Aqueous \(\mathrm{A}\left(C_{\mathrm{A} 0}=1 \text { mol/liter }\right)\) with physical properties close to water \(\left(\rho=1000 \mathrm{kg} / \mathrm{m}^{3}, \mathscr{D}=10^{-9} \mathrm{m}^{2} / \mathrm{s}\right)\) reacts by a first-order homogeneous reaction \(\left(\mathrm{A} \rightarrow \mathrm{R}, k=0.2 \mathrm{s}^{-1}\right)\) as it flows at \(100 \mathrm{mm} / \mathrm{s}\) through a tubular reactor \(\left(d_{t}=50 \mathrm{mm}, L=5 \mathrm{m}\right) .\) Find the conversion of \(\mathrm{A}\) in the fluid leaving this reactor.

Chemical Reaction Engineering (CRE) is the field that quantitatively analyzes chemical reactions to design and operate reactor systems effectively. CRE looks closely at finding the optimal conditions to enhance the yield of products while minimizing costs and ensuring safety. It intertwines with kinetics, the study of the rates of chemical processes, to understand how factors like temperature, pressure, and concentration impact the transformation of reactants into products.

A core aspect of CRE is the understanding of reaction mechanisms and rate laws, such as the first-order reaction in our exercise. Knowing that the rate of reaction is directly proportional to the reactant concentration in a first-order process helps engineers to predict how fast a reactant will be consumed and design reactors accordingly. Effective design and operation of reactors, such as the tubular reactor from our example, require a robust knowledge of these principles for optimization and scale-up from laboratory to industrial production.

The equation given:

• \( \mathrm{Rate} = k[A] \)

shows a direct connection between rate and concentration, indicating its first-order nature. This foundational equation also sets the stage for more complex calculations essential for designing and optimizing reactors.

Conversion of reactants refers to the extent to which reactants transform into products in a chemical process. It is a critical parameter that indicates the efficiency of a chemical reactor. For students and engineers, understanding conversion is pivotal for designing reactors and evaluating their performance.

Mathematically, conversion is expressed as:

• \( X = \frac{[A]_0 - [A]_e}{[A]_0} \)

where

• \( X \)

is the conversion factor,

• \( [A]_0 \)

is the initial concentration of reactants, and

• \( [A]_e \)

is the concentration of reactants at the exit of the reactor. High conversion rates usually mean more efficient reactor systems, but this must be balanced against the cost and feasibility of achieving such conversions.

In the context of our exercise, determining conversion is essential for measuring how successful the tubular reactor is at transforming Reactant A into Product R. By calculating the conversion, engineers can make informed decisions on process improvements and optimizations.

Residence time, represented by the Greek letter tau (\( \tau \)), is a key concept in CRE, denoting the average time a reactant particle spends in a reactor. It is a crucial factor that influences conversion, as it directly affects the amount of time reactants are exposed to reaction conditions. The longer the residence time, the more opportunity for reactants to convert to products, up to a certain limit depending on the reaction kinetics.

The formula used to calculate residence time is:

• \( \tau = \frac{L}{v} \)

where

• \( L \)

is the length of the reactor and

• \( v \)

is the linear velocity of fluid flow through the reactor. For a tubular reactor like the one in our exercise, residence time aids in predicting the final conversion of reactants at the reactor outlet.

By combining the concepts of residence time and reaction kinetics, as shown in the solution provided:

• \( \ln[A]_e = -k\tau + \ln[A_0] \)

students can understand the interplay between the physical dimensions and flow characteristics of the reactor and the chemical reaction that takes place within its confines. This knowledge culminates in the ability to design reactors that optimize production yields and efficiency.