Double Potential Step Method for Measuring Rate Constants of Dimerization Reactions Michael L. Olmsteadl and Richard S. Nicholson Chemistry Department, Michigan State University, East Lansing, Mich. 48823 RECENTLY we published a mathematical description of the cyclic voltammetry experiment when the electrolysis product undergoes a homogeneous dimerization reaction (1). Although cyclic voltammetry should be useful for studying this electrode process, in some instances a double potential step method may be advantageous (2). In particular, the potential step method probably is applicable to faster reactions, with the added advantage that by proper selection of stepping potentials the influence of electron transfer kinetics in principle can be eliminated as a variable. As a consequence of the use of constant potential, however, the potential step method is less useful diagnostically than cyclic voltammetry, and therefore the two techniques should be regarded as complementary. From a mathematical point of view the double potential step method is essentially a limiting case of cyclic voltammetry. Thus, it is a simple matter to transpose the analysis cited above for cyclic voltammetry to the case of a double potential step forcing function. Hence, the purpose of this Note is to present theory of the double potential step method for a succeeding dimerization.
The mathematical approach is identical with that for cyclic voltammetry ( I ) , except a double potential step (2) is employed in place of the triangular wave, and the present treatment is restricted to linear diffusion. We label duration of the first potential step r , and assume that its amplitude corresponds to the limiting current region for reduction of 0. Similarly, for t > r we assume that the potential corresponding to the second step is such that the surface concentration of R is essentially zero. The problem is then formulated in terms of a dimensionless time coordinate defined by
y
=
(1)
kzCo*t
where CO*is the initial concentration of depolarizer. At t equal r the second potential step is applied, and the dimensionless switching time, y,, is defined by
y , = kKo*r
(2)
With this particular definition of dimensionless time, current is given by [see analogy with y and x(y), Reference ( I ) ]
i For y
=
(3)
n F A d k 2 C o * D oCo*+(y)
< y , the function +(y) can be expressed analytically as
THEORY
Equations 1 and 4 substituted in Equation 3 give the well known expression for purely diffusion-limited potentiostatic electrolysis (3).
The mechanism being considered is (1) O+neeR 2R
-P
dimer
1
Present address, Bell Telephone Laboratories, Murray Hill,
N. J. 07971.
(1) M. L. Olmstead, R. G. Hamilton, and R. S. Nicholson, ANAL. CHEM., 41,260 (1969). (2) W. M. Schwarz and I. Shain, J . Phys. Chem., 69,30 (1965).
(3) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience, New York, 1954, p 51.
Table I. Absolute Values of the Function +(y)
-YlYs
0.5
1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45
1.550 1.130
1.50
1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00
0.885
0.732 0.621 0.540 0.475 0.424 0.382 0.347 0.317 0.291 0.269 0.250 0.233 0.218 0.204 0.192 0.181
0.6 1.316 0.983 0.774 0.639 0.543 0.471 0.415 0.370 0.333 0.303 0.276 0.254 0.235 0.218 0.203 0.190 0.178 0.167 0.158
~-
0.8 1.076 0.787 0.621 0.512 0.435 0.377 0.332 0.296 0.266 0.242 0.221 0.203 0.187 0.174 0.162 0.151 0.142 0.133 0.140 0.125
0.7 1.210 0.874 0.689 0.569 0.483 0.419 0.369 0.329 0.296 0.269 0.245 0.225 0.208 0,193 0.180 0.168 0.158 0.148
Ya
0.9 0.986 0.717 0.565 0.466 0.396 0.343 0.302 0.269 0.242 0.220 0.200 0.184 0.170 0.157 0.147 0.137 0.128 0.120 0.114
1.0 0.903 0.658 0.519 0.428 0.363 0.314 0.277 0.246 0.222 0.201 0.183 0.168 0.155 0.144 0.134 0.125 0.117 0.110 0.104
1.1 0.836 0.609 0.479 0.395 0.335 0.290 0.255 0.227 0.204 0.185 0.169 0.155
0.143 0.133 0.123 0.115 0.108 0.101 0.095
1.2 0.777 0.565 0.445 0.367 0.311 0.269 0.236 0.211 0.189 0.172 0.157 0.144 0.132 0.123 0.114 0.107 0.100 0.094
1.3 0.726 0.528 0.416 0.342 0.290 0.251 0.221 0.196 0.176 0.159 0.146 0.133 0.123 0.114 0.106 0.099 0.093 0.087 0.088 0.082
1.4 0.682 0.495 0.390 0.321 0.272 0.235 0.207 0.184 0.165 0.150 0.136 0.125 0.115 0.107 0.099 0.093 0.087 0.081 0.077
1.5 0.642 0.466 0.367 0.301 0.255 0.220 0.194 0.172 0.155 0.140
0.128 0.117 0.108 0.100 0.093 0.086 0.081 0.076 0.072
1.6 0.607 0.440 0.346 0.285 0.241 0.208 0.183 0.162 0.146 0.132 0.120 0.110 0.102 0.094 0.087 0.082 0.076 0.072 0.067
1.7 0.574 0.417 0.327 0.269 0.228 0.196 0.173 0.153 0.138 0.124 0.114 0.104 0.096 0.088
1.8 0.546 0.396 0.311 0.255 0.216 0.187 0.163 0.145 0.130 0.118 0.107 0.098 0.090 0.084 0.078 0.072 0.068 0.064
1.9 0.520 0.366 0.295 0.238 0.205 0.174 0.155
0.136 0.123 0.110 0.102 0.092 0.086 0.078 0.074 0.068 0.064
0.082 0.077 0.072 0.067 0.060 0.064 0.060 0.057
2.0 3.0 0.496 0.339 0.359 0.244 0.282 0.190 0.231 0.155 0.195 0.131 0.169 0.112 0.148 0.098 0.131 0.087 0.118 0.078 0.106 0.070 0.0970.064 0.089 0.058 0.082 0.054 0.0760.050 0.070 0.046 0.066 0.043 0.061 0.041 0.057 0.037 0.054 0.035
VOL. 41, NO. 6, MAY 1969
851
‘::I
For y > y 8 and finite k2 the function +(y) cannot conveniently be expressed analytically, and therefore is expressed numerically in tabular form. These numerical values of +(y) were obtained by exactly the procedures described previously ( I ) , following minor modification of the computer programs to replace the potential scans by potential steps.
