Problem 168 - A Complete Resource Book in Chemistry for JEE Main

Question

The chemical kinetics of the reaction \(\mathrm{a} \mathrm{A}+\mathrm{bB} \rightarrow\) \(\mathrm{C}\) at \(298 \mathrm{~K}\) were followed. The initial rates were recorded rates were recorded under different initial conditions and are summarized as follows. $$ \begin{array}{lll} \hline \begin{array}{l} \text { Initial conc. } \\ {[\mathrm{A}]_{0}(\mathrm{~mol} / \mathrm{L})} \end{array} & \begin{array}{l} \text { Initial conc. } \\ {[\mathrm{B}]_{0}(\mathrm{~mol} / \mathrm{L})} \end{array} & \begin{array}{l} \text { Initial rate } \\ \text { (mol } / \mathrm{L} \mathrm{s}) \end{array} \\ \hline 0.1 & 0.1 & 2.4 \times 10^{-3} \\ 0.2 & 0.1 & 4.8 \times 10^{-3} \\ 0.4 & 0.1 & 9.7 \times 10^{-3} \\ 0.1 & 0.2 & 9.6 \times 10^{-3} \\ 0.1 & 0.4 & 3.8 \times 10^{-2} \\ \hline \end{array} $$ Which of the following statements is incorrect? (a) The rate constant \(\mathrm{k}\) is governed by the activation energy of the reaction (b) Reaction rate \(=\mathrm{k}[\mathrm{A}][\mathrm{B}]^{2}\) (c) In the chemical equation of \(\mathrm{aA}+\mathrm{bB} \rightarrow \mathrm{C}\), a is 0 and \(\mathrm{b}\) is 3 . (d) The overall order of reaction is third order.

Step-by-Step Solution

Answer

Statement (c) is incorrect.

Step 1: Analyzing the Rate Data for Dependence on [A]

First, examine how the rate changes with different initial concentrations of [A], while keeping [B] constant. By looking at trials 1-3 with constant [B] = 0.1 M: - When [A] doubles from 0.1 M to 0.2 M, the rate doubles from \(2.4 \times 10^{-3}\) to \(4.8 \times 10^{-3}\).- When [A] doubles again from 0.2 M to 0.4 M, the rate doubles from \(4.8 \times 10^{-3}\) to \(9.7 \times 10^{-3}\).This indicates a first-order dependence on [A].

Step 2: Analyzing the Rate Data for Dependence on [B]

Next, we observe the rate change with different initial concentrations of [B], while [A] = 0.1 M remains constant (trials 1,4, and 5): - When [B] doubles from 0.1 M to 0.2 M, the rate quadruples from \(2.4 \times 10^{-3}\) to \(9.6 \times 10^{-3}\).- When [B] doubles again from 0.2 M to 0.4 M, the rate quadruples from \(9.6 \times 10^{-3}\) to \(3.8 \times 10^{-2}\). This indicates a second-order dependence on [B].

Step 3: Determining the Rate Law and Overall Reaction Order

Based on the analysis from steps 1 and 2, the rate law can be written as: \[ \text{Rate} = k[A]^1[B]^2 \]The reaction has first-order dependence on [A] and second-order dependence on [B], leading to an overall reaction order of 3 (1 + 2 = 3).

Step 4: Evaluating the Statements

(a) The rate constant \(k\) is indeed governed by the activation energy, so this statement is correct.(b) The proposed rate law matches the observed data (first-order in [A] and second-order in [B]), thus this statement is correct.(c) In the reaction \(\text{aA} + \text{bB} \rightarrow \text{C}\), based on the rate law, the order in terms of experiment does not correspond to stoichiometric coefficients directly, so this requires careful analyst check.(d) The overall order calculated is third order, so this is correct.

Step 5: Identifying the Incorrect Statement

The incorrect statement is (c) because in the rate law compared with initial concentrations and rates, \(a\) and \(b\) do not specify the reaction orders directly, especially \(a = 0\) and \(b = 3\). The correct orders derived from the kinetics are first order in [A] and second order in [B], yielding the correct rate law for this data consistently instead which differs from statement (c).

Key Concepts

Rate Law

Understanding the rate law of a chemical reaction is essential for predicting how the concentration of reactants affects the reaction rate. The rate law is an equation that relates the rate of a reaction to the concentration of its reactants. In simpler terms, it helps us comprehend how quickly or slowly a reaction proceeds under varying conditions. The general form of a rate law is:
\[\text{Rate} = k[A]^x[B]^y\]where:

• \(k\) is the rate constant,
• \([A]\) and \([B]\) are concentrations of reactants A and B,
• \(x\) and \(y\) are the orders of the reaction with respect to A and B,

Each reaction has its unique rate law, which is determined through experiments. By knowing the rate law, chemists can predict how changes in concentration affect the reaction rate.
Identifying the correct rate law involves analyzing experimental data to discern patterns in how the reaction rate changes with variation in reactant concentrations. In the original exercise, the rate was found to be first order with respect to [A] and second order with respect to [B]. Consequently, the rate law was concluded to be:
\[\text{Rate} = k[A]^1[B]^2\]Mastering rate laws enables students to not just predict outcomes but to understand the underlying kinetics of chemical reactions.

Reaction Order

Reaction order is a crucial concept in the realm of chemical kinetics as it provides insight into the mechanism of a chemical reaction. A reaction's order tells us the power to which the concentration of a reactant is raised in the rate law. It is the sum of the exponents of the concentration terms in the rate law equation.
The reaction order can be:

• Zero Order: The rate of reaction is independent of the concentration of reactants.
• First Order: The rate changes linearly with a change in concentration of a reactant.
• Second Order: The rate changes quadratically with a change in concentration of a reactant.
• Overall Order: The sum of all individual orders of reactants in the reaction.

In the provided example, it was found that the reaction is first order with respect to [A] and second order with respect to [B]. Therefore, the overall order of the reaction is the sum of these individual orders: 1 + 2 = 3.

The overall reaction order is pivotal in understanding how the reaction rate will alter in response to changes in concentration, providing valuable insight into the dynamics of the reaction mechanism. By understanding these concepts, students can better predict reaction behavior and design experiments accordingly.

Activation Energy

Activation energy is the minimum amount of energy required for a chemical reaction to occur. It is a critical component that determines the rate at which a reaction takes place. Without sufficient energy to exceed this threshold, reactants will not convert into products, regardless of their concentration.
The activation energy can be visualized as a barrier that reactants need to overcome to transform into products. The higher the activation energy, the slower the reaction, as fewer molecules will have the necessary energy to initiate the reaction. Conversely, a lower activation energy results in a faster reaction because a larger number of molecules can surpass the energy barrier.
The rate constant \(k\) in the rate law is directly influenced by the activation energy. This relationship is mathematically described by the Arrhenius equation:
\[k = A e^{-\frac{E_a}{RT}}\]where:

• \(E_a\) is the activation energy,
• \(R\) is the universal gas constant,
• \(T\) is the temperature in Kelvin,
• \(A\) is the frequency factor, representing the likelihood of a reaction occurring.

Understanding activation energy provides insight into how conditions such as temperature can impact reaction rates. By manipulating variables like temperature, chemists can control the speed of a reaction to suit desired outcomes. This concept underscores the intricate balance between reactant energy and reaction feasibility, highlighting the dynamic nature of chemical kinetics.