Problem 66 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life of a substance in a first-order reaction is 15 minutes. The rate constant is (a) \(2.46 \times 10^{2} \mathrm{~min}^{-1}\) (b) \(4.62 \times 10^{-2} \mathrm{~min}^{-1}\) (c) \(3 \times 10^{-5} \mathrm{~min}^{-1}\) (d) \(3 \times 10^{-9} \mathrm{~min}^{-1}\)

Step-by-Step Solution

Answer

The rate constant is (b) \( 4.62 \times 10^{-2} \mathrm{~min}^{-1} \).

Step 1: Understand the Formula for Half-Life

For a first-order reaction, the half-life \( t_{1/2} \) is related to the rate constant \( k \) by the formula: \[ t_{1/2} = \frac{0.693}{k} \] In this case, the half-life is given as 15 minutes.

Step 2: Rearrange the Formula to Solve for Rate Constant

To find the rate constant, \( k \), rearrange the half-life formula: \[ k = \frac{0.693}{t_{1/2}} \] Substitute \( t_{1/2} = 15 \) minutes into the equation.

Step 3: Perform the Calculation

Substituting the given half-life into the rearranged formula: \[ k = \frac{0.693}{15} \] Calculate \( k \): \[ k = 0.0462 \text{ min}^{-1} \]

Step 4: Match the Calculation with Given Options

From the options given, b) \( 4.62 \times 10^{-2} \mathrm{~min}^{-1} \) matches the calculated rate constant of \( 0.0462 \text{ min}^{-1} \).

Key Concepts

Half-Life Formula

In chemical kinetics, understanding the concept of half-life is crucial, particularly for first-order reactions. The half-life of a reaction is the time it takes for half of the reactant to transform into products. For first-order reactions, this is especially straightforward thanks to the derived formula: \[ t_{1/2} = \frac{0.693}{k} \]where \( t_{1/2} \) is the half-life and \( k \) is the rate constant. This formula originates from the natural logarithm base (approximately 0.693), which derives from the exponential nature of decay in first-order reactions. This means, no matter how much of a substance you have initially, after one half-life, only half will remain.

• This characteristic makes calculations predictable and consistent for reactions following first-order kinetics.
• The formula shows an inverse relationship between \( t_{1/2} \) and \( k \), indicating that a faster reaction (larger \( k \)) corresponds to a shorter half-life.

Let's see how this affects real-world calculations: when given a half-life, you can directly find the rate constant by rearranging the formula to solve for \( k \). Understanding and applying this formula allows us to predict how long it will take for a significant portion of a reactant to disappear.

Rate Constant Calculation

Calculating the rate constant, \( k \), for a first-order reaction is a straightforward process once you have the reaction's half-life. Using the half-life formula mentioned earlier, rearrange it to solve for \( k \):\[ k = \frac{0.693}{t_{1/2}} \] In our example, the half-life is provided as 15 minutes. By substituting directly into the formula:\[ k = \frac{0.693}{15} \]By performing this calculation, we find \( k = 0.0462 \, \text{min}^{-1} \), which matches the option given as b) \( 4.62 \times 10^{-2} \, \text{min}^{-1} \).

• **Rearranging the formula** - Always ensure to correctly switch around an equation to solve for your unknown variable.
• **Unit Consistency** - Keep track of your units. Since we used minutes for half-life, the rate constant is in \( \text{min}^{-1} \).
• **Quick Checks** - See if there's an option which directly matches your calculation to verify the answer quickly.

Chemical Kinetics

Chemical kinetics is the study of rate reactions and the factors that affect these rates. It helps us understand how reactions happen and allows us to manipulate conditions to control the reaction speed.The principles of kinetics can be grouped into several key concepts:

• **Reaction Order** - Determines how the concentration of reactants affects the rate. In a first-order reaction, the rate depends linearly on the concentration of one reactant.
• **Rate Constant (\( k \))** - A crucial value for determining the speed of a reaction. It's specific to each reaction and influenced by conditions like temperature.
• **Temperature Effect** - Typically, increasing temperature increases the rate constant, hence speeding up the reaction due to particles having more energy and colliding more frequently.
• **Concentration and Volume** - Notably impacts multi-reactant systems yet in first-order reactions, only a single reactant concentration directly affects the rate.

Chemical kinetics not only enables us to understand theoretical aspects but also has practical applications.
It aids in industries, environmental systems, and even biological fields, maintaining safety and efficiency by predicting how changes affect reactions. By mastering kinetics, you gain powerful insights into reaction dynamics.