Problem 203 - A Complete Resource Book in Chemistry for JEE Main

Question

The rate law for a reaction between the substances A and is given by Rate \(=[A]^{n}[B]^{m}\) on doubling the concentration of \(\mathrm{A}\) and halving the-concentration of \(\mathrm{B}\), the ratio of the new rate to the earlier rate of the reaction will be as \(\quad\) [2003] (a) \(1 / 2^{(\mathrm{m}+\mathrm{n})}\) (b) \((\mathrm{m}+\mathrm{n})\) (c) \((\mathrm{n}-\mathrm{m})\) (d) \(2^{(\mathrm{n}-\mathrm{m})}\)

Step-by-Step Solution

Answer

The ratio of the new rate to the initial rate is \( 2^{(n-m)} \), so the answer is (d).

Step 1: Write the Expression for the Initial Rate

The rate of the reaction can be written as \( R_1 = k[A]^n[B]^m \), where \( k \) is the rate constant, \( [A] \) is the concentration of A, and \( [B] \) is the concentration of B.

Step 2: Apply Changes to Concentrations

The concentration of \( A \) is doubled and the concentration of \( B \) is halved. Therefore, the new concentrations are \( [2A] \) and \( \left[\frac{B}{2}\right] \).

Step 3: Write the Expression for the New Rate

With the new concentrations, the rate becomes \( R_2 = k[2A]^n\left[\frac{B}{2}\right]^m \). Expand the expression to \( k \cdot 2^n [A]^n \cdot \frac{1}{2^m} [B]^m \).

Step 4: Simplify the New Rate

Simplify the expression for \( R_2 \) to \( R_2 = k \cdot 2^{n-m} [A]^n [B]^m \).

Step 5: Determine the Ratio of the New Rate to the Initial Rate

Compute the ratio \( \frac{R_2}{R_1} = \frac{k \cdot 2^{n-m} [A]^n [B]^m}{k [A]^n [B]^m} = 2^{n-m} \).

Step 6: Match the Ratio with Given Options

The ratio equals \( 2^{n-m} \). Compare with the options to find that (d) \( 2^{(n-m)} \) matches this ratio.

Key Concepts

Reaction Kinetics

Reaction kinetics is the study of how chemical reactions occur and at what speed or rate. It focuses on understanding the factors that influence the speed of a reaction. One main aspect of reaction kinetics is the mechanism by which reactants are converted into products. By studying these mechanisms, chemists can determine the step-by-step changes that occur during the conversion. Reaction kinetics also involves analyzing how different conditions, such as temperature, pressure, and concentration, affect the reaction rate.

Through experiments, data is collected to create models. These models help predict the outcomes of reactions under various conditions. In our example, changing concentrations of reactants show how sensitive the reaction rate is to such adjustments. This underscores the importance of reaction kinetics in practical applications, such as designing reactors or optimizing industrial chemical processes.

Rate Equation

A rate equation is a mathematical expression that describes the rate of a chemical reaction. It is influenced by the concentration of reactants and sometimes products. The general form of a rate equation reflects these dependencies through exponents that indicate the order of the reaction relative to each reactant.

For our example, the rate equation is given as \( \text{Rate} = k[A]^n[B]^m \). Here, \( k \) is the rate constant, and \( [A] \) and \( [B] \) are the concentrations of reactants A and B. The exponents \( n \) and \( m \) denote the order of the reaction with respect to each reactant. These values can be determined experimentally and define how sensitive the rate is to changes in concentration. A higher exponent means a greater impact on the rate when that particular reactant's concentration is altered.

Concentration Effect

The concentration effect refers to how changes in the concentration of reactants impact the rate of reaction. When concentrations are modified, the rate equation can predict the new rate. In our example, by doubling the concentration of A and halving the concentration of B, we explore how these particular changes affect the reaction rate.

The rate law tells us that the new reaction rate, following the changes in concentrations, is expressed by adjusting the exponents in the rate equation: \( 2^n \) for doubling A, and \( \frac{1}{2}^m \) for halving B. These adjustments reflect directly in the new rate calculation. Consequently, by altering concentrations, one can increase or decrease the speed of reaction, which is critical in both lab experiments and industrial processes.

Rate Determination Steps

Determining the rate of a reaction involves a few key steps. First, establish the initial rate using a known set of concentrations and the corresponding rate equation. Our example begins with \( R_1 = k[A]^n[B]^m \).

Next, apply the intended changes to concentrations, recalculating the rate with the new values: \( R_2 = k[2A]^n\left[\frac{B}{2}\right]^m \). Simplifying this to \( R_2 = k \cdot 2^{n-m}[A]^n[B]^m \), allows comparison to the original rate. Finally, compute the ratio of the new rate to the initial rate, reducing the equation to a form that can quickly identify the effect of the changes. This ratio, \( 2^{n-m} \), succinctly captures the impact of concentration changes on reaction kinetics.

The careful following of these steps aids in both understanding and predicting how a reaction will respond to experimental adjustments.