Problem 142 - A Complete Resource Book in Chemistry for JEE Main

Question

Reaction \(\mathrm{A}_{2}+\mathrm{B}_{2} \rightarrow 2 \mathrm{AB}\) is completed according to the following mechanism. \(\mathrm{A}_{2} \rightleftharpoons 2 \mathrm{~A}\) \(\mathrm{A}+\mathrm{B}_{2} \rightarrow \mathrm{AB}+\mathrm{B} \quad\) (slow step) \(\mathrm{A}+\mathrm{B} \rightarrow \mathrm{AB}\) The order of reaction is (a) 1 (b) \(3 / 2\) (c) \(1 / 2\) (d) 2

Step-by-Step Solution

Answer

The reaction order is 2.

Step 1: Identify the Rate-Determining Step

In the given reaction mechanism, identify the slow step as it is the rate-determining step. The slow step provided is \( \mathrm{A} + \mathrm{B}_2 \rightarrow \mathrm{AB} + \mathrm{B} \). This is the step that controls the overall reaction rate.

Step 2: Determine the Reaction Order for Rate-Determining Step

For the reaction \( \mathrm{A} + \mathrm{B}_2 \rightarrow \mathrm{AB} + \mathrm{B} \), the rate is dependent on the concentration of \( \mathrm{A} \) and \( \mathrm{B}_2 \). Hence, the rate law for this slow step can be written as: \( \text{Rate} = k[\mathrm{A}][\mathrm{B}_2] \). This step indicates that the reaction is first-order with respect to \( \mathrm{A} \) and first-order with respect to \( \mathrm{B}_2 \).

Step 3: Add the Orders to Find the Total Reaction Order

The total reaction order is the sum of the powers of the concentrations in the rate law equation. Since the rate law is \( \text{Rate} = k[\mathrm{A}][\mathrm{B}_2] \), the order with respect to \( \mathrm{A} \) is 1, and with respect to \( \mathrm{B}_2 \) is 1. Thus, the overall reaction order is \( 1 + 1 = 2 \).

Key Concepts

Rate-Determining Step

The rate-determining step is an important concept in reaction mechanisms. It is the slowest step in the series of steps that make up the reaction pathway. Think of it like the bottleneck in a production line; no matter how fast all other steps are, the slowest worker (or step) determines the overall speed of the process. In chemical terms, this step largely governs how quickly the reaction occurs. In our given mechanism, the slow step is \( \mathrm{A} + \mathrm{B}_2 \rightarrow \mathrm{AB} + \mathrm{B} \), and it limits the speed of the entire reaction. By focusing on this step, chemists can derive crucial information about the kinetics of the reaction.

Reaction Order

Reaction order tells us how the concentration of reactants affects the speed of the reaction. It helps us determine which reactants have more influence on the rate. In this exercise, we found the reaction order from the rate-determining step. The rate law derived from this step is \( \text{Rate} = k[\mathrm{A}][\mathrm{B}_2] \), indicating that the reaction is first-order with respect to both \( \mathrm{A} \) and \( \mathrm{B}_2 \). This means doubling the concentration of \( \mathrm{A} \) will double the reaction rate, and likewise for \( \mathrm{B}_2 \).

• First-order reactions change rate proportional to the change in concentration of one reactant.
• Overall reaction order is the sum of all individual orders in the rate law.

Understanding reaction order is vital as it influences how we predict reactions in different conditions.

Rate Law

The rate law is an equation that links the rate of a reaction to the concentration of its reactants. It is a powerful tool to predict how changes in concentration affect reaction time. For the identified rate-determining step \( \mathrm{A} + \mathrm{B}_2 \rightarrow \mathrm{AB} + \mathrm{B} \), the rate law is given by \( \text{Rate} = k[\mathrm{A}][\mathrm{B}_2] \). Here, \( k \) is the rate constant, which is a factor that accounts for the effect of temperature and other conditions on the rate.
Key takeaways from the rate law are:

• The exponents in the rate law indicate the order of reaction with respect to each reactant.
• The rate constant \( k \) does not change with concentration but can vary with temperature.
• The rate law helps in calculating the reaction rate given specific concentrations of reactants.

Remember, the rate law is specific to a particular reaction and must be experimentally determined.