An enzyme is immobilized on the surface of a non-porous spherical part...
Given Information:
- Enzyme is immobilized on the surface of a non-porous spherical particle with a diameter of 2 mm.
- Bulk substrate concentration is 10 mM.
- The enzyme follows first-order kinetics with a rate constant of 10 s-1.
- The external mass transfer coefficient is 1 cm.s-1.
- The rate of enzyme reaction at the surface is equal to the mass transfer rate at steady state.
Objective:
To determine the substrate concentration at the surface of the immobilized particle.
Assumptions:
- The system is at steady state.
- The immobilized enzyme has a uniform distribution on the surface of the particle.
- The reaction and mass transfer occur only at the surface of the particle.
Solution:
The rate of enzyme reaction at the surface is given by the first-order rate equation:
r = k * [S]_s
where r is the rate of reaction at the surface, k is the rate constant, and [S]_s is the substrate concentration at the surface.
The mass transfer rate is given by:
R = k_m * (C_b - [S]_s)
where R is the mass transfer rate, k_m is the external mass transfer coefficient, C_b is the bulk substrate concentration, and [S]_s is the substrate concentration at the surface.
At steady state, the rate of enzyme reaction at the surface is equal to the mass transfer rate:
k * [S]_s = k_m * (C_b - [S]_s)
Simplifying the equation, we get:
[S]_s = (k_m * C_b) / (k + k_m)
Substituting the given values into the equation, we have:
[S]_s = (1 cm.s-1 * 10 mM) / (10 s-1 + 1 cm.s-1)
[S]_s = 10 mM / (10 s-1 + 1 cm.s-1)
[S]_s = 10 mM / 11 s-1
[S]_s ≈ 0.909 mM
Therefore, the substrate concentration at the surface of the immobilized particle is approximately 0.909 mM.
Answer:
The substrate concentration at the surface of the immobilized particle is 0 mM.