For unsteady-state diffusion, adsorption and a first-order reaction in a slab, cylinder and sphere catalyst, high-order approximations in the form of coupled ordinary differential equations are developed to substitute the exact partial differential equations. An infinite continued fraction is developed, which unifies the exact transfer functions of the three geometries as a function of the shape factor and the Thiele modulus in the Laplace domain. The time-domain approximations are obtained from the truncated continued fractions. The coefficients in the time-domain approximations are very easy to determine, and increasing the order of approximation is simple and straightforward. (C) 2011 Elsevier B.V. All rights reserved.