Let's solve the problem step by step: --- ## **Given Data** - **Michaelis constant:** \( K_m = .2 \) mg/cm³ - **Maximum rate:** \( V_m' = .1 \) mg guanidine/(cm²·s) - **Bulk substrate concentration:** \( S_b = 1 \) mg/cm³ We are to find, for surface concentrations \( S_s = 1, .8, .6, .4, .2, .1 \) mg/cm³: 1. Observed reaction rate (\( r_{obs} \)) 2. Effectiveness factor (\( \eta \)) 3. Mass transfer coefficient (\( k_L \)) --- ## **Step 1: Michaelis-Menten Kinetics** The enzyme-catalyzed reaction at the surface is given by: \[ r = \frac{V_m' S_s}{K_m + S_s} \] Where: - \( r \): Reaction rate per unit surface area (mg/(cm²·s)) - \( S_s \): Substrate concentration at the surface --- ## **Step 2: Mass Transfer Rate** Substrate is transported from the bulk to the surface via a liquid film: \[ \text{Mass transfer rate} = k_L (S_b - S_s) \] At steady state, mass transfer rate = reaction rate: \[ k_L (S_b - S_s) = \frac{V_m' S_s}{K_m + S_s} \] --- ## **Step 3: Calculate Observed Reaction Rate (\( r_{obs} \))** \[ r_{obs} = \text{mass transfer rate} = k_L (S_b - S_s) \] But we don't know \( k_L \) yet. Let's find \( k_L \) first. --- ## **Step 4: Calculate Mass Transfer Coefficient (\( k_L \))** Rearrange the steady-state equation: \[ k_L (S_b - S_s) = \frac{V_m' S_s}{K_m + S_s} \] \[ k_L = \frac{V_m' S_s}{(K_m + S_s)(S_b - S_s)} \] --- ## **Step 5: Calculate Effectiveness Factor (\( \eta \))** Effectiveness factor is the ratio of the observed rate to the rate if the surface concentration were equal to the bulk: \[ \eta = \frac{r_{obs}}{r_{max}} \] where \[ r_{max} = \frac{V_m' S_b}{K_m + S_b} \] and \[ r_{obs} = \frac{V_m' S_s}{K_m + S_s} \] --- ## **Step 6: Plug in Values for Each \( S_s \)** Let's calculate for each \( S_s \): ### **Case 1: \( S_s = 1 \) mg/cm³** #### a) **Observed Reaction Rate** \[ r_{obs} = \frac{V_m' S_s}{K_m + S_s} = \frac{.1 \times 1}{.2 + 1} = \frac{.1}{1.2} = .0833 \text{ mg/(cm²·s)} \] #### b) **Effectiveness Factor** \[ r_{max} = \frac{.1 \times 1}{.2 + 1} = .0833 \] \[ \eta = \frac{.0833}{.0833} = 1.00 \] #### c) **Mass Transfer Coefficient** \[ k_L = \frac{.1 \times 1}{(.2 + 1)(1 - 1)} \] But \(S_b - S_s = \), so **\( k_L \) is undefined/infinite** (no mass transfer limitation). --- ### **Case 2: \( S_s = .8 \) mg/cm³** #### a) **Observed Reaction Rate** \[ r_{obs} = \frac{.1 \times .8}{.2 + .8} = \frac{.08}{1.} = .08 \text{ mg/(cm²·s)} \] #### b) **Effectiveness Factor** \[ \eta = \frac{.08}{.0833} = .96 \] #### c) **Mass Transfer Coefficient** \[ k_L = \frac{.1 \times .8}{(.2 + .8)(1 - .8)} = \frac{.08}{1. \times .2} = \frac{.08}{.2} = .4 \text{ cm/s} \] --- ### **Case 3: \( S_s = .6 \) mg/cm³** #### a) \( r_{obs} = \frac{.1 \times .6}{.2 + .6} = \frac{.06}{.8} = .075 \) mg/(cm²·s) #### b) \( \eta = \frac{.075}{.0833} = .90 \) #### c) \( k_L = \frac{.1 \times .6}{(.2 + .6)(1 - .6)} = \frac{.06}{.8 \times .4} = \frac{.06}{.32} = .1875 \) cm/s --- ### **Case 4: \( S_s = .4 \) mg/cm³** #### a) \( r_{obs} = \frac{.1 \times .4}{.2 + .4} = \frac{.04}{.6} = .0667 \) mg/(cm²·s) #### b) \( \eta = \frac{.0667}{.0833} = .80 \) #### c) \( k_L = \frac{.1 \times .4}{(.2 + .4)(1 - .4)} = \frac{.04}{.6 \times .6} = \frac{.04}{.36} = .111 \) cm/s --- ### **Case 5: \( S_s = .2 \) mg/cm³** #### a) \( r_{obs} = \frac{.1 \times .2}{.2 + .2} = \frac{.02}{.4} = .05 \) mg/(cm²·s) #### b) \( \eta = \frac{.05}{.0833} = .60 \) #### c) \( k_L = \frac{.1 \times .2}{(.2 + .2)(1 - .2)} = \frac{.02}{.4 \times .8} = \frac{.02}{.32} = .0625 \) cm/s --- ### **Case 6: \( S_s = .1 \) mg/cm³** #### a) \( r_{obs} = \frac{.1 \times .1}{.2 + .1} = \frac{.01}{.3} = .0333 \) mg/(cm²·s) #### b) \( \eta = \frac{.0333}{.0833} = .40 \) #### c) \( k_L = \frac{.1 \times .1}{(.2 + .1)(1 - .1)} = \frac{.01}{.3 \times .9} = \frac{.01}{.27} = .037 \) cm/s --- ## **Step 7: Summary Table** | \( S_s \) (mg/cm³) | \( r_{obs} \) (mg/cm²·s) | \( \eta \) | \( k_L \) (cm/s) | |--------------------|--------------------------|------------|---------------------| | 1. | .0833 | 1.00 | ∞ (undefined) | | .8 | .080 | .96 | .40 | | .6 | .075 | .90 | .188 | | .4 | .0667 | .80 | .111 | | .2 | .050 | .60 | .0625 | | .1 | .0333 | .40 | .037 | --- ## **Key Points** - **Observed reaction rate** is calculated by Michaelis-Menten at the surface. - **Effectiveness factor** shows how much the observed rate is reduced by mass transfer limitations. - **Mass transfer coefficient** can be back-calculated from the steady-state equation. Let me know if you need more details or a specific explanation for any step!