Problem 75 - A Complete Resource Book in Chemistry for JEE Main

Question

The reaction \(\mathrm{A} \longrightarrow \mathrm{B}\) follows first order kinetics. The time taken for \(0.8\) mole of \(\mathrm{A}\) to produce \(0.6\) mole of is 1 hour. What is the time taken for conversion of \(0.9\) mole of A to produce \(0.675\) mole of \(\mathrm{B} ?\) (a) 2 hour (b) 1 hour (c) \(0.5\) hour (d) \(0.25\) hour

Step-by-Step Solution

Answer

The time taken is 1 hour.

Step 1: Understanding First Order Kinetics

In first order kinetics, the rate of decomposition of substance A is proportional to its concentration. The rate equation is given by \( r = k[A] \), where \( k \) is the rate constant and \([A]\) is the concentration of A.

Step 2: Determine the Rate Constant

The first order kinetic equation can be expressed as \( [A] = [A_0]e^{-kt} \). For the first scenario, we have \( [A_0] = 0.8 \) moles and \( [A] = 0.2 \) moles after 1 hour (since 0.6 moles of B were produced). Applying the equation:\( 0.2 = 0.8 e^{-k \times 1} \) Solve for \( k \):\( k = -\ln(0.2/0.8) = -\ln(0.25) \).

Step 3: Calculate the Required Time for Second Scenario

For the conversion of 0.9 moles of A to 0.675 moles of B, the remaining A is 0.225 moles (since \(0.9 - 0.675 = 0.225\)). Now use the first-order equation again:\( 0.225 = 0.9 e^{-kt} \)Substitute \( k = -\ln(0.25) \) from Step 2:\( \ln\left(\frac{0.225}{0.9}\right) = -(-\ln(0.25)) \times t \)Solve for \( t \).

Step 4: Simplify and Solve for Time

Simplify the equation: \( \ln\left(0.25\right) = \ln(0.25) \times t \)\( t = \frac{\ln(0.25)}{\ln(0.25)} = 1 \) hour.

Key Concepts

Rate Constant

In a chemical reaction, the rate constant, often denoted by the symbol \( k \), plays a vital role in determining the speed at which a reaction occurs. For first-order reactions, the rate constant signifies the proportionality factor in the relationship between the rate of reaction and the concentration of the reactant. The general rate equation for first-order kinetics is \( r = k[A] \), where \([A]\) represents the concentration of the reactant at a specific time.
To determine \( k \), we can rearrange the first-order kinetic expression \( [A] = [A_0]e^{-kt} \). Here, \( [A_0] \) represents the initial concentration of reactant A, while \( [A]\) is the concentration at time \( t \).
By substituting known values of initial and final concentrations along with the time, we can solve for \( k \). This constant provides insight into how quickly the reaction proceeds and is crucial for predicting future concentration changes as time progresses.

Reaction Rate

The rate of a chemical reaction is an essential concept that measures how fast reactants are converted into products. In first-order kinetics, the reaction rate is directly proportional to the concentration of a single reactant. This means that as the concentration of the reactant decreases over time, the reaction rate slows down.
**Key points in understanding reaction rate:**

• The mathematical expression for a first-order reaction is \( r = k[A] \). This indicates that the rate \( r \) is the product of the rate constant \( k \) and the reactant concentration \([A]\).
• As the reaction progresses, the concentration of \( A \) typically decreases, leading to a decrease in the reaction rate.
• We assess reaction speed through experiments by measuring how fast the concentration of a reactant decreases or how fast a product forms.

Understanding the relationship between the rate and concentration helps in predicting how long it takes for a reactant to reach a specific concentration, thereby calculating the time needed for a particular conversion as seen in first-order processes.

Concentration

Concentration is a fundamental concept in chemistry that describes the amount of a substance in a given volume. In the context of first-order reactions, concentration significantly impacts the reaction rate. Initial concentrations set the stage for how the reaction unfolds over time.
The concentration of a reactant, \([A]\), decreases exponentially with time in a first-order reaction according to the formula \( [A] = [A_0]e^{-kt}\). This mathematical expression captures the exponential decline in reactant concentration as the reaction proceeds.
**Role of concentration in first-order kinetics:**

• Beginning with a higher initial concentration \([A_0]\) leads to a faster reaction at the start.
• The concentration drops at a rate determined by both the rate constant \( k \) and the current concentration \([A]\).
• Assessing changes in concentration over time allows calculation of the reaction's half-life, the time it takes for the concentration to reduce to half its initial value, which is constant in first-order reactions.

By mastering the concept of concentration in kinetics, one can predict how the amount of reactant varies with time, and effectively determine reaction characteristics, such as the time required for specific transformations.