✍ Scribed by David I. Dornfeld; Dennis H. Evans
No coin nor oath required. For personal study only.
The first derivation of a transition time equation for ehronopotentiometry with cylindrical electrodes was that of PETERS AND LINGANE 1. These authors utilized asymptotic expansions of the Bessel functions whose quotient appears in the Laplace transform of the concentration in the solution of the cylindrical form of the Fick's law equation with chronopotentiometric boundary conditions. Inverse transforms were obtained term by term giving a transition time equation whose validity was supported by satisfactory agreement with experimental data 1.
Nevertheless, it soon became apparent that the Peters and Lingane equation fails when the dimensionless parameter, O, becomes large 2 (O=D~r~/ro where D is the diffusion coefficient of the electroactive species (cm2/sec), z is the transition time (see) and r0 is the radius of the cylindrical electrode (cm)). In the reduction of hydrogen ion, which has an exceptionally large D (8.6. lO -5 cruZ/see at 25 ° in I M KC1), at an electrode of 0.0252 cm radius, the Peters and Lingane equation failed for r greater than about 4 sec (0=0.7) even with two additional terms added to the series2, a. The improvement afforded by eactl new derived term is so slight that further extensions of the series in the Peters and Lingane equation seem unwarranted.
An equation valid for large values of 0 may be derived by the methods of CARSLAW AND JAEGER 4. The inversion theorem is applied directly to the Laplace transform of the concentration. The Bessel functions are written in the form of their series definitions, and term by term integration is performed. Using this procedure we have added another term to the equation given by CARSLAW AND JAEGER 4. The result is:
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