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 \(\mathrm{A}\) 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 reaction for \( \mathrm{A} \) is half.

Step 1: Understanding the Problem

We need to find the order of reaction with respect to A given that when the concentration of \( \mathrm{A} \) is increased four times, the rate of reaction doubles.

Step 2: Rate Law Expression

The rate of reaction for a chemical reaction \( \mathrm{A} \rightarrow \mathrm{B} \) is given by the rate law: \( \text{Rate} = k [\mathrm{A}]^n \), where \( k \) is the rate constant and \( n \) is the order of reaction we need to determine.

Step 3: Applying Given Conditions

According to the problem, when \( [\mathrm{A}] \) is increased by four times, the rate doubles. Therefore, the expressions for the initial and final rates are: \[ \text{Initial Rate} = k [\mathrm{A}]^n \] \[ \text{Final Rate} = 2 \times \text{Initial Rate} = k (4[\mathrm{A}])^n \].

Step 4: Set Up Equation

Plug in the condition of the rate doubling into the rate law expressions: \[ 2k[\mathrm{A}]^n = k (4[\mathrm{A}])^n \].

Step 5: Simplify the Equation

Divide both sides of the equation by \( k[\mathrm{A}]^n \): \[ 2 = 4^n \].

Step 6: Solve for the Order of Reaction

Solve the equation \( 2 = 4^n \) by expressing 4 as \( 2^2 \), so \[ 2 = (2^2)^n = 2^{2n} \]. Equating the powers of 2 gives: \[ 1 = 2n \]. Solving for \( n \) gives \( n = \frac{1}{2} \).

Step 7: Determine the Answer

Since the order of reaction \( n \) is \( \frac{1}{2} \), the correct option is (d) half.

Key Concepts

Rate Law

When dealing with chemical reactions, the rate law helps us understand how the concentration of reactants affects the rate of the reaction. It is expressed as:

• \( \text{Rate} = k [\text{A}]^n \)

Here, \( k \) is the rate constant, which is specific to each reaction, and \( n \) is the order of reaction with respect to the reactant \( \text{A} \). This equation tells us how changes in concentration impact the rate at which the reaction proceeds.
If you alter the concentration of \( \text{A} \), the rate law helps predict how the reaction rate will change. For example, if \( n = 1 \), the reaction is first-order with respect to \( \text{A} \), meaning the rate changes linearly with concentration. If the order is 0, the rate is constant regardless of concentration. Different orders give different patterns in how the rate changes.
It's important to not confuse "rate law" with "rate equation," as the latter also incorporates other parameters like temperature and catalysts, not solely concentration.

Rate of Reaction

The rate of reaction refers to how quickly a reaction progresses. For a reaction like \( \text{A} \rightarrow \text{B} \), understanding the rate is crucial for predicting how long it will take for reactants to convert into products. The rate of reaction can be expressed in terms of concentration change over time. For instance:

• \( \text{Rate} = \frac{-d[\text{A}]}{dt} = \frac{d[\text{B}]}{dt} \)

This equation tells us how the concentration of \( \text{A} \) decreases and \( \text{B} \) increases with time. The minus sign indicates that \( [\text{A}] \) is decreasing.
When we say that the rate of reaction doubles, it means that the speed at which products form or reactants disappear is twice as fast. This could result from changes in conditions such as concentration, temperature, or the presence of a catalyst. Understanding these factors helps in controlling and optimizing reactions for desired outcomes, especially in industrial processes.

Chemical Kinetics

Chemical kinetics is the study of reaction speeds and the factors affecting them. It provides insights into:

• The rate of reactions
• Their mechanisms
• The effect of different variables

One of the major components of chemical kinetics is formulating and understanding the rate law. This involves experimentally determining how concentrations change over time and how these changes affect rate laws, like order of reactions.
Kinetics studies are important because they help chemists understand not just the speed of reactions, but also the pathways they follow. This information is crucial for applications like developing new chemical processes, pharmaceuticals, or materials with specific properties. By manipulating different conditions like temperature, pressure, and concentration, scientists aim to achieve desired rates and yields.
Overall, chemical kinetics serves as a bridge connecting the microscopic world of molecules and atoms with the macroscopic observations of reaction speed.

Reaction Mechanism

A reaction mechanism describes the step-by-step process by which reactants transform into products. Unlike the rate law, which provides a mathematical relationship of reaction rates with concentrations, the reaction mechanism gives a detailed molecular view of how reactions occur.
Reaction mechanisms often consist of a series of elementary steps, each representing a simple reaction event. These can involve:

• Bond formation and breakage
• Formation of reaction intermediates
• Collision and alignment of molecules

Each step of the mechanism can have its own intermediate and transition state, and it's important to remember that many reactions do not happen in a single step.
Understanding the mechanism can provide valuable insights into how altering conditions might affect the outcome of the reaction. For example, it can help predict whether a catalyst will lower the activation energy or if a change in temperature might influence certain steps more than others. Insights gained from mechanisms support the design and optimization of chemical processes, making them an essential aspect of industrial chemistry and research. In a nutshell, while the rate law can describe how fast a reaction proceeds, the reaction mechanism reveals the underlying story of how that speed is achieved.