Problem 76 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

For irreversible elementary reactions in parallel: \(\mathrm{A} \stackrel{K_{1}}{\longrightarrow} \mathrm{R}\) and \(\mathrm{A} \stackrel{K_{2}}{\longrightarrow} \mathrm{S}\), the rate of disappearance of reactant ' \(\mathrm{A}\) ' is (a) \(\left(k_{1}-k_{2}\right) C_{\mathrm{A}}\) (b) \(\left(k_{1}+k_{2}\right) C_{\mathrm{A}}\) (c) \(1 / 2\left(k_{1}+k_{2}\right) C_{\mathrm{A}}\) (d) \(k_{1} C_{\mathrm{A}}\)

Step-by-Step Solution

Answer

The rate of disappearance of reactant A is (b) \((k_{1}+k_{2}) C_{\mathrm{A}}\).

Step 1: Understanding the Reaction

The given reactions are elementary and irreversible. This means that the rates of the reactions are solely dependent on the concentration of the reactant A and the rate constants of the individual reactions.

Step 2: Write the Rate Equations

For individual reactions, the rate can be defined as the product of the rate constant and the concentration of the reactant. Thus for each reaction: Rate of A to R is given by \(k_{1}C_{\text{A}}\) and Rate of A to S is \(k_{2}C_{\text{A}}\).

Step 3: Add the Rates for A's Disappearance

Since A is reacting to both R and S, the overall rate of disappearance of A is the sum of the rates of the individual reactions: \(k_{1}C_{\text{A}} + k_{2}C_{\text{A}}\).

Step 4: Simplify the Combined Rate

Combine the terms by factoring out \(C_{\text{A}}:\) \((k_{1} + k_{2})C_{\text{A}}\).

Key Concepts

Rate of Reaction

Understanding the rate of a chemical reaction is essential for grasping the dynamics of how substances transform. The rate of reaction refers to how fast the concentration of a reactant or product changes over time. It's a measure that can reveal a lot about the conditions and factors affecting a reaction.

For the given elementary reactions, the rate of disappearance of reactant 'A' is a key focus. Contrary to complex reactions, for elementary reactions, we can directly correlate the rate with the concentration of the reactants involved. In the example provided, the disappearance of 'A' occurs through two pathways, and by summing up the rates at which 'A' converts to products 'R' and 'S', we get the overall rate of disappearance of 'A'.

Thus, in mathematical terms, the rate of disappearance of 'A' is not simply the rate constant times the concentration, \(kC_{\text{A}}\), but the sum of such terms for all the competing reactions. So, with two parallel reactions, the total rate will be the sum of two rate expressions: \(k_{1}C_{\text{A}} + k_{2}C_{\text{A}}\).

Chemical Kinetics

Chemical kinetics is the study of the rates of chemical processes and the factors that influence them. It involves understanding how various variables such as temperature, pressure, concentration, and catalysts impact the speed at which reactions occur. By analyzing the kinetics of a reaction, scientists can optimize conditions for industrial processes, control reaction pathways, and predict product formation.

Reaction Rates and Concentration

The kinetics of the elementary reactions discussed here is heavily influenced by the concentration of reactant 'A'. As a fundamental principle, the rate law for an elementary reaction is proportional to the concentration of the reactants raised to the power of their stoichiometric coefficients. Since the reactions are first-order with respect to 'A', the rate laws are simply \(k_{1}C_{\text{A}}\) and \(k_{2}C_{\text{A}}\) respectively for the two pathways.

Rate Constants

The rate constant is a proportionality factor that is unique to each chemical reaction at a given temperature. Represented by the symbol \(k\), it provides a quantitative measure of the speed of a reaction. In other words, it tells you how rapidly a reaction will occur when the reactants are present at a certain concentration.

Looking at the given exercise, these constants, \(k_{1}\) and \(k_{2}\), represent the intrinsic reactivity of reactant 'A' towards forming products 'R' and 'S', respectively. Rate constants are crucial because they allow us to compare the relative speeds of different reactions and understand the likelihood of each pathway. It's interesting to note that these constants do not depend on the concentration of 'A' but are influenced by temperature and the presence of catalysts.

Reaction Mechanism

The reaction mechanism is like a detailed script of a play that describes the step-by-step sequence of elementary reactions that lead to the final product of a complex reaction. Elementary reactions are the individual steps in this script, each with its own specific rate law and rate constant.

In the case of the provided irreversible elementary reactions, the mechanism is straightforward: reactant 'A' can transform into product 'R' or 'S' without any intermediate steps. Unlike more complicated mechanisms that can involve intermediates and multiple steps, these reactions are direct and simple. However, understanding the mechanism is still critical because it reveals the path taken by the reactant to become the product, telling us not just the 'what' but the 'how' of the reaction process.