Problem 11 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

For the chemical reaction: \(\mathrm{A} \rightarrow\) products, the rate of disappearance of \(\mathrm{A}\) is a given by $$ r_{A}=-\frac{\mathrm{d} C_{\mathrm{A}}}{\mathrm{d} t}=\frac{K_{1} \cdot C_{\mathrm{A}}}{1+K_{2} \cdot C_{\mathrm{A}}} $$ At low concentration of A, \(C_{A}\), the reaction is of the first order with the rate constant (a) \(\frac{K_{1}}{K_{2}}\) (b) \(K_{1}\) (c) \(\frac{K_{2}}{K_{1}}\) (d) \(\frac{K_{1}}{K_{1}+K_{2}}\)

Step-by-Step Solution

Answer

At low concentrations of A, the reaction is of the first order with the rate constant (b) \(K_1\).

Step 1: Understanding the Rate Expression

The rate of disappearance of A, denoted as \(r_A\), is given by the rate expression \(r_A=-\frac{\mathrm{d} C_{\mathrm{A}}}{\mathrm{d} t}=\frac{K_{1} \cdot C_{\mathrm{A}}}{1+K_{2} \cdot C_{\mathrm{A}}}\). This rate expression is non-linear and suggests that the rate of the reaction depends on the concentration of A, \(C_A\), through a complex relationship involving the constants \(K_1\) and \(K_2\).

Step 2: Recognizing the Low Concentration Condition

At low concentration of A, the term \(K_2 \cdot C_A\) in the denominator of the rate expression becomes very small compared to 1. We can make an approximation that \(1 + K_2 \cdot C_A \approx 1\) when \(C_A\) is low.

Step 3: Simplifying the Rate Expression

With the above approximation, the rate expression simplifies to \(r_A = -\frac{\mathrm{d} C_{\mathrm{A}}}{\mathrm{d} t} \approx K_{1} \cdot C_{\mathrm{A}}\) at low concentrations of A. This indicates that the reaction follows first-order kinetics with respect to A, and the rate constant is simply \(K_1\).

Key Concepts

Rate of Reaction

The rate of reaction is a fundamental concept in chemical kinetics, representing how quickly a reactant is consumed or a product is formed over time. Mathematically, it's expressed as the change in concentration of a reactant or product per unit time. For a reaction where substance A transforms into products, this rate can be described by the equation \( r_{A} = -\frac{\mathrm{d} C_{\mathrm{A}}}{\mathrm{d} t} \), where \( C_{A} \) is the concentration of A and \( t \) is time.

It is crucial to distinguish between the rate at which A disappears, termed the rate of disappearance, from the rate of appearance of the products. The negative sign in the rate expression indicates that the concentration of A decreases over time. In chemical kinetics, understanding the rate of reaction is essential to predicting how long a reaction will take and how conditions such as temperature and concentration affect the reaction speed.

When analyzing reaction rates, it's important to consider initial rates, which are measured right after the reaction begins, before any significant changes in concentration occur. This helps avoid complications arising from the reverse reaction or the depletion of reactants affecting the rate.

In sum, the rate of reaction provides valuable insight into the dynamics of chemical processes and is a key factor in the design and control of chemical reactions in industrial and laboratory settings.

First-Order Reaction

A first-order reaction is characterized by a rate that is directly proportional to the concentration of one reactant. This means that as the concentration of the reactant doubles, so does the rate of the reaction. Mathematically, it is represented as \( r = k \cdot C \), where \( k \) is the reaction rate constant and \( C \) is the concentration of the reactant.

In the context of our exercise, at low concentrations of A, the complex rate expression simplifies, and the reaction behaves as if it were first-order with respect to A. This is because the term involving \( K_{2} \) becomes negligible, leading to a simplified rate expression \( r_{A} \approx K_{1} \cdot C_{A} \). Therefore, the rate solely depends on the concentration of A and the constant \( K_{1} \).

First-order kinetics are commonly encountered in processes such as radioactive decay and simple chemical reactions. They have the unique property that the half-life of the reactant, which is the time it takes for half of it to react, remains constant regardless of the starting concentration. This simplifies the analysis of reaction progress and the calculation of reactant lifetimes.

Reaction Rate Constant

The reaction rate constant, denoted by \( k \), is a measure of the intrinsic speed of a chemical reaction. It is a critical factor in the rate law equation that correlates the rate of reaction to the concentrations of reactants. The value of the rate constant depends on various factors such as temperature, presence of a catalyst, and the particular reaction mechanism.

In the provided exercise, we encounter two constants, \( K_{1} \) and \( K_{2} \). When the concentration of A is low, \( K_{1} \) emerges as the rate constant for our first-order reaction, as the rate can be approximated by \( r_{A} = K_{1} \cdot C_{A} \). This informs us about how rapidly the concentration of A decreases under these conditions.

It's important to understand that the rate constant is not affected by changes in concentration, but is sensitive to temperature changes—described by the Arrhenius equation—and the presence of catalysts that provide alternative reaction pathways with lower activation energies. Thus, the reaction rate constant is a pivotal part of the puzzle when studying the kinetics of a reaction and tailoring conditions for desired reaction speeds in practical applications.