Let's go through each part step by step: --- ### (a) Sketch the Exit Concentration \( C_A \) vs. Space Time \( \tau \) First, we need the CSTR design equation for a first-order reaction at steady state: \[ F_{in}C_{A} - F_{out}C_A - Vr_A = \] For a CSTR, \( F_{in} = F_{out} = F \), and \( r_A = -kC_A \): \[ F(C_{A} - C_A) = VkC_A \] Since space time \( \tau = \frac{V}{F} \): \[ C_{A} - C_A = \tau k C_A \] \[ C_{A} = C_A(1 + k\tau) \] \[ C_A = \frac{C_{A}}{1 + k\tau} \] If you plot \( C_A \) versus \( \tau \), the curve starts at \( C_{A} \) when \( \tau = \) and decreases monotonically, approaching zero as \( \tau \) goes to infinity. The curve is hyperbolic, flattening out as \( \tau \) increases. --- ### (b) Plot the Fractional Conversion \( X_A \) vs. Damköhler Number \( Da \) Fractional conversion is defined as: \[ X_A = \frac{C_{A} - C_A}{C_{A}} \] Substitute the expression for \( C_A \): \[ X_A = \frac{C_{A} - \frac{C_{A}}{1 + k\tau}}{C_{A}} \] \[ X_A = 1 - \frac{1}{1 + k\tau} \] The Damköhler number is defined as \( Da = k\tau \): \[ X_A = 1 - \frac{1}{1 + Da} \] If you plot \( X_A \) versus \( Da \), at low \( Da \) (small \( \tau \) or slow reaction), \( X_A \) is close to zero, meaning little conversion. As \( Da \) increases, \( X_A \) approaches 1, meaning almost complete conversion. The curve rises rapidly at first and then levels off as it approaches 1. --- ### (c) Relationship Between Reaction Rate and Exit Concentration The rate law is \( -r_A = kC_A \). If you plot \( -r_A \) (y-axis) versus \( C_A \) (x-axis), you get a straight line through the origin with slope \( k \). This means the rate constant \( k \) is the slope of this line. A higher \( k \) makes the line steeper, indicating a faster reaction for the same concentration of \( C_A \). --- If you need help with the actual sketches or further explanation of the shapes of the curves, let me know!