Problem 190 - A Complete Resource Book in Chemistry for JEE Main

Question

The rate of a first order reaction at \(20 \mathrm{~min}\) is \(0.55 \mathrm{~mol}\) \(\mathrm{L}^{-1} \min ^{-1}\) and \(0.055 \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~min}^{-1}\) at 40 min after initi- ation. Find half life of the reaction in minutes.

Step-by-Step Solution

Answer

The half-life of the reaction is approximately 66.3 minutes.

Step 1: Understand the Rate Law for First Order Reactions

In first order reactions, the rate of the reaction is proportional to the concentration of a single reactant. The rate law is expressed as \( r = k[A] \) where \( r \) is the rate of the reaction, \( k \) is the rate constant, and \( [A] \) is the concentration of the reactant.

Step 2: Calculate the Rate Constant \( k \)

For a first-order reaction, we use the formula: \( r = k[A] \). At \( t = 20 \) min, \( r_1 = 0.55 \) mol L\(^{-1}\) min\(^{-1}\). At \( t = 40 \) min, \( r_2 = 0.055 \) mol L\(^{-1}\) min\(^{-1}\). The change in time \( \Delta t \) is \( 40 - 20 = 20 \) min.

Step 3: Use the Formula for First Order Reactions

The concentration relation for a first-order reaction is \([A]_t = [A]_0 e^{-kt}\). From this, the expression for two different times can be set: \( \frac{r_2}{r_1} = e^{-k20} \). Solve for \( k \): \( \frac{0.055}{0.55} = e^{-20k} \). Simplify and solve for \( k \) to find \( k = \frac{1}{20} \ln 0.1 \).

Step 4: Find Half-Life Using Rate Constant \( k \)

The half-life of a first-order reaction is defined as \( t_{1/2} = \frac{0.693}{k} \). Substitute the value of \( k \) obtained in Step 3 into this formula to calculate the half-life. Since \( k = \frac{1}{20} \ln(0.1) \), substitute in to find \( t_{1/2} \approx 66.3 \) minutes.

Key Concepts

Rate Law

In chemical kinetics, understanding the rate law is crucial for analyzing how fast a reaction occurs. The rate law for a first order reaction tells us that the rate is directly proportional to the concentration of a single reactant. This can be expressed with the formula:

• \( r = k[A] \)

where \( r \) represents the reaction rate, \( k \) is the rate constant, and \( [A] \) is the reactant concentration. This means as the concentration of the reactant increases, the reaction rate increases at the same rate. This proportionality is a defining feature of first order reactions, offering a straightforward way to predict reaction rates based on concentration.

Rate Constant

The rate constant, designated as \( k \), is a critical component in the rate law equation. For first order reactions, \( k \) quantifies how rapidly the reaction proceeds. It's a unique value for every reaction and can change with temperature. To find \( k \) in a first order reaction, we use the known rate equation \( r = k[A] \). In our scenario, we calculated \( k \) using the change in reactions rates over time:

• First, determine the proportion of rates at two different times: \( \frac{r_2}{r_1} \).
• This ratio is equal to \( e^{-k \Delta t} \), where \( \Delta t \) is the time difference.
• Solve the equation \( \frac{0.055}{0.55} = e^{-20k} \) for \( k \).
• Through calculations, \( k \) is derived as \( \frac{1}{20} \ln(0.1) \).

Finding \( k \) allows us to deeply understand and predict the behavior of the reaction over time.

Half-Life of Reaction

Half-life is a fascinating concept in chemistry, especially when dealing with first order reactions. It refers to the time it takes for half of the reactant to be consumed in the reaction. For first order reactions, the half-life does not depend on the initial concentration, which means no matter how much reactant you start with, the half-life remains constant. This is different from reactions of other orders. The half-life \( t_{1/2} \) is calculated using the formula:

• \( t_{1/2} = \frac{0.693}{k} \)

By substituting \( k \) into this formula, we find \( t_{1/2} \approx 66.3 \) minutes. This ability to determine a constant half-life makes first order reactions particularly predictable and easier to work with in real-world applications.

First Order Reactions Formula

The fundamental formula linking concentration and time in a first order reaction is crucial for calculating reactant concentrations at different times. The formula is expressed as:

• \([A]_t = [A]_0 e^{-kt} \)

where:

• \([A]_t\) is the concentration at time \( t \)
• \([A]_0\) is the initial concentration
• \( k \) is the rate constant

With this relation, you can predict how much reactant remains at any point in time, making it invaluable in scenarios where reaction progress needs to be monitored or controlled. In our specific exercise, we use this formula to equate the ratios of concentrations at two different times, helping us solve for \( k \), then further exploring the reaction's dynamics.