Problem 14 - A Complete Resource Book in Chemistry for JEE Main

Question

For a chemical reaction \(\mathrm{A} \longrightarrow \mathrm{B}\), the rate of reaction doubles when the concentration of A is in creased four times. The order of reaction for \(\mathrm{A}\) is (a) zero (b) one (c) two (d) half

Step-by-Step Solution

Answer

The order of the reaction is half (option d).

Step 1: Understanding the Rate Equation

The rate of a chemical reaction can be expressed in terms of the concentration of its reactants. For a reaction \( \mathrm{A} \longrightarrow \mathrm{B} \), the rate can be written as \( \text{Rate} = k [A]^n \), where \( k \) is the rate constant and \( n \) is the order of the reaction with respect to \( A \).

Step 2: Identifying Changes in Rate and Concentration

According to the problem, when the concentration of \( A \) is increased four times, the rate of the reaction doubles. This can be expressed as: \( 2 \times \text{Rate}_0 = k(4[A]_0)^n \), where \( \text{Rate}_0 = k[A]_0^n \).

Step 3: Setting Up the Equation

We have \( 2 \times k[A]_0^n = k(4[A]_0)^n \). Cancel the rate constant \( k \) and the initial concentration \( [A]_0^n \), which gives us the equation: \( 2 = 4^n \).

Step 4: Solving for the Order of Reaction

To find \( n \), solve \( 2 = 4^n \). Recognize that \( 4 \) is \( 2^2 \), so \( 4^n = (2^2)^n = 2^{2n} \). Hence, \( 2 = 2^{2n} \). Since the bases are the same, equate the exponents: \( 1 = 2n \).

Step 5: Finding the Value of n

Solve \( 1 = 2n \) for \( n \): \( n = \frac{1}{2} \). Thus, the order of the reaction with respect to \( A \) is \( \frac{1}{2} \).

Key Concepts

Rate Equation

The rate equation is a mathematical expression that describes how the rate of a chemical reaction depends on the concentration of its reactants. It provides crucial insight into how different factors influence the speed of a reaction. For the reaction \( A \longrightarrow B \), the rate equation can be expressed as:

• \( \text{Rate} = k [A]^n \)

Here, \( k \) is the rate constant, which is unique for each reaction and varies with temperature. The term \( [A]^n \) signifies the concentration of reactant \( A \) raised to the power of \( n \), the reaction order.

This order, symbolized by \( n \), reveals how sensitive the reaction rate is to changes in the concentration of \( A \).

• If \( n = 0 \), the reaction rate does not depend on the concentration of \( A \).
• If \( n = 1 \), the rate is directly proportional to the concentration of \( A \).
• If \( n = 2 \), the rate is proportional to the square of \( A \).
• For fractional \( n \), as we found in this exercise, it indicates that the relationship is not straightforward and may involve complex reaction mechanisms.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed, or rate, of chemical reactions and the factors affecting them. It allows us to understand and predict how different conditions impact reaction rates, thus offering insights for both industrial applications and theoretical chemistry.One important component of chemical kinetics is investigating the effects of changing reactant concentrations.

In our example, increasing the concentration of \( A \) by four times results in the reaction rate doubling. This information helps determine the order of the reaction, revealing how the system responds to changes in concentration and guiding us in deducing the mechanism of the reaction.

Understanding kinetics also involves evaluating temperature, catalysts, and pressure, although these aren't the focus here. Through experiments and observations, chemical kinetics gives chemists tools to optimize reactions for efficiency or to control industrial processes.

Reaction Mechanism

A reaction mechanism is a step-by-step sequence of elementary reactions by which a chemical change occurs. It's like the detailed roadmap of what actually happens during a reaction.Each individual process within the mechanism can have its own rate and order, contributing to the overall reaction order seen in the general rate equation.

In our scenario, the reaction of \( A \rightarrow B \) with a fractional order of \( \frac{1}{2} \) suggests that the mechanism could include intermediate steps not captured by a single-step process.

This order implies a more complex interaction than simply colliding particles.

• It might indicate a rate-determining step that involves another form of interaction or a multi-step process.
• It also suggests potential for alternative pathways that may contribute to the overall reaction's rate.
• Understanding such mechanisms allows chemists to manipulate reactions for desired outcomes, like increased yield or reduced unwanted by-products.

In summary, deciphering reaction mechanisms helps in revealing the hidden intricacies of chemical processes.