Series reactions A → B → C on successive spheres

- Exact solution for the two-step series reaction on successive spherical surfaces is presented.
- Reaction rate phenomena are discussed for various sphere radius ratios and separations.
- A rate maximum is obtained at an optimum sphere center-center separation.
- Reaction rate results are used to verify numerical values from the literature.

The twin spherical harmonic expansion method with iterative solution of the coefficient equations is used to generate a rigorous analytical solution for the rate of series reaction, A → B → C, occurring, respectively, on two successive spheres of radius a1 and a2, a center-to-center distance d apart. To investigate the influences of the intersphere diffusion and geometry, diffusion-controlled reactions are considered. Results are presented as a series expansion of the dimensionless reaction rate R in terms of the dimensionless center-to-center sphere separation d¯(=d/(a1+a2)) reciprocals, and for various sphere radius ratios γ(=a1/a2). When the sphere radius ratio γ is less than unity, a maximum in the series reaction rate is found for a d¯ of about 1.05. Also an exact value of the series dimensionless reaction rate of ln2 is obtained in the limit γ → 0 (very large a2 or very small a1) for spheres in contact. Results suggest that the plots of reaction rates for contacting spheres can be extrapolated versus γ to the ln2 limit at γ → 0, and that the rate maximum effect is large in the γ → 0 limit.

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