0.4
0.2-
RESULTS AND DISCUSSION
Figure 1 is a plot of the function +(y) for y , equals 2 [for comparison, Figure 1 includes +(y) for cases in which there is a first order succeeding reaction (2), and no succeeding reaction]. The fact that the current function in Figure 1 for y < y. is independent of kz, whereas for y > y , +(y) is a function of kz (or more generally, y J , permits a number of possible correlations between current and the rate constant. For example, from data for several values of y , a family of working curves directly analogous with those of Schwarz and Shain for the first order reaction could be constructed (2). Any given correlation is essentially arbitrary, however, and might not be preferable in every situation. Thus, the most satisfactory way to present data for the present case is to give values of +(y) for reasonable values of y,. These values of y , are dictated by the fact that to observe optimum influence of the chemical reaction, T should approximately equal the half-life of the reaction, so that values of y, near unity are most useful. Thus, values of +(y) for y > y a are presented in Table I for values of y , in the range 0.5 < y 8 < 3. From these data, curves like Figure 1 can be constructed,
#Y)
0.0-
-0.2-0.4-0.6-
I
0
I
I
I
I
2
3
I I 4
Y ’
Figure 1. Variation of the current function, $,(y), for: ( A ) , first-order succeeding reaction, y = klt: (B), succeeding dimerization, y = kzCo*t: (C) no coupled chemical reaction, y=t and correlations with experimental data appropriate for a given situation can be developed easily. RECEIVED for review November 6, 1968. Accepted February 24, 1968. Work supported by the National Science Foundation and United States Army Research Office-Durham.
Rapid Group Radiochemical Separations for Activation Analysis of Steels Barbara A. Thompson and,Philip D. LaFleur Analytical Chemistry Division, National Bureau of Standards, Washington, D. C. 20234
AMONGthe Standard Reference Materials (SRM’s) certified by the National Bureau of Standards, there are about 100 different types of steel and cast iron ( I ) . Concentrations of 10-20 minor constituents are commonly reported and each must be determined by at least two independent analytical methods before certification. Of the elements whose concentrations are usually certified, Mn, Cu, Ni, Cr, V, Mo, W, Co, Ti, and As could readily be determined by activation analysis if sensitivity were the only consideration. These elements are normally present in steels at levels of a few tenths or hundredths of a per cent which is, for most of them, one or more orders of magnitude higher than is required for detection by neutron activation analysis. In fact, Albert (2) and Malvano and Grosso ( 3 ) have reported the determination of most of these elements in pure iron at the ppm level. (1) “Catalog and Price List of Standard Materials Issued by the National Bureau of Standards,” National Bureau of Standards Miscellaneous Publication 260, U. S . Government Printing Office,Washington, D. C., 1968, p 3-11. (2) P. Albert, Proceedings, 1961 International Conference on Modern Trends in Activation Analysis, College Station, Texas, 1961, p 86. (3) R. Malvano and P. Grosso, Anal. Chim. Acta, 34, 253 (1966). 852
ANALYTICAL CHEMISTRY
The work of Albert and his group utilized extensive radiochemical separations while that of Malvano and Grosso was largely nondestructive. Both of these groups were concerned with the analysis of pure iron; however, most of the steels certified by NBS contain between 0.5 and 2 % manganese and 0.1% or more of copper. The radiation from these constituents completely masks most of the short-lived activities and presents a serious interference for those of intermediate half-life. Additional interferences arise from the iron matrix and from chromium in some of the high chromium steels. Some chemistry is thus necessary in order to obtain the desired information with the high precision and accuracy required for these analytical standards. The detailed radiochemical separation procedures used by Albert have the potential for high precision and accuracy, but are very time consuming and, thus, very expensive. By separating the elements to be determined into groups instead of separating each element individually, and utilizing the high resolution of semiconductor detectors where appropriate, the advantages of nondestructive analysis and radiochemical separations can be combined. Most of the group separation procedures which have been
