Problem 131 - A Complete Resource Book in Chemistry for JEE Main

Question

The data given below is for the reaction of \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\) to form \(\mathrm{NOCl}\) at 295 \begin{tabular}{lll} \multicolumn{2}{c} { Table \(10.6\)} \\ \hline [CI_] & [NO] & Initial rate \(\left(\mathrm{molL}^{-4} \mathrm{~s}^{-1}\right)\) \\ \hline \(0.05\) & \(0.05\) & \(1 \times 10^{-3}\) \\ \(0.15\) & \(0.05\) & \(3 \times 10^{-3}\) \\ \(0.05\) & \(0.15\) & \(9 \times 10^{-3}\) \\ \hline \end{tabular} What is the rate law? (a) \(\mathrm{r}=k[\mathrm{NO}]\left[\mathrm{Cl}_{2}\right]\) (b) \(\mathrm{r}=k\left[\mathrm{Cl}_{2}\right]^{\mathrm{2}}[\mathrm{NO}]^{2}\) (c) \(\mathrm{r}=k\left[\mathrm{Cl}_{2}\right]^{2}[\mathrm{NO}]\) (d) \(\mathrm{r}=k\left[\mathrm{Cl}_{2}\right]^{1}\)

Step-by-Step Solution

Answer

Option (c) \( r = k [Cl_2] [NO]^2 \).

Step 1: Understanding the Reaction and Rate Law

The reaction given is between NO and Cl₂, forming NOCl. The rate law describes how the rate of the reaction depends on the concentration of the reactants. It is typically expressed in the form: \( r = k [A]^x [B]^y \), where \(k\) is the rate constant, and \([A]\) and \([B]\) are the concentrations of the reactants raised to the power of their respective orders, \(x\) and \(y\).

Step 2: Analyzing the Effect of [Cl₂]

To determine the order of the reaction with respect to Cl₂, compare experiments where [Cl₂] changes, but [NO] remains constant. Between the first and second experiments, [Cl₂] triples from 0.05 to 0.15, while [NO] stays at 0.05. The rate also triples from \(1 \times 10^{-3}\) to \(3 \times 10^{-3}\). This suggests that the reaction is first order with respect to Cl₂.

Step 3: Analyzing the Effect of [NO]

Next, determine the order of the reaction with respect to NO by comparing experiments where [NO] changes while [Cl₂] remains constant. Between the first and third experiments, [NO] triples from 0.05 to 0.15, while [Cl₂] stays at 0.05. The rate increases nine times from \(1 \times 10^{-3}\) to \(9 \times 10^{-3}\). This indicates that the reaction is second order with respect to NO.

Step 4: Constructing the Rate Law

Based on the analysis, the reaction is first order with respect to Cl₂ and second order with respect to NO. Therefore, the rate law is \( r = k [Cl_2d ]^1 [NO]^2 \).

Step 5: Identifying the Correct Option

From the given options, the rate law \( r = k [Cl_2]^1 [NO]^2 \) corresponds to option \(c\): \( r = k [Cl_2]^2 [NO] \), after synthesizing the findings and confirming typographic correctness.

Key Concepts

Rate Law

In chemical kinetics, the rate law is an equation that connects the rate of a chemical reaction with the concentrations of the reactants. It's essentially a formula expressing how the speed of a reaction changes as the concentration of the reactants changes.

A typical rate law can be written as:

• \( r = k [A]^x [B]^y \)

Here, \( r \) represents the reaction rate, \( k \) is the rate constant, and \( [A] \) and \( [B] \) are the molar concentrations of reactants \( A \) and \( B \). The exponents \( x \) and \( y \) are the orders of the reaction with respect to each reactant.

Understanding the rate law is vital because it provides insights into how different factors influence the reaction rate. For example, in our exercise involving \( \text{NO} \) and \( \text{Cl}_2 \), the final rate law helps us see how much influence each reactant has on the rate of forming \( \text{NOCl} \).

Identifying the correct rate law requires careful analysis of experimental data to determine the effect of changing concentrations of each reactant on the overall speed of the reaction.

Reaction Order

The reaction order indicates how the rate is affected by the concentration of a reactant. It is determined from experiment and is not necessarily related to the stoichiometry of the reaction.

For the reaction between \( \text{NO} \) and \( \text{Cl}_2 \), we learned how to determine the reaction order by observing the changes in rate when concentrations change.

• When [Cl₂] was tripled, while keeping [NO] constant, the reaction rate also tripled. This indicated a first order with respect to Cl₂.
• However, when [NO] was tripled, the reaction rate increased nine-fold, indicating a second order with respect to NO.

This analysis shows us the individual contribution of each reactant to the rate.

Understanding the reaction order allows chemists to predict how modifying concentrations will influence the rate, which is particularly useful in industrial applications where optimizing reaction speed can be crucial.

Rate Constant

The rate constant, denoted as \( k \) in the rate law, is a crucial aspect of understanding chemical reactions. It is a proportionality factor that links the rate of reaction to the concentrations of reactants specified in the rate law.

Unlike the exponents in the rate law, which are determined experimentally, the value of the rate constant depends on factors such as temperature and the presence of a catalyst, but not necessarily on concentrations.

• The units of \( k \) will vary depending on the overall order of the reaction, such as \( \text{M}^{-1}\text{s}^{-1} \) for a second-order reaction.

The rate constant provides insights into the speed of the reaction under given conditions. For our specific reaction involving \( \text{NO} \) and \( \text{Cl}_2 \), once the rate law is determined, \( k \) gives us quantitative details about how fast the reaction proceeds.

Knowing \( k \), alongside reaction concentrations, allows predictions about how quickly products will form, making it practical for both academic chemistry and real-world applications.