Question

Using Equation (3.6), derive a material balance for a reactant in a closed system.

Short answer

Question: Derive the material balance for a reactant in a closed system using Equation (3.6). Answer: The material balance for a reactant in a closed system using Equation (3.6) is derived as follows: \begin{align*} \frac{d[A]}{dt} + r\nu_A = k[A] \end{align*} Where, d[A]/dt is the rate of change of the reactant's concentration with respect to time, r is the chemical reaction rate, ν_A is the stoichiometric coefficient of the reactant, and k is the reaction rate constant.

Step-by-step solution

Understand Equation (3.6)

Equation (3.6) is the general equation for calculating the rate of change of a reactant in a closed system. It's defined as follows: \begin{align*} \text{Rate of accumulation of reactant} = \text{Rate of input} - \text{Rate of output} + \text{Rate of generation} - \text{Rate of consumption} \end{align*} We'll be using this equation to derive the material balance for the reactant.

Identify rate of input and rate of output in a closed system

In a closed system, there is no input or output of reactant. The reactant is either consumed or generated within the system. So, in this case, the rate of input and rate of output are both zero. Now, the equation becomes: \begin{align*} \text{Rate of accumulation of reactant} = \text{Rate of generation} - \text{Rate of consumption} \end{align*}

Analyze the reactant's rate of generation and rate of consumption

Consider a reactant A that participates in a chemical reaction. The rate of generation of A is given by the product of the reaction rate constant (k) and the concentration of A ([A]). The rate of consumption of A is given by the product of the chemical reaction rate (-r) and the stoichiometric coefficient of A (ν_A). Now, the equation becomes: \begin{align*} \text{Rate of accumulation of A} = k[A] - (-r\nu_A) \end{align*}

Derive the material balance for the reactant

We're now ready to derive the material balance for reactant A. Let's substitute the rate of accumulation of A with the rate of change of its concentration with respect to time: \begin{align*} \frac{d[A]}{dt} = k[A] - (-r\nu_A) \end{align*} Now, rearrange the equation to get the material balance for reactant A: \begin{align*} \frac{d[A]}{dt} + r\nu_A = k[A] \end{align*} This is the material balance for a reactant in a closed system using Equation (3.6).

Key concepts

Closed System Dynamics

In chemical engineering, a closed system is essential when discussing chemical reactions and material balances. A closed system is defined as one where no matter enters or leaves; only energy can be transferred across the system's boundaries. This implies that all reactants involved in a reaction are confined within the system. For chemical reactions occurring within a closed system, it is crucial to understand that no additional reactants are introduced, nor are any products removed during the reaction.
In practical applications, this concept allows engineers and chemists to focus solely on the changes happening inside the system.
This simplification paves the way for a clear material balance analysis, making it easier to track reactant conversion to products and study the conservation of mass.
By understanding closed system dynamics, the complexity of monitoring inputs and outputs is eliminated, narrowing the focus to the reaction's intrinsic properties like generation and consumption rates.

Rate of Reaction

The rate of reaction is a fundamental concept in chemical kinetics that describes how quickly reactants are converted to products in a chemical reaction. In a closed system, this concept is particularly valuable because it dictates how the concentration of reactants and products change over time.
The rate of reaction is influenced by various factors, including temperature, pressure, the concentration of reactants, and the presence of catalysts.
Mathematically, it is expressed using the reaction rate constant and the concentration of reactants, showing that as the concentration of reactants decreases, the rate of reaction may also decrease.
Understanding the rate of reaction allows chemists and engineers to predict how a reaction will proceed, which is critical for designing reactors and optimizing reaction conditions to achieve desired product yields.

Chemical Reaction Stoichiometry

Stoichiometry is a key aspect of chemical reactions that involves the quantitative relationship between reactants and products. In a closed system, stoichiometry ensures that the correct proportions of reactants combine to produce the expected products.
This relationship is governed by the balanced chemical equation, which provides the stoichiometric coefficients used to calculate the rate of generation and consumption of each reactant and product involved.
Stoichiometric coefficients are not just numbers; they represent the precise amount of each substance needed for the reaction to proceed as designed.
By applying stoichiometry, chemists can ensure reactants are used efficiently, minimizing waste and maximizing product output. Understanding stoichiometry is essential when setting up reactions, as even a small deviation in the amounts of reactants can significantly impact the reaction's outcome.

Rate of Generation and Consumption

The rate of generation and consumption is a crucial element in understanding material balances in chemical reactions. For any reactant or product in a closed system, the system's material balance equation simplifies into rates of generation and consumption without accounting for inputs and outputs.
The rate of generation refers to how much of a substance is produced per unit of time, while the rate of consumption indicates how much is used up in the same period.
For example, in a reaction involving reactant A, if A is being produced and consumed at different rates, it is crucial to balance these rates to predict how the concentration of A will change with time.
Evaluating these rates, typically described with reaction rate constants and stoichiometric coefficients, allows scientists to derive equations that reflect the system's dynamics accurately.
By understanding the rate of generation and consumption, chemists can effectively manipulate reaction conditions to optimize the production of desired compounds.