Problem 165 - A Complete Resource Book in Chemistry for JEE Main

Question

The hypothetical reaction, \(\mathrm{A}_{2}+\mathrm{B}_{2} \rightarrow 2 \mathrm{AB}\) follows the following mechanism: \(\mathrm{A}_{2}=\mathrm{A}+\mathrm{A}\) (fast) \(\mathrm{A}+\mathrm{B}_{2} \longrightarrow \mathrm{AB}+\mathrm{B} \quad\) (slow) \(\mathrm{A}+\mathrm{B} \longrightarrow \mathrm{AB} \quad\) (fast) The order of the overall reaction is: (a) \(\underline{1}\) (b) 2 (c) \(3 / 2\) (d) 0

Step-by-Step Solution

Answer

The order of the overall reaction is 2 (option b).

Step 1: Identify Slow Step

In a reaction mechanism, the slowest step determines the rate of the overall reaction. Here, the slow step is \( \mathrm{A} + \mathrm{B}_{2} \rightarrow \mathrm{AB} + \mathrm{B} \). This step dictates the reaction rate.

Step 2: Determine Rate Law from Slow Step

For the slow step, the rate law is given by the reactants involved: \( \text{Rate} = k[\mathrm{A}][\mathrm{B}_{2}] \). This is because the rate of a reaction is proportional to the concentration of the reactants in the slow step.

Step 3: Relate Rate Law to Overall Reaction

The rate law for the slow step can be used to determine the order of the reaction. The reaction order is defined by the sum of the powers of the concentration terms in the rate law. Here, it is \(1 + 1 = 2\).

Step 4: Determine Answer from Reaction Order

The overall reaction order derived from the rate law is 2, corresponding to option (b). This reflects the power sum in the rate law from the slow step's reactants.

Key Concepts

Reaction Rate

The reaction rate in a chemical process defines how quickly or slowly a reaction takes place. It is essentially the change in concentration of reactants or products per unit time. Understanding this concept is crucial for predicting how a reaction will behave under different conditions.

Several factors can influence the reaction rate, including temperature, concentration of reactants, surface area, and the presence of a catalyst. A higher concentration of reactants will generally increase the rate of reaction because there are more particles available to collide with each other.

• Temperature: Increasing temperature typically speeds up reactions because particles move faster and collide more frequently.
• Concentration: More particles in a given volume increases the likelihood of collisions, raising the rate.
• Surface Area: More surface area allows more particles to react at once, quicker reactions.
• Catalyst: These substances increase reaction rate without being consumed in the process.

The reaction rate ultimately depends on the slowest step in the mechanism (also known as the rate-determining step), as it acts like a bottleneck that controls the pace of the overall reaction.

Rate Law

The rate law is an essential concept that expresses the relationship between the reaction rate and the concentrations of reactants. It is a mathematical equation that lets us predict how changes in concentration affect the reaction rate.

The general form of a rate law for a reaction \( A + B \rightarrow Products \) is:
\[ \text{Rate} = k[A]^m[B]^n \]
Here, \( k \) is the rate constant, and \( m \) and \( n \) represent the orders of the reaction with respect to reactants \( A \) and \( B \). The exponents indicate the dependency of the rate on the concentration of each reactant.

• \( m \) and \( n \) are typically determined experimentally.
• \( k \), the rate constant, changes with temperature and must be determined under specific conditions.

The rate law derived from the rate-determining step in the reaction mechanism provides insight into which reactants influence the rate the most. Thus, knowing the rate law helps chemists control and optimize reaction conditions.

Reaction Order

In chemical kinetics, the reaction order tells us the power to which the concentration of a reactant is raised in the rate law. It gives insight into how reactant concentrations affect the rate of reaction. The overall reaction order is the sum of the individual orders for each reactant involved in the rate law.

For example, if the rate law of a reaction is \( \text{Rate} = k[A]^2[B]^1 \), then:

• The order with respect to \( A \) is 2.
• The order with respect to \( B \) is 1.
• The overall reaction order is \( 2 + 1 = 3 \).

Reaction order is determined experimentally and can be an integer, zero, or even a fraction. Knowing the reaction order, scientists can predict how a change in concentration will quantitatively affect the rate of the reaction. In the case of the hypothetical reaction with a slow step of \( \mathrm{A} + \mathrm{B}_2 \rightarrow \mathrm{AB} + \mathrm{B} \), the reaction order adds up to 2 according to the rate law derived.