Let's break down and address the first task: **writing the full set of ODEs (Ordinary Differential Equations)** for the given reactions and kinetics. --- ## **Step 1: Write the Reactions** Given: 1. \( \mathbf{A \rightarrow B} \) (R1) 2. \( \mathbf{A \rightarrow C} \) (R2) 3. \( \mathbf{B \rightarrow D} \) (R3) --- ## **Step 2: Write the Rate Expressions** Given kinetics: - \( r_1 = k_1 C_A \) - \( r_2 = k_2 C_A^{1.5} \) - \( r_3 = k_3 C_B \) Where: - \(C_A, C_B, C_C, C_D\) are the concentrations of A, B, C, D. - \(k_1, k_2, k_3\) are rate constants. --- ## **Step 3: Write the Mole Balances (ODEs)** Assuming a plug flow reactor (PFR) or batch reactor (no spatial variation): ### **For A:** A is consumed in both R1 and R2. \[ \frac{dC_A}{dt} = -r_1 - r_2 = -k_1 C_A - k_2 C_A^{1.5} \] --- ### **For B:** B is formed from R1 and consumed in R3. \[ \frac{dC_B}{dt} = r_1 - r_3 = k_1 C_A - k_3 C_B \] --- ### **For C:** C is formed from R2. \[ \frac{dC_C}{dt} = r_2 = k_2 C_A^{1.5} \] --- ### **For D:** D is formed from R3. \[ \frac{dC_D}{dt} = r_3 = k_3 C_B \] --- ## **Step 4: Final Set of ODEs** \[ \boxed{ \begin{aligned} \frac{dC_A}{dt} &= -k_1 C_A - k_2 C_A^{1.5} \\ \frac{dC_B}{dt} &= k_1 C_A - k_3 C_B \\ \frac{dC_C}{dt} &= k_2 C_A^{1.5} \\ \frac{dC_D}{dt} &= k_3 C_B \\ \end{aligned} } \] --- ## **Explanation** - Each equation represents the rate of change of concentration for each species. - The negative signs indicate consumption, while positive signs indicate formation. - The rates are substituted from the given kinetics. --- ### **Next Steps** You can use these ODEs for further tasks: 1. **Selectivity profile** (e.g., C vs. B) 2. **Optimal reactor length/time** 3. **Hot spot analysis** (add energy balance) 4. **Effect of reducing particle diameter** (affects heat/mass transfer) --- **Let me know which subtask you'd like to solve next, or if you need explanations for the rest!**