Problem 221 - A Complete Resource Book in Chemistry for JEE Main

Question

For a first order reaction, (A) \(\rightarrow\) products, the concentration of A changes from \(0.10 \mathrm{M}\) to \(0.025\) Min 40 minutes. The rate of reaction when the concentration of \(\mathrm{A}\) is \(0.01 \mathrm{M}\), is: (a) \(3.47 \times 10^{-5} \mathrm{M} / \mathrm{min}\) (b) \(3.47 \times 10^{-4} \mathrm{M} / \mathrm{min}\) (c) \(1.73 \times 10^{-5} \mathrm{M} / \mathrm{min}\) (d) \(1.73 \times 10^{-4} \mathrm{M} / \mathrm{min}\)

Step-by-Step Solution

Answer

The rate is \(3.47 \times 10^{-4} \mathrm{M/min}\), option (b).

Step 1: Understand First Order Reaction

A first-order reaction is one where the rate depends linearly on the concentration of a single reactant. The rate law for a first-order reaction is given by:\[ \frac{d[A]}{dt} = -k[A] \]where \(k\) is the rate constant and \([A]\) is the concentration of A.

Step 2: Use Integrated Rate Law for First Order

For a first-order reaction, the integrated rate law is:\[ \ln \left( \frac{[A]_0}{[A]} \right) = kt \]where \([A]_0\) is the initial concentration, \([A]\) is the concentration at time \(t\), and \(k\) is the rate constant.

Step 3: Calculate Rate Constant (k)

Given: \([A]_0 = 0.10\, \text{M}\), \([A] = 0.025\, \text{M}\), and \(t = 40\, \text{min}\).Use the equation:\[ \ln \left( \frac{0.10}{0.025} \right) = k \times 40 \]\[ \ln(4) = k \times 40 \]\[ k = \frac{\ln(4)}{40} \approx 0.0347\, \text{min}^{-1} \]

Step 4: Determine the Reaction Rate at Given Concentration

Now, find the rate of reaction when \([A] = 0.01\, \text{M}\) using the expression:\[ \text{Rate} = k[A] = 0.0347 \times 0.01 \]\[ \text{Rate} = 3.47 \times 10^{-4}\, \text{M/min} \]

Step 5: Choose the Correct Answer

Compare the calculated rate with the given options:(a) \(3.47 \times 10^{-5} \mathrm{M/min}\)(b) \(3.47 \times 10^{-4} \mathrm{M/min}\)(c) \(1.73 \times 10^{-5} \mathrm{M/min}\)(d) \(1.73 \times 10^{-4} \mathrm{M/min}\)The correct answer is (b) \(3.47 \times 10^{-4} \mathrm{M/min}\).

Key Concepts

Rate of Reaction

In chemistry, the rate of reaction is a measure of how quickly reactants are converted into products. For a first order reaction, like the one given in the original exercise, the rate of the reaction depends on the concentration of one reactant alone. This is expressed mathematically as:\[ \text{Rate} = k[A] \]Where:

• \( [A] \) is the concentration of reactant \( A \)
• \( k \) is the rate constant

A first order reaction has a direct linear relationship between the concentration of the reactant and the rate. This means if the concentration of \( A \) doubles, the rate of the reaction also doubles. Understanding the rate of reaction helps in controlling chemical processes, such as in industrial synthesis or even in biological systems. It is crucial because it can affect the yield and efficiency of the reactions. Keep in mind: In a practical situation, knowing the rate helps us adjust concentrations to optimize outcomes, whether speeding up or slowing down a reaction.

Integrated Rate Law

The integrated rate law for a first order reaction is key to understanding how the concentration of reactants changes over time. It is expressed as:\[ \ln \left( \frac{[A]_0}{[A]} \right) = kt \]In this equation:

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

This integrated rate law reveals the logarithmic relationship between the initial concentration and the concentration at any given time. The time it takes for half of the reactants to react (half-life) is constant and can be easily calculated for first order reactions. The integrated rate law is a powerful tool for chemists, as it allows us to backtrack and find out vital details like rate constants or predict future concentration levels of reactants. It's used widely in labs to monitor reaction progress without continuous concentration measurements.

Rate Constant

The rate constant, represented by \(k\), plays an essential role in determining the speed of a reaction for a given concentration. It is a proportionality factor in the rate law equation and is unique for every reaction under specific conditions, such as temperature and pressure. In a first order reaction:\[ k = \frac{\ln \left( \frac{[A]_0}{[A]} \right)}{t} \]Where:

• \( \ln \left( \frac{[A]_0}{[A]} \right) \) is the natural logarithm of the initial concentration divided by the concentration at time \(t\)
• \(t\) is the time elapsed

The rate constant directly impacts the rate of reaction: a larger \(k\) means a quicker reaction. Calculating \(k\) accurately is fundamental for reaction modeling and control. It allows chemists to understand how specific changes in their experimental settings might influence the speed of a reaction, ensuring predictability and control in chemical processes.