Let's solve the problem step by step for both the **PFR (Plug Flow Reactor)** and **CSTR (Continuous Stirred Tank Reactor)**. --- ## **Given Data** - **Initial concentration of A,** \( C_{A0} = 2 \) mol/L - **Reactor volume,** \( V = 3 \) L - **Rate law:** \( r = 0.2 C_A \) (units: mol/L·min) - This is a **first order** reaction. - **To find:** - (a) Feed rate (L/min) for **PFR** to get \( C_A = 0.1 \) mol/L at outlet. - (b) For **PFR**, find outlet \( C_A \) if feed rate is 0.1 L/min. - (c) Repeat (a) and (b) for **CSTR**. --- # **(a) PFR: What feed rate (L/min) gives outlet \( C_A = 0.1 \) mol/L?** ### **Step 1: PFR Design Equation for First Order Reaction** The design equation for a PFR is: \[ V = \int_{C_{A0}}^{C_{A}} \frac{dC_A}{-r} \] Given \( r = 0.2 C_A \), so \( -r = -0.2 C_A \). \[ V = \int_{C_{A0}}^{C_{A}} \frac{dC_A}{-0.2 C_A} = \frac{1}{0.2} \int_{C_{A0}}^{C_{A}} \frac{dC_A}{C_A} = 5 \left[ \ln(C_A) \right]_{C_{A0}}^{C_A} \] \[ V = 5 [\ln(C_A) - \ln(C_{A0})] = 5 \ln\left(\frac{C_A}{C_{A0}}\right) \] But in terms of flow rate (\( F \)), for a PFR: \[ \frac{dC_A}{dt} = F (C_{A0} - C_A) = \text{total amount reacted per unit time} \] But for constant volumetric flow rate (\( F \)), the space time (\( \tau \)) is: \[ \tau = \frac{V}{F} \] And for a first order reaction in PFR: \[ \frac{C_A}{C_{A0}} = \exp(-k \tau) \] where \( k = 0.2 \) min\(^{-1}\). So, \[ \frac{C_A}{C_{A0}} = \exp\left(-k \frac{V}{F}\right) \] \[ \Rightarrow \ln\left(\frac{C_A}{C_{A0}}\right) = -k \frac{V}{F} \] \[ \Rightarrow \frac{V}{F} = -\frac{1}{k} \ln\left(\frac{C_A}{C_{A0}}\right) \] \[ \Rightarrow F = \frac{V}{-\frac{1}{k} \ln\left(\frac{C_A}{C_{A0}}\right)} \] --- ### **Step 2: Plug in the values** Given: - \( C_{A0} = 2 \) mol/L - \( C_A = 0.1 \) mol/L - \( k = 0.2 \) min\(^{-1}\) - \( V = 3 \) L So, \[ F = \frac{V}{-\frac{1}{k} \ln\left(\frac{C_A}{C_{A0}}\right)} \] \[ \ln\left(\frac{C_A}{C_{A0}}\right) = \ln\left(\frac{0.1}{2}\right) = \ln(0.05) = -2.9957 \] \[ -\frac{1}{k} \ln\left(\frac{C_A}{C_{A0}}\right) = -\frac{1}{0.2} \times (-2.9957) = 5 \times 2.9957 = 14.9785 \] So, \[ F = \frac{3}{14.9785} = 0.2003\ \text{L/min} \] **Final answer (a) for PFR:** \[ \boxed{F = 0.20\ \text{L/min}} \] --- # **(b) PFR: What is outlet \( C_A \) if feed rate is 0.1 L/min?** Given \( F = 0.1\ \text{L/min} \): \[ \tau = \frac{V}{F} = \frac{3}{0.1} = 30\ \text{min} \] \[ \frac{C_A}{C_{A0}} = \exp(-k \tau) = \exp(-0.2 \times 30) = \exp(-6) = 0.0024788 \] \[ C_A = C_{A0} \times 0.0024788 = 2 \times 0.0024788 = 0.00496\ \text{mol/L} \] **Final answer (b) for PFR:** \[ \boxed{C_A = 0.005\ \text{mol/L}} \] --- # **(c) CSTR: Repeat (a) and (b) for CSTR** ## **CSTR Design Equation** At steady state: \[ F (C_{A0} - C_A) = V \times r \] \[ F (C_{A0} - C_A) = V \times k C_A \] \[ F C_{A0} - F C_A = V k C_A \] \[ F C_{A0} = F C_A + V k C_A = C_A (F + V k) \] \[ C_A = \frac{F C_{A0}}{F + V k} \] --- ### **(a) CSTR: What feed rate gives \( C_A = 0.1 \) mol/L?** \[ C_A = \frac{F C_{A0}}{F + Vk} \] Rearrange for \( F \): \[ C_A (F + V k) = F C_{A0} \] \[ C_A F + C_A V k = F C_{A0} \] \[ C_A V k = F (C_{A0} - C_A) \] \[ F = \frac{C_A V k}{C_{A0} - C_A} \] Plug in values: - \( C_A = 0.1 \) mol/L - \( V = 3 \) L - \( k = 0.2 \) min\(^{-1}\) - \( C_{A0} = 2 \) mol/L \[ F = \frac{0.1 \times 3 \times 0.2}{2 - 0.1} = \frac{0.06}{1.9} = 0.0316\ \text{L/min} \] **Final answer (a) for CSTR:** \[ \boxed{F = 0.032\ \text{L/min}} \] --- ### **(b) CSTR: What is outlet \( C_A \) if feed rate is 0.1 L/min?** \[ C_A = \frac{F C_{A0}}{F + Vk} \] Plug in values: - \( F = 0.1 \) L/min - \( V = 3 \) L - \( k = 0.2 \) - \( C_{A0} = 2 \) \[ C_A = \frac{0.1 \times 2}{0.1 + 3 \times 0.2} = \frac{0.2}{0.1 + 0.6} = \frac{0.2}{0.7} = 0.286\ \text{mol/L} \] **Final answer (b) for CSTR:** \[ \boxed{C_A = 0.29\ \text{mol/L}} \] --- # **Summary Table** | Subpart | Reactor | Given/Find | Feed rate (L/min) | Outlet \( C_A \) (mol/L) | |---------|---------|----------------------|-------------------|--------------------------| | (a) | PFR | \( C_A = 0.1 \) | 0.20 | 0.1 | | (b) | PFR | \( F = 0.1 \) | 0.1 | 0.005 | | (a) | CSTR | \( C_A = 0.1 \) | 0.032 | 0.1 | | (b) | CSTR | \( F = 0.1 \) | 0.1 | 0.29 | --- ## **Boxed Final Answers** ### **(a) PFR:** \[ \boxed{F = 0.20\ \text{L/min}} \] ### **(b) PFR:** \[ \boxed{C_A = 0.005\ \text{mol/L}} \] ### **(a) CSTR:** \[ \boxed{F = 0.032\ \text{L/min}} \] ### **(b) CSTR:** \[ \boxed{C_A = 0.29\ \text{mol/L}} \] --- **Let me know if you want the step-by-step calculations in a specific format or have further questions!**