Let's break down the problem into **step-by-step solutions** for each part: --- ## 1. **Reciprocal Rate vs Conversion Relationship (Second-Order Exothermic Reaction)** For a **second-order reaction**: \[ -r_A = k C_A^2 \] Assuming the feed is pure A: - \( C_{A} \) is the initial concentration - \( X \) is the conversion The concentration of A at any point: \[ C_A = C_{A}(1-X) \] So, \[ -r_A = k [C_{A}(1-X)]^2 = k C_{A}^2 (1-X)^2 \] The reciprocal rate: \[ \frac{1}{-r_A} = \frac{1}{k C_{A}^2 (1-X)^2} \] --- ## 2. **Volume Required for 85% Conversion** Given equation for CSTR: \[ V = \frac{F_{A} X}{-r_A} \] Where: - \( X = .85 \) (for 85% conversion) - \( F_{A} \) is the molar flow rate of A - \( -r_A \) at exit \[ C_A = C_{A}(1-X) = C_{A}(1-.85) = .15C_{A} \] \[ -r_A = k (.15C_{A})^2 = k (.0225) C_{A}^2 \] \[ V = \frac{F_{A} \cdot .85}{k (.0225) C_{A}^2} \] **This is the volume required for 85% conversion in a CSTR with a second-order reaction.** --- ## 3. **Plug Flow Reactor (PFR) for First-Order Reaction** Given: \[ C_A = C_{A} \exp(-k \tau) \] where \( \tau \) is the **space time** (reactor volume / volumetric flow rate). ### If \( \tau = 120 \) s, calculate the reactor volume: **Assume you are given \( C_{A} \), \( k \), and the desired conversion.** \[ X = 1 - \frac{C_A}{C_{A}} = 1 - \exp(-k \tau) \] You can rearrange to solve for \( k \) or check the final concentration for a given \( k \). \[ C_A = C_{A} \exp(-k \cdot 120) \] --- ## 4. **Step-by-Step Calculation Example** ### **Assume**: - \( C_{A} = 1 \) mol/L - \( k = .02 \) s\(^{-1}\) - \( \tau = 120 \) s #### **Calculate \( C_A \):** \[ C_A = 1 \cdot \exp(-.02 \cdot 120) \] \[ C_A = \exp(-2.4) \] \[ C_A \approx .0907 \text{ mol/L} \] #### **Calculate Conversion \( X \):** \[ X = 1 - \frac{C_A}{C_{A}} = 1 - .0907 = .909 \text{ or } 90.9\% \] --- ### **Final Answers Summarized** **A. Volume for 85% conversion in CSTR, second-order:** \[ V = \frac{F_{A} \cdot .85}{k (.0225) C_{A}^2} \] **B. PFR First-Order, \( \tau = 120 \) s, \( k = .02 \) s\(^{-1} \):** \[ C_A = .0907 \text{ mol/L}, \quad X = 90.9\% \] --- **If you have specific values for \( F_{A} \), \( C_{A} \), or \( k \), plug them in for more precise results!**