J. Chem. Soc., Faraday Trans. I, 1982, 78, 1561-1567 Tube Electrode and Electron Spin Transient Signals B Y w. JOHN ALBERY* AND RICHARD G. Resonance COM P T O N ~ Department of Chemistry, Imperial College of Science and Technology, South Kensington, London SW7 2AY Received 6th July, 1981 A tube electrode is positioned so that the electrode is just above an electron spin resonance cavity; radicals generated on the electrode are transported by laminar flow into the cavity. By solving the time-dependent convective-diffusion equation for the transport, the variation of the e.s.r. signal with time as the current is stepped on the electrode can be calculated. Theory and experiment are shown to be in good agreement. We have previ~uslyl-~ described how a tube electrode coupled to an e.s.r.spectrometer can be used to study the kinetics of radical species generated at the electrode. The electrode, which is a section of the wall of the tube, is situated just above the e.s.r. cavity. The species are carried by laminar flow from the electrode into the e.s.r. cavity where they are detected and their concentration measured from the e.s.r. signal. The variation of this signal with flow rate characterises whether the radical is stable,l whether it is decomposing by simple first-2 or ~econd-~ order kinetics or whether a more complicated kinetic scheme operate^.^ In this paper we consider the transient signal seen when the current at the electrode is changed. In particular, the case for a galvanostatic step is considered and theory and experiment are shown to be in good agreement.THEORY The convective-diffusion equation describing the concentration of a stable radical is where v, is the velocity of flow at the centre of the tube, ro is the radius of the tube, r is the radial distance from the centre of the tube, and x is the distance down the tube. The following variables are used to express eqn (1) in dimensionless variables t Present address: Department of Inorganic, Physical and Industrial Chemistry, The University of Liverpool, Liverpool L69 3BX. 51 1561 FAR 11562 TUBE ELECTRODE A N D E.S.R. TRANSIENT SIGNALS where xE is the length of the electrode and 1 is the total length of the electrode and cavity. Providing' that eqn (1) can be written (lD/uori) 4 1 av a2w az a t 2 ax (3) ---- - where the concentration c has been normalized with The relevant boundary conditions are at z = o , v = o t + c o , V+O t = O as and at for a t and for av a t The functionflz) describes the variation of the flux at the surface of the electrode XdXE, %=AT) x>&, -=o.A T ) = - We extend our previous function M3 to describe the transient e.s.r. signal S, normalised with respect to the current, i, at the electrode where vf is the volume flow rate (V, = xrEu0/2) and n is the number of electrons transferred in the electrode reaction. M , varies between 0 at z = 0 and M , the steady-state e.s.r. detection effi~iency,~ as z -, co. The e.s.r. signal S is given by1 where S* is the signal for one mole of spins at the centre of the cavity. Then from eqn (6) and (7) We now take the Laplace transform of eqn (3) with respect to time, and obtainW.J. ALBERY AND R. G . COMPTON 1563 where p is the transform variable. Eqn (9) has the same form as eqn (4) of ref. (2) which describes radical decomposition by first-order kinetics. The boundary conditions for the two equations become identical if c' in table 1 of ref. (2) is replaced byfip). Thus using the series solution previously obtained2 we can write where b, has been defined in ref. (2) and I 7=0 7 /- 0.5 1-0 1.5 2.0 I ... $ ... I I 1 I 7 FIG. 1.-Calculated e.s.r. signal transient due to a galvanostatic step caused by a step in the current at the electrode (as shown in the inset). So far we have not specified the current transient on the electrode. For the particular case of the galvanostatic step shown in fig. 1, f(z) = - 1 and s o f i p ) = - l/p.Hence (14) 51-21564 TUBE ELECTRODE AND E.S.R. TRANSIENT SIGNALS A power series such as eqn (14) cannot be inverted term by term. Consequently we adopt the method of rational approximation^.^ That is we take the first n terms of eqn (14), J,(p), and express them as a ratio of two polynomials of degrees n - 1 and n, which in turn yield n terms that can be inverted analytically. The procedure is repeated until increasing n makes no significant difference to the result. The particular procedure employed was Levin transformation6t7 of g ( p ) , as described in the Appendix. The function h(z) in eqn (12) is given by eqn (A 10). A complete programme was written to evaluate this function. Good convergence was obtained as n increased for all but small values of z (z c 0.3).For z = 0.2 no convergence was found even taking n as large as 17. Similar problems have been found with ring-disc transients. In contrast only a small number of terms were needed for larger values of z, for instance n = 5 for z > 0.8. Clearly the method of rational approximations is a powerful and efficient method of performing inverse Laplace transformation when analytical methods fail. In fig. 1 the converged values of h(z) are plotted to show a signal transient as a function of time. It is apparent that the transient reaches the steady-state value when z - 2.4. Assuming that D - 10+ cm2 s-l we find that this value of z corresponds to a real-time value of ca. 1 s for our apparatus at the fastest flow rate available.These times will be shorter for unstable species since their concentration profiles extend less far into the cavity. EXPERIMENTAL The basic apparatus and technique have been described previously.’