The enzyme, urease, is immobilized in Ca-alginate beads 2 mm in diameter. When the urea concent

step 1 So here first under the given condition the surface concentration of urea is 200 mg per liter th

The enzyme, urease, is immobilized in Ca-alginate beads 2...

The enzyme, urease, is immobilized in Ca-alginate beads $2 \mathrm{~mm}$ in diameter. When the urea concentration in the bulk liquid is $0.5 \mathrm{~m} M$ the rate of urea hydrolysis is $v=10 \mathrm{mmoles-1- \textrm {h } .}$ Diffusivity of urea in $\mathrm{Ca}$-alginate beads is $D_{e}=1.5 \times 10^{-5} \mathrm{~cm}^{2} / \mathrm{sec}$, and the Michaelis constant for the enzyme is $K_{m}^{\prime}=0.2 \mathrm{~m} M$. By neglecting the liquid film resistance on the beads (i.e., $\left.\left[\mathrm{S}_{0}\right]=\left[\mathrm{S}_{\mathrm{s}}\right]\right)$ determine the following: a. Maximum rate of hydrolysis $V_{m}$, Thiele modulus $(\phi)$, and effectiveness factor $(\eta)$. b. What would be the $V_{w}, \phi$, and $\eta$ values for a particle size of $\mathrm{Dp}=4 \mathrm{~mm}$ ?

The enzyme, urease, is immobilized in Ca-alginate beads $2 \mathrm{~mm}$ in diameter. When the urea concentration in the bulk liquid is $0.5 \mathrm{~m} M$ the rate of urea hydrolysis is $v=10 \mathrm{mmoles-1- \textrm {h } .}$ Diffusivity of urea in $\mathrm{Ca}$-alginate beads is $D_{e}=1.5 \times 10^{-5} \mathrm{~cm}^{2} / \mathrm{sec}$, and the Michaelis constant for the enzyme is $K_{m}^{\prime}=0.2 \mathrm{~m} M$. By neglecting the liquid film resistance on the beads (i.e., $\left.\left[\mathrm{S}_{0}\right]=\left[\mathrm{S}_{\mathrm{s}}\right]\right)$ determine the following:
a. Maximum rate of hydrolysis $V_{m}$, Thiele modulus $(\phi)$, and effectiveness factor $(\eta)$.
b. What would be the $V_{w}, \phi$, and $\eta$ values for a particle size of $\mathrm{Dp}=4 \mathrm{~mm}$ ?

Key Concepts

Michaelis-Menten Kinetics

This concept describes the relationship between the rate of enzymatic reactions and substrate concentration. It introduces key parameters such as the maximum rate (Vmax) and the Michaelis constant (Km), which respectively represent the maximum catalytic activity of the enzyme and the substrate concentration at half of Vmax. Understanding these parameters is essential for analyzing and modeling enzyme-catalyzed reactions, especially when enzymes are immobilized, as it establishes the baseline kinetics before any diffusional limitations are taken into account.

Immobilized Enzyme Catalysis

Immobilized enzyme catalysis involves attaching or entrapping enzymes within or onto a solid support, such as Ca-alginate beads. This technique is widely used to improve enzyme stability, allow for reuse, and simplify separation from the reaction mixture. However, immobilization often introduces diffusional limitations due to the porous nature of the support, which may result in a reaction rate within the particle that is less than the intrinsic kinetic rate observed in a well-mixed system.

Thiele Modulus

The Thiele modulus is a dimensionless parameter that compares the intrinsic reaction rate to the rate of substrate diffusion within a porous catalyst or immobilized enzyme particle. It quantifies the extent to which internal diffusion limitations affect the overall reaction rate. A higher Thiele modulus indicates that diffusion is slower relative to the reaction, possibly leading to significant concentration gradients within the particle and thus lower overall effectiveness.

Effectiveness Factor

The effectiveness factor is defined as the ratio of the actual reaction rate observed in an immobilized system to the theoretical reaction rate if the entire catalyst particle were exposed to the bulk substrate concentration. This factor accounts for internal mass transfer limitations, providing a measure of how effectively the immobilized enzyme is used. It is intrinsically linked to the Thiele modulus, as larger values of the Thiele modulus generally lead to lower effectiveness factors.

Influence of Particle Size on Diffusion

The particle size of an immobilized enzyme system critically influences diffusion distances within the particle. Larger particles generally exhibit longer diffusion paths for substrates and products, which can exacerbate internal mass transfer limitations and lower the effectiveness factor. Understanding this relationship is crucial in designing and scaling up immobilized enzyme systems to ensure that these diffusional effects do not compromise the overall catalytic efficiency.

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