Let's solve each part step-by-step. --- # (a) **CSTR Design Equation and \( C_A \) vs. \( \tau \)** ### **Step 1: Write the CSTR Mole Balance** At steady state: \[ \text{In} - \text{Out} + \text{Generation} = \] For component \( A \): \[ F_{in} C_{A} - F_{out} C_A + V(-kC_A) = \] With \( F_{in} = F_{out} = F \) and \( \tau = \frac{V}{F} \): \[ F(C_{A} - C_A) - kVC_A = \] \[ C_{A} - C_A = k\tau C_A \] ### **Step 2: Rearrange for \( C_A \) as a Function of \( \tau \)** \[ C_{A} = C_A (1 + k\tau) \] \[ C_A = \frac{C_{A}}{1 + k\tau} \] ### **Step 3: Sketch and Behavior** - At \( \tau = \): \( C_A = C_{A} \) - As \( \tau \to \infty \): \( C_A \to \) (asymptotic approach) **Plot**: \( y \)-axis is \( C_A \), \( x \)-axis is \( \tau \). The curve starts at \( C_{A} \) and decreases asymptotically toward . --- # (b) **Fractional Conversion \( X_A \) vs. Damköhler Number \( Da \)** ### **Step 1: Define Fractional Conversion** \[ X_A = \frac{C_{A} - C_A}{C_{A}} \] ### **Step 2: Substitute \( C_A \) from Part (a)** \[ X_A = 1 - \frac{C_A}{C_{A}} = 1 - \frac{1}{1 + k\tau} \] ### **Step 3: Define Damköhler Number** \[ Da = k\tau \] So, \[ X_A = 1 - \frac{1}{1 + Da} \] \[ X_A = \frac{Da}{1 + Da} \] ### **Step 4: Plot Behavior** - At \( Da \to \), \( X_A \to \) - At \( Da \to \infty \), \( X_A \to 1 \) **Plot**: \( y \)-axis is \( X_A \), \( x \)-axis is \( Da \). The curve starts at , approaches 1 asymptotically. --- # (c) **Reaction Rate \( -r_A \) vs. Exit Concentration \( C_A \)** ### **Step 1: Write the Rate Law** \[ -r_A = k C_A \] ### **Step 2: Plot** Plot \( -r_A \) (y-axis) vs. \( C_A \) (x-axis). - This is a straight line through the origin with slope \( k \). ### **Step 3: Explanation** - The slope of the line is the first-order rate constant \( k \). - The larger \( k \), the steeper the line. --- ## **Summary of Answers** ### **(a)** \[ \boxed{C_A = \frac{C_{A}}{1 + k\tau}} \] - Starts at \( C_{A} \), decreases to as \( \tau \to \infty \). ### **(b)** \[ \boxed{X_A = \frac{Da}{1 + Da} = \frac{k\tau}{1 + k\tau}} \] - Starts at , increases to 1 as \( Da \to \infty \). ### **(c)** - Plot \( -r_A \) vs. \( C_A \): straight line through origin, slope \( k \). - The rate constant \( k \) is the slope of this line. --- ## **Sketches (Description)** 1. **\( C_A \) vs. \( \tau \):** - Hyperbolic decay from \( C_{A} \) to . 2. **\( X_A \) vs. \( Da \):** - Rises from to 1, approaching 1 asymptotically. 3. **\( -r_A \) vs. \( C_A \):** - Straight line through origin, slope = \( k \). If you need actual graphs or a more detailed algebraic derivation, let me know!