Reactant A decomposes in an isothermal batch reactor \(\left(C_{\mathrm{A} 0}=100\right)\) to produce wanted \(R\) and unwanted \(S\), and the following progressive concentration readings are recorded: $$\begin{array}{l|rrrrrrrrrrr} C_{\mathrm{A}} & (100) & 90 & 80 & 70 & 60 & 50 & 40 & 30 & 20 & 10 & (0) \\ C_{\mathrm{R}} & (0) & 1 & 4 & 9 & 16 & 25 & 35 & 45 & 55 & 64 & (71) \end{array}$$ Additional runs show that adding \(R\) or \(S\) does not affect the distribution of products formed and that only \(\mathrm{A}\) does. Also, it is noted that the total number of moles of \(\mathrm{A}, \mathrm{R},\) and \(\mathrm{S}\) is constant. (a) Find the \(\varphi\) versus \(C_{\mathrm{A}}\) curve for this reaction. With a feed of \(C_{A 0}=100\) and \(C_{A f}=10,\) find \(C_{R}\) (b) from a mixed flow reactor, (c) from a plug flow reactor, (d) and (e): Repeat parts (b) and (c) with the modification that \(C_{\mathrm{A} 0}=70\)

An isothermal batch reactor is a type of reactor used in chemical processes where the temperature is kept constant (isothermal) throughout the reaction. These reactors are typically vessels where chemicals are placed to react over time without continuous inflow or outflow of materials.

In our scenario, reactant A decomposes in an isothermal batch reactor to form products R and S. The concentration of reactant A is observed to decrease, while the concentrations of the products R and S increase proportionally because the reaction takes place at a constant volume, and the total number of moles remains consistent.

To analyze the kinetics of the reaction and the distribution of products formed, we require understanding of reaction rates and the relations between reactant and product concentrations over time. The data provided for different concentrations of A and R will help in plotting the \( \varphi \) versus \( C_{\mathrm{A}} \) curve to better understand this relationship and predict the behavior in different types of reactors.

Reactant decomposition is the process where a single reactant breaks down into multiple products. In the given scenario, reactant A decomposes to produce desirable product R and undesirable product S.

It is important to note from the additional runs of the experiment that neither product R nor S affects the decomposition of A, indicating that the decomposition is independent of the concentration of the products once they are formed.

Keeping track of how A decomposes into R and S is key to optimizing the process to maximize yield of R and minimize the formation of S. The data presented in the problem allows for the calculation of composition and yield at any given time, which is crucial for designing the optimal reactor system for this reaction.

Mole balance is a fundamental principle used in reaction engineering to account for the moles of reactants and products in a chemical reaction. It is based on the law of conservation of mass.

For the given problem, the total number of moles is constant, which suggests that there is no net production or loss of moles. When reactant A decreases, the combined moles of products R and S increase equivalently. The mole balance equation is given by: \( C_{\mathrm{A}0} - C_{A} = C_{R} + C_{S} \), where \( C_{\mathrm{A}0} \) is the initial concentration of reactant A, \( C_{A} \) is its concentration at any time, and \( C_{R} \) and \( C_{S} \) are the concentrations of products R and S, respectively.

The mole balance helps us understand the stoichiometry of the reaction and calculate the composition of the reaction mix at any point, which is critical in reactor design and analysis.

A Mixed Flow Reactor, or Continuous Stirred Tank Reactor (CSTR), is a common type of industrial reactor design. It is characterized by the contents of the reactor being well mixed, resulting in uniform composition throughout the vessel. Additionally, the reactor operates in a steady-state with continuous flow of reactants in and products out.

Given our reactant decomposition, the key aspect of a CSTR is that the exit concentration of reactants and products is the same as the concentration within the reactor. This is different from batch reactions where concentrations change with time. The mole balance equation used for a CSTR under steady-state conditions relates the inflow, outflow, reaction rate, and volume.

To solve for the exit concentration of R from a CSTR, one would use the previously established \( \varphi \) vs \( C_{\mathrm{A}} \) relationship to find the steady-state concentration of R based on the inflow concentration of A.

A Plug Flow Reactor (PFR) is another reactor design used extensively in chemical engineering. In a PFR, reactants move through the reactor as a 'plug', with little back mixing, and composition changes occur along the length of the reactor, unlike the uniform composition in a CSTR.

The design of a PFR allows for the variation of reactant concentration along the reactor length, with a unique reaction rate at each position. Calculating the exit concentration of a product in a PFR generally involves setting up a material balance over a differential volume and integrating it along the reactor's length.

For our specific chemical reaction, since the concentration of reactants changes along the reactor, we utilize the integral of the \( \varphi \) vs \( C_{\mathrm{A}} \) curve within the limits of initial and final \( C_{\mathrm{A}} \) values to calculate the concentration of R at the reactor exit. This method considers the progressive conversion of A to R and S as it flows through the PFR.