Problem 62 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Rate of a reaction \(A+B \rightarrow\) products, is given below as a function of different initial concentrations of \(A\) and \(B\) : [1982-4 Marks] 119 \(\begin{array}{lll}{[A](\operatorname{mol} / l)} & {[B](\mathrm{mol} / l)} & \text { Initial rate }(\mathrm{mol} / l / \mathrm{min}) \\ 0.01 & 0.01 & 0.005 \\ 0.02 & 0.01 & 0.010 \\ 0.01 & 0.02 & 0.005\end{array}\) 1

Step-by-Step Solution

Answer

The rate law is \( \text{Rate} = k[A] \).

Step 1: Write the rate law equation

The rate law for this reaction can be expressed as \( \text{Rate} = k[A]^m[B]^n \), where \( k \) is the rate constant, and \( m \) and \( n \) are the orders of the reaction with respect to reactants \( A \) and \( B \) respectively.

Step 2: Use the experimental data

Using the data provided, set up the equations based on the initial rate and concentrations: 1. \( 0.005 = k (0.01)^m (0.01)^n \) 2. \( 0.010 = k (0.02)^m (0.01)^n \) 3. \( 0.005 = k (0.01)^m (0.02)^n \)

Step 3: Determine the order with respect to A

Divide equation 2 by equation 1: \[ \frac{0.010}{0.005} = \frac{k (0.02)^m (0.01)^n}{k (0.01)^m (0.01)^n} \2 = \left( \frac{0.02}{0.01} \right)^m \m = 1\] Thus, the reaction is first order with respect to \( A \).

Step 4: Determine the order with respect to B

Divide equation 3 by equation 1: \[ \frac{0.005}{0.005} = \frac{k (0.01)^m (0.02)^n}{k (0.01)^m (0.01)^n} \1 = \left( \frac{0.02}{0.01} \right)^n = 0\] Thus, the reaction is zero order with respect to \( B \).

Step 5: Combine the results to write the rate law

Based on the findings from Steps 3 and 4, the rate law is \( \text{Rate} = k[A]^1[B]^0 \). This simplifies to \( \text{Rate} = k[A] \).

Key Concepts

Rate Law

A rate law in chemical kinetics describes how the concentration of reactants affects the rate of a chemical reaction. It is expressed with an equation that includes the rate constant and the concentrations of the reactants raised to some power. This general form can be written as \( \text{Rate} = k[A]^m[B]^n \), where \( k \) is the rate constant, and \( [A] \) and \( [B] \) represent the concentrations of the reactants. The exponents \( m \) and \( n \) depict the reaction order with respect to each reactant.

In simple terms, the rate law helps to predict how changing the amount of each reactant will affect the speed at which the reaction occurs. It is derived from experimental data rather than the stoichiometric coefficients in the balanced equation, which many students often confuse. Understanding the rate law allows scientists to control reaction rates, vital in industrial applications, pharmaceuticals, and other fields.

Reaction Order

Reaction order is a crucial concept in understanding the dynamics of chemical reactions. It indicates the dependency of the reaction rate on the concentration of one or more reactants. The order with respect to a reactant is defined by the exponent in the rate law equation. For instance, in \( \text{Rate} = k[A]^1[B]^0 \), the reaction is first order in \( A \) and zero order in \( B \).

This tells us that the rate of reaction increases linearly with an increase in \( A \), while changes in \( B \) have no effect on the rate. Overall, the reaction order is significant as it provides insight into the mechanism of the reaction and often influences the strategy for controlling the rate of product formation. To determine the reaction order, one must rely on experimental data since it cannot be determined directly from the balanced chemical equation.

Rate Constant

The rate constant, represented by \( k \), is a proportionality factor in the rate law equation. It provides valuable information about the speed of the reaction. Unlike the reaction rate or the concentration, the rate constant remains unaffected by changes in concentrations. It is influenced by temperature and the presence of catalysts.

The units of \( k \) can vary depending on the overall order of the reaction and can help verify the correctness of derived rate laws. For the rate law \( \text{Rate} = k[A] \), the units of \( k \) would be \( \text{L mol}^{-1} \text{min}^{-1} \) for a first-order reaction. Understanding the rate constant is vital as it provides comprehensive insights into the kinetics of a reaction and helps predict the behavior of the reaction under different conditions.