United States Army Research Office-Durham (Contract No. DA-31-124-ARO-D-308). One of us (M.L.O.) thanks the Phillips Petroleum Company for the fellowship during the summer of 1968.
Precipitation Titrations with Electrochemically Generated Lanthanum Ion Potentiometric Titration of Fluoride and Turbidimetric Titration of Oxalate D. J. Curran and K. S . Fletcher 111' Dppartment of Chemistry, University of Massachusetts, Amherst, Mass. 01002 Lanthanum ion electrochemically generated from a lanthanum hexaboride anode has been used as a precipitant for fluoride and oxalate ions. A lanthanum fluoride membrane electrode was used for end point detection in the fluoride work and a turbidimetric end point technique was used in the oxalate titrations. The pH conditions for the fluoride reaction were studied and 0.5 to 2.0 mg of fluoride in about 100 ml of solution have been determined with a precision and accuracy of a part per thousand at a pH electrochemically controlled at 5.0. The techniques of pretitration and titration back to a predetermined end point potential were used. An equation for the fluoride titration curve is discussed. Approximately 0.5- to 2.6-mg samples of oxalate ion in acidic solution (approximately 20-1111 volumes) have been titrated with a precision and accuracy in the 1 to 2% range.
IN A RECENT REPORT (1) a lanthanum hexaboride anode was used to generate known amounts of lanthanum ion in solution. The current efficiency (defined as the ratio of the number of moles of lanthanum generated at constant current as found by chemical analysis of the electrolysis solution to the number of moles of lanthanum ion predicted in solution from Faraday's law on the basis of pure LaBG of ideal stoichiometry) exceeded 100% and was a function only of the pH of the solution. Application was made to the determination of several metal ions by back titration of excess EDTA added to the solution of metal ion with electrochemically generated lanthanum ion. Lanthanum forms stoichiometric precipitates with a number of anions and this paper reports on the potentiometric titration of fluoride ion and the turbidimetric titration of oxalate ion with electrogenerated lanthanum ion. Present address, Research Center, The Foxboro Co, Foxboro, Mass. 02035 (1) D. J. Curran and K. S. Fletcher 111, ANAL.CHEM., 40, 1809 (1968).
Classical methods for fluoride analyses usually involve time consuming laboratory procedures. The recent development of a solid state electrode with a selective potentiometric response to fluoride activity has provided a simple procedure for fluoride determinations (2). Because the potential of this electrode has a theoretical response of 59.16 mV per decade in activity of fluoride at 25 "C, are lative error of 11 in activity of fluoride would require reliability of the potential measurement to be +0.25 mV. While the response stability of this electrode has been reported to be &0.1 mV (3), relative errors in the range 0.1 to 1% are difficult to obtain by direct potentiometry because the determination of concentration from a measurement of activity requires independent knowledge of activity coefficients to at least this degree of accuracy. Further, response stability of the fluoride and reference electrodes, liquid junction potentials, ionic strength changes, and pH effects all render analyses to accuracies better than 1 % doubtful. Lingane has recently described the potentiometric titration of fluoride with lanthanum nitrate as titrant and a commercially available lanthanum fluoride membrane electrode for end point detection (4). This work indicates that relative errors of 1 0 . 1 % are possible for the titration of 7.6-mg samples of fluoride ion in 100 ml of solution if the equivalence point is measured to ~ k 0 . 3mV. In the present study, conditions for the titration of fluoride ion using electrogenerated lanthanum ion and a fluoride ion selective membrane electrode for end point detection are described. An equation for the titration curve has been developed and 0.5- to 2-mg samples of fluoride ion in 105 ml of solution have been titrated with a precision and accuracy of about =t0.1%by use of the techniques of pretitration and titration to a predetermined end point potential.
_(2) M. S . Frant and J. W. Rose, Jr., Science, 154, 1553 (1966). (3) T. S. Light, Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, March 1967. (4) J. J. Lingane, ANAL.CHEW., 39, 881 (1967). VOL. 41,NO. 2, FEBRUARY 1969
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