Question 1. diffusion & reaction in spherical coordinates. consider a porous spherical catalyst particle with radius r, under a constant stream of a reactant a. within the particle, reactor occurs at the surface of tortuous porous passages. assume that the reaction occurs homogeneously within the porous sphere, and is first order with respect to a, in which ra-k' a ca where a is constant representing the available catalytic surface area per unit volume. the surface concentration within the porous sphere, atr-r, is given as car. a) derive the governing equation for this problem, either from a shell-balance or from governing equations. b) what are the boundary conditions? c) place both the governing equation as well as boundary conditions in relevant dimensionless variables: use г for concentration, and n for radius. d) what is the relevant thiele modulus for this problem? e) find the concentration profile γ as a function of η within the particle find the total molar flow rate of species a across the surface of the particle. f) g) find an expression for the effectiveness factor of this catalyst particle. h) make a plot of the effectiveness factor as a function of the thiele modulus you defined. hint: for solving ode problems in spherical coordinates, the transformation y(r)-fr) is often useful, to simplify the equation.