Problem 86 - A Complete Resource Book in Chemistry for JEE Main

Question

From the following data for the reaction between \(\mathrm{A}\) and \(\mathrm{B}\) $$ \begin{array}{cccc} \hline[\mathrm{A}] & {[\mathrm{B}]} & \text { Initial rate } & \left(\mathrm{mol} \mathrm{L}^{-1} \mathbf{s}^{-1}\right) \\ \mathrm{molL}^{-1} & \mathrm{molL}^{-1} & \mathbf{3 0 0} \mathrm{K} & \mathbf{3 2 0} \mathrm{K} \\ \hline 2.5 \times 10^{-4} & 3.0 \times 10^{-5} & 5.0 \times 10^{-4} & 2.0 \times 10^{-3} \\ 5.0 \times 10^{-4} & 6.0 \times 10^{-5} & 4.0 \times 10^{-3} & \- \\ 1.0 \times 10^{-3} & 6.0 \times 10^{-5} & 1.6 \times 10^{-2} & \- \\ \hline \end{array} $$ Calculate the rate equation (a) \(\mathbf{r}=k[\mathrm{~B}]^{1}\) (b) \(\mathrm{r}=k[\mathrm{~A}]^{2}\) (c) \(\mathbf{r}=k[\mathrm{~A}]^{2}[\mathrm{~B}]^{1}\) (d) \(\mathrm{r}=\mathrm{A}[\mathrm{A}][\mathrm{B}]\)

Step-by-Step Solution

Answer

The rate equation is \( r = k [A]^2 [B]^1 \) which corresponds to option (c).

Step 1: Understand the Problem

We need to determine the rate equation from the given options by using the experimental data provided. We have a reaction between species A and B, and we need to use this data to predict the correct rate equation which relates the concentration of reactants to the reaction rate.

Step 2: Analyze Concentration Changes of [A]

Let's evaluate how the initial rate of the reaction changes with the concentration of A when B is constant. In experiments 2 and 3, increase the concentration of A from \(5.0 \times 10^{-4}\) to \(1.0 \times 10^{-3}\) mol/L and the initial rate changes from \(4.0 \times 10^{-3}\) to \(1.6 \times 10^{-2}\) mol L\(^{-1}\) s\(^{-1}\). This shows a fourfold increase, suggesting that the rate is proportional to \([A]^2\).

Step 3: Analyze Concentration Changes of [B]

Now analyze Experiments 1 and 2 where [A] is constant. Increase [B] from \(3.0 \times 10^{-5}\) to \(6.0 \times 10^{-5}\) mol/L. The initial rate increases from \(5.0 \times 10^{-4}\) to \(4.0 \times 10^{-3}\) mol L\(^{-1}\) s\(^{-1}\), indicating that the rate is directly proportional to \([B]^1\).

Step 4: Combine Findings

Based on the analysis, the rate of the reaction can be expressed as: \( r = k [A]^2 [B]^1 \). This matches with option (c).

Step 5: Conclusion

The rate equation is \(r = k [A]^2 [B]^1\) based on the data analysis, confirming that the reaction depends on \([A]^2\) and \([B]^1\). Thus, the correct answer is (c).

Key Concepts

Reaction Kinetics

Reaction kinetics is the branch of chemistry that studies the speed or rate at which chemical reactions occur. It's like understanding how fast the ingredients in your cake mix turn into a cake! This field helps us predict how different factors affect the speed of chemical reactions.

This area of study allows us to elaborate on mechanisms by which a reaction occurs. Key factors that influence reaction kinetics include temperature, concentration of reactants, and presence of a catalyst. Understanding these influences helps us determine how quickly or slowly a reaction will proceed and under what conditions.

• In practical terms, reaction kinetics involves measuring changes in concentration of reactants or products over time.
• This involves using mathematical equations to model these changes and predict future reaction behavior.

When analyzing kinetics, chemists aim to determine the rate law, which is an equation that details how the rate depends on concentrations of reactants. This equation provides critical insight into the reaction mechanism.

Concentration Effect

Concentration effect refers to the impact of reactant concentration on the rate of a reaction. Think of it as adjusting the number of bakers making cookies: more bakers can produce cookies faster! In chemical reactions, altering the concentration of reactants changes the rate at which the reaction proceeds.

In our exercise, increasing the concentration of A and observing a significant increase in rate indicates that the reaction rate is significantly affected by the concentration of A. Similarly, when we increased the concentration of B, there was a corresponding increase in reaction rate, though not as pronounced as A.

• The general rule is that an increase in reactant concentration typically leads to an increase in the reaction rate.
• In the case of reactants A and B, our detailed examination showed that \( [A]^2 \) means rate is more sensitive to changes in A than B.

Understanding the concentration effect helps us in controlling reactions for desired speed and yield.

Rate Law Determination

Determining the rate law is like cracking a chemical mystery—it involves piecing together evidence from experiments to find out how reactant concentrations influence reaction rates. Rate law is a mathematical expression that relates the rate of a reaction to the concentration of its reactants.

The rate law can typically be written as \( r = k [ ext{A}]^m [ ext{B}]^n \) where \( r \) is the rate of reaction, \( k \) is the rate constant, and \( m \) and \( n \) are the orders of reaction for reactants A and B respectively.

• In our problem, experimentation showed the rate to be proportional to \( [A]^2 \) suggesting an order of 2 for A.
• Similarly, the first-order relation for B was determined as \( [B]^1 \), resulting in the overall rate law \( r = k [A]^2 [B]^1 \).

Finding the rate law aids chemists in predicting how the reaction will respond to different concentrations, pressures, or temperatures, enabling them to optimize conditions for industrial, research, or environmental purposes.