Let's break down and explain the solution step-by-step for **Question 2** regarding the spherical catalyst and concentration profile. --- ## **2. Spherical Catalyst - Concentration Profile Derivation** ### **a) Derivation of the Governing ODE** #### **Context** - We have a spherical porous catalyst. - There is diffusion and first-order reaction occurring inside the particle. - We are to derive an ODE for steady-state concentration, \( C(r) \), as a function of radius \( r \). #### **Step 1: Mass Balance in Spherical Coordinates** The general steady-state diffusion-reaction equation in spherical coordinates (no accumulation, no convection) is: \[ \frac{1}{r^2} \frac{d}{dr} \left( r^2 D_e \frac{dC}{dr} \right) - kC = \] Where: - \( D_e \): Effective diffusivity - \( k \): First-order reaction rate constant #### **Step 2: Simplify the Equation** Assuming \( D_e \) is constant: \[ \frac{d}{dr} \left( r^2 \frac{dC}{dr} \right) - \frac{k}{D_e} r^2 C = \] Divide both sides by \( r^2 \): \[ \frac{d^2C}{dr^2} + \frac{2}{r} \frac{dC}{dr} - \frac{k}{D_e} C = \] This is the required ODE. --- ### **b) Boundary Conditions for Integration** #### **Symmetry at Center (\( r = \)):** - At the center, the flux must be finite (no singularity). - This requires the derivative to be zero: \[ \left. \frac{dC}{dr} \right|_{r=} = \] #### **External Surface (\( r = R \)):** - The surface concentration is specified (constant external environment): \[ C(R) = C_s \] --- ### **c) Exact Analytical Solution for \( C(r) \)** #### **Step 1: General Solution** The ODE: \[ \frac{d^2C}{dr^2} + \frac{2}{r} \frac{dC}{dr} - \frac{k}{D_e} C = \] Let: \[ \phi^2 = \frac{k}{D_e} \] The general solution for such an equation is: \[ C(r) = \frac{A \sinh(\phi r)}{r} + \frac{B \cosh(\phi r)}{r} \] #### **Step 2: Apply Symmetry Condition at \( r = \)** - For finite \( C \) at \( r = \), only the \(\sinh\) term survives because \(\cosh(\phi r)/r\) diverges as \( r \to \). - Thus, \( B = \). #### **Step 3: Apply Surface Condition at \( r = R \)** \[ C(R) = C_s \] So, \[ C(R) = A \frac{\sinh(\phi R)}{R} = C_s \implies A = C_s \frac{R}{\sinh(\phi R)} \] #### **Step 4: Final Solution** \[ C(r) = C_s \frac{R}{r} \frac{\sinh(\phi r)}{\sinh(\phi R)} \] #### **Final Boxed Answer** \[ \boxed{ C(r) = C_s \frac{R}{r} \frac{\sinh\left(\phi r\right)}{\sinh\left(\phi R\right)} } \] where \( \phi = \sqrt{\frac{k}{D_e}} \). --- ### **Summary Table** | Step | Equation / Condition | |----------------|-----------------------------------------------------------| | Governing ODE | \( \frac{d^2C}{dr^2} + \frac{2}{r} \frac{dC}{dr} - \frac{k}{D_e} C = \) | | Center (\( r= \)) | \( \left. \frac{dC}{dr} \right|_{r=} = \) | | Surface (\( r=R \)) | \( C(R) = C_s \) | | Solution | \( C(r) = C_s \frac{R}{r} \frac{\sinh(\phi r)}{\sinh(\phi R)} \) | | \(\phi\) definition | \( \phi = \sqrt{\frac{k}{D_e}} \) | --- **This is the step-by-step derivation and final answer for the spherical catalyst shell balance concentration profile problem.**