Question

For a reaction mixture with a density that is linearly dependent on the conversion, with \(\rho=\rho_{0}\left(1+\varepsilon \zeta_{A}\right)\), derive a relation for the concentration of species \(\mathrm{A}\left(\mathrm{c}_{A}\right)\) as a function of the relative degree of conversion \(\zeta_{A}\).

Short answer

Question: Derive a relation for the concentration of species A (\(c_{A}\)) as a function of the relative degree of conversion (\(\zeta_{A}\)) using the given density formula, \(\rho = \rho_{0}(1+\varepsilon \zeta_{A})\). Answer: The concentration of species A, \(c_{A}\), can be expressed as a function of the relative degree of conversion, \(\zeta_{A}\), using the derived equation: $$ c_{A} = \frac{\rho_{0} \zeta_{A} (1+\varepsilon \zeta_{A})}{M_{A}} $$

Step-by-step solution

Define the given formula and relation

We are given the linear relationship between the density of the reaction mixture and the degree of conversion. The density formula is: $$\rho=\rho_{0}(1+\varepsilon \zeta_{A})$$ Where: - \(\rho\) is the density of the mixture, - \(\rho_0\) is the initial density, - \(\varepsilon\) is a constant, and - \(\zeta_A\) is the relative degree of conversion for species A. We want to derive a relation between the concentration of species A, \(c_A\), and \(\zeta_A\). We'll start by introducing a formula that relates concentration and density.

Introduce the mass balance relation

A mass balance equation relating concentration and density is provided by the following formula: $$c_{A}=\frac{\rho_{A}}{M_{A}}$$ Where: - \(c_{A}\) is the concentration of species A, - \(\rho_A\) is the density of species A, and - \(M_{A}\) is the molar mass of species A. Notice that we can express the density of species A as a function of the general density of the reaction mixture \(\rho\) by multiplying by the degree of conversion: $$ \rho_A = \rho \zeta_A $$ So, our mass balance equation becomes: $$ c_{A} = \frac{\rho \zeta_{A}}{M_{A}} $$

Substitute the given density equation into the mass balance equation

Now we will substitute the given density equation for the reaction mixture into the mass balance equation: $$ c_{A} = \frac{\rho_{0}(1+\varepsilon \zeta_{A}) \cdot \zeta_{A}}{M_{A}} $$

Simplify the equation for c_A as a function of zeta_A

Our final step is to simplify the equation to get \(c_{A}\) expressed as a function of \(\zeta_{A}\), which is important to isolate the variables of interest and make a general equation that can be easily solved. $$ c_{A} = \frac{\rho_{0} \zeta_{A} (1+\varepsilon \zeta_{A})}{M_{A}} $$ This is the equation we were looking for, which represents the concentration of species A, \(c_{A}\), as a function of the relative degree of conversion, \(\zeta_{A}\).

Key concepts

Chemical Reaction Engineering

Chemical Reaction Engineering (CRE) is a discipline of chemical engineering that focuses on the design and operation of chemical reactors to optimally change raw materials into desired products. It is essential for transforming the laboratory synthesis of chemical compounds into a viable industrial process.

Key aspects of CRE include the determination of reaction kinetics, which give the rates at which reactions occur, and reactor design, which involves selecting the right type of reactor and the conditions, such as temperature and pressure, to maximize product yield and minimize costs. A deep understanding of both the physical and chemical phenomena involved in reactions is crucial for CRE. This includes knowledge in areas like mass and energy balances, thermodynamics, and transport processes.

Concentration of Reactants

The concentration of reactants is a fundamental concept in chemistry and plays a pivotal role in chemical reaction engineering. It refers to the amount of a substance present in a given volume of solution. In the context of the given problem, it's expressed as \(c_A\), the concentration of species A.

The concentration is a critical factor influencing reaction rates; when reactant concentrations are higher, reactions may proceed more quickly. In practical applications, understanding and controlling the concentration of reactants can be key to optimizing the performance of a chemical process. For this reason, chemists and chemical engineers often express concentration in terms of molarity or molality, which allows for the precise quantification of reactant amounts in reaction mixtures.

Molar Mass Balance

The molar mass balance is a way of expressing the conservation of mass in terms of moles, which is particularly useful in chemical reactions. It pertains to the principle that the total amount of each element must be conserved in a chemical process. This balance is essential for converting mass flow rates to molar flow rates and vice versa, which is crucial in designing and analyzing chemical reactors.

In the given exercise, the molar mass balance plays a central role in deriving the relationship between the concentration of species A and its degree of conversion. The balance incorporates the density of the species and its molar mass, helping to relate these physical properties to the concentration, which is a key step in understanding how reactant concentration changes as the reaction proceeds.

Degree of Conversion

The degree of conversion, often represented by the Greek letter \(\zeta\), is a measure of the progress of a chemical reaction. It is a dimensionless quantity that ranges from 0 (no reaction) to 1 (complete reaction). Essentially, it's the fraction of reactant that has been converted into product(s). The degree of conversion is vital for determining reaction yields and for designing and controlling reactors.

In practical applications, engineers often need to know not just whether a reaction can occur, but to what extent it does occur. Calculating the degree of conversion can help in optimizing the reaction conditions to achieve the most efficient production process possible. This concept allows engineers to adjust the process conditions such as temperature or catalyst concentration to influence the conversion rate, and thus, the overall effectiveness of the reaction.