For an immobilized-enzyme system with Damköhler number NDa ≫ 1 (reaction much faster than mass transfer), the overall rate is controlled by mass transfer. Using symbols Csb (bulk substrate), Cs (surface substrate), k_s (mass transfer coefficient), and a (particle interfacial area), which expression correctly describes the observed rate r_p?

Introduction:
When the Damköhler number is very large (NDa ≫ 1), intrinsic enzyme kinetics are so rapid that the surface concentration of substrate adjusts to match the mass transfer supply. In this regime, the bottleneck is the hydrodynamic transport of substrate from the bulk liquid to the enzyme surface, so the measurable overall rate mirrors the flux across the liquid film rather than the enzyme's intrinsic kinetics.

Given Data / Assumptions:

• NDa ≫ 1 implies mass transfer control.
• External mass transfer described by r_p = k_s a (C_sb − C_s).
• Enzyme surface is at concentration C_s that is generally below C_sb under consumption.
• Isothermal, steady hydrodynamic conditions with well-defined k_s and interfacial area a.

Concept / Approach:
Under mass transfer control, overall uptake equals the film flux. The classic film theory gives flux J = k_s (C_sb − C_s). Multiplying by specific area a yields the volumetric rate r_p = k_s a (C_sb − C_s). Any expression that omits the driving force (C_sb − C_s) or replaces it with sums/products does not represent a diffusion-controlled flux.

Step-by-Step Solution:
Step 1: Recognize the control regime: transport from bulk to surface limits.Step 2: Write the film-flux relation: J = k_s (C_sb − C_s).Step 3: Convert to volumetric rate by multiplying by a: r_p = k_s a (C_sb − C_s).Step 4: Confirm dimensions: k_s [m/s], a [m^2/m^3], concentration difference [mol/m^3] ⇒ r_p [mol/(m^3·s)].

Verification / Alternative check:
Experimental signatures include rate increases with agitation (higher k_s) or reduced particle size (higher a), supporting the mass transfer form and independence from intrinsic V_max in this regime.

Why Other Options Are Wrong:

• ksaCsb or ksaCs: Miss the driving difference; predict non-zero flux at equilibrium (C_sb = C_s), which is unphysical.
• ksa(C_sb + C_s): Uses a sum, not a gradient; lacks physical basis.
• Squared driving force: Not part of first-order film theory and not supported empirically for dilute solutions.

Common Pitfalls:
Assuming intrinsic kinetic parameters (K_m, V_max) matter under NDa ≫ 1; in this limit, changes in agitation or particle size give the strongest rate response.

Final Answer:
rp = ksa(Csb-Cs)