A specific enzyme acts as catalyst in the fermentation of reactant A. At a given enzyme concentration in the aqueous feed stream ( 25 liter/min) find the volume of plug flow reactor needed for \(95 \%\) conversion of reactant \(\mathrm{A}\left(C_{\mathrm{A} 0}=2 \mathrm{mol} / \text { liter }\right) .\) The kinetics of the fermentation at this enzyme concentration is given by $$\mathrm{A} \stackrel{\text { enzme }}{\longrightarrow} \mathrm{R}, \quad- r_{\mathrm{A}}=\frac{0.1 \mathrm{C}_{\mathrm{A}}}{1+0.5 \mathrm{C}_{\mathrm{A}}} \frac{\mathrm{mol}}{\text { liter } \cdot \mathrm{min}}$$

Enzyme catalysis plays a crucial role in various biochemical processes, including the fermentation reaction described in the exercise. Enzymes are biological catalysts that speed up chemical reactions without being consumed in the process. They provide an alternative pathway with lower activation energy, allowing reactions to proceed more rapidly at ambient temperatures.

Enzyme-catalyzed reactions follow a specific mechanism where the enzyme binds to the substrate (reactant A in the exercise) to form an enzyme-substrate complex. This complex then undergoes a chemical reaction to form the product (R), releasing the enzyme. The rate of these reactions often follows Michaelis-Menten kinetics, which describes the reaction rate as a function of substrate concentration and enzyme activity. The given rate law in the exercise,
\[\begin{equation}-r_{\text{A}} = \frac{0.1C_{\text{A}}}{1 + 0.5C_{\text{A}}}\end{equation}\],
represents a simplified version of such kinetics.

In chemical reaction engineering, a plug flow reactor (PFR) is a model for a tubular reactor where the reaction mixture moves as a 'plug' through a cylindrical pipe. The key assumption is that the fluid particles passing through the reactor do not mix with each other longitudinally, only radially.

This means that every element of the flow that enters the reactor has the same residence time, and there is no back-mixing of the product with reactants. Because of this, concentration and reaction rate can change along the length of the reactor but remain constant across any cross-section perpendicular to the direction of flow. The PFR is ideal for processes that have rapid kinetics and for achieving high conversions.

The exercise's request to calculate the volume of a PFR is a common application of reaction design principles, specifically in sizing reactors to achieve a desired conversion.

Reaction kinetics involves studying the rate of a chemical reaction and the factors that affect it. It's essential in designing reactors and processes in the chemical industry. The fundamental equation for reaction rate is usually expressed in terms of reactant concentration and rate constants.

The given exercise demonstrates reaction kinetics by incorporating the rate law for the enzymatic reaction:
\[\begin{equation}-r_{\text{A}} = \frac{0.1C_{\text{A}}}{1 + .5C_{\text{A}}}\end{equation}\].
The negative sign indicates that concentration of reactant A decreases over time. The rate law equation underscores how the reaction rate depends on the concentration of A and also highlights the saturation effect due to enzyme kinetics.

Understanding and applying these kinetics principles allows engineers to predict the behavior and size reactors appropriately, like calculating the necessary volume of a PFR for a given conversion.

Fermentation is a biochemical process often involving microorganisms or enzymes to convert organic compounds into simpler products. In the context of the exercise, reactant A undergoes fermentation to produce product R, catalyzed by an enzyme.

It's a crucial process for manufacturing a wide range of products, from food and beverages to pharmaceuticals and biofuels. Fermentation can be aerobic or anaerobic, depending on whether the process requires oxygen. The exercise hints at an anaerobic process, typical for many fermentation-based production processes due to the nature of the microorganisms involved.

Understanding the kinetics and reactor design for fermentation processes is important for optimizing production and scaling up from lab to industrial scale. The calculation for the plug flow reactor volume needed for a given conversion of reactant A exemplifies the application of reaction kinetics to process design in the fermentation industry.