. The aqueous sol- utions used contained 3 mmol dmP3 NNN”’-tetramethyl-p-phenylenediamine (TMPD) and 0.10 mol dm-3 K,SO,. The K,SO, was of AnalaR standard. TMPD was B.D.H. LR grade recrystallized as described by Michaelis.8 All solutions were made up in doubly distilled deionized water. The electrode was made of platinum and had the following geometrical parameters (uide supra), x,/cm = 0.29 and r,/cm = 0.045. All potentials were measured with respect to the saturated calomel electrode. The e.s.r. spectrometer used was a Bruker ER200tt.Transients were obtained by holding the magnetic field at the value corresponding to the maximum in the central peak of the spectrum. RESULTS AND DISCUSSION The system studied was the oxidation of TMPD: NMe2 NMe2 Steady-state studies on the tube electrode showed that the electron transfer was reversible and that the diffusion coefficient of TMPD was 4.9 x cm2 s-l. The e.s.r. spectrum obtained was in good agreement with the literature.g$ lo The spectrum was overmodulated in order to increase the sensitivity. Fig. 2 shows a log-log plot of the steady-state signal ( S / i ) against 5. The observed gradient of -2/3 is that expected from a stable radical.’ All the transients recorded at different flow rates were normalised onto one curve using eqn (2). The resulting plot is shown in fig.3. The solid line shown is theW. J. ALBERY A N D R. G. COMPTON 1565 1 o-2 to-' Vf/cm3 s-' FIG. 2.-Variation of the normalised e.s.r. signal (S/i) as a function of flow rate. The slope of -2/3 is that expected for a stable radical. f r-o-'-our /" 9' 7'' theoretical line derived above. Reasonable agreement is found between theory and experiment. Fig. 4 shows the time taken to reach half of the steady-state value as a function of flow rate. The theoretical line shown was calculated using the curve in fig. 1 and eqn (2), together with the experimental value for the diffusion coefficient. Again reasonable agreement is found between theory and experiment. Potentiostatic transients were also recorded using the same chemical system. The1566 TUBE ELECTRODE A N D E.S.R.TRANSIENT SIGNALS FIG. 4.-Variation with flow rate of the time taken for the signal to reach half of its steady-state value. The solid line is the theoretical line calculated using D = 4.9 x cm2 s-l. resulting signal transient was indistinguishable from galvanostatic transients measured at the same flow rate and having the same steady current value. This is because the large size of the cavity relative to the electrode means that although a larger number of radicals is produced during the potentiostatic step, this increase is small compared with the total number of radicals present in the cavity during the time that the transient is observed. The theory and experiments presented here therefore allow the e.s.r. tube electrode apparatus to be used for transient measurements.The advantage of such measurements over steady-state measurements is that surface processes such as adsorption can be studied. This approach is similar to that used in ring-disc electrode studies.ll9 l2 Using the observed transient and the transient calculated above it is possible to calculate the amount of radicals adsorbed at the electrode. For the existing tube e.s.r. apparatus one or more monolayers would be required for the method to be applicable. The severity of this restriction arises from the relatively long transit times between electrode and cavity. The sensitivity would be significantly increased if the electrode were placed at the centre of the cavity. The development of an in situ tube electrode will be reported in a forthcoming paper and this development should allow smaller quantities of adsorbed radicals to be studied.We thank the S.R.C. for financial support. APPENDIX This Appendix describes the inversion of g ( p ) in eqn (14) where and ck-l Jk-l a, =W. J. ALBERY AND R. G. COMPTON 1567 Each of the partial sums gn(p), where k=m gn(P> = x P”-’ k = l is transformed using the Levin U, transformation6$ ’ to yield the rational approximation U , where Un = Gn(P)/Hn(P) n i-1 G,(p) = E atpi-’ and The coefficients a, and pi can be found from the coefficients a,, a2, a3. . . , a,,, of the series (A 1) by means of the following equations6* (n + 2 ai= i ( ) J-1 J-l and On inverting eqn (A 4) we find where qm is given by (A 9) and Carrying out the integration in eqn (13) we may finally write for h(z) in eqn (12) H ~ ( P ) = d[Hn(p)I/d~- W. J. Albery, B. A. Coles and A. M. Couper, J. Electroanal. Chem., 1975, 65, 901. W. J. Albery, R. G. 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