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) \(3 \times 10^{-4} \mathrm{~min}^{-1}\)

Short answer

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

Step-by-step solution

Understand the Concept of Half-life

For a first-order reaction, the half-life is the time required for the concentration of a reactant to decrease by half. The formula relating half-life \( t_{1/2} \) and the rate constant \( k \) is \( t_{1/2} = \frac{0.693}{k} \).

Set Up the Equation Using Given Data

We know the half-life \( t_{1/2} \) is 15 minutes, so we substitute it into the formula: \( 15 = \frac{0.693}{k} \).

Solve for the Rate Constant \( k \)

Rearrange the equation to solve for \( k \): \( k = \frac{0.693}{15} \).

Calculate the Rate Constant

Perform the division to find that \( k \approx 0.0462 \).

Identify the Correct Option

Compare the calculated rate constant \( k \approx 0.0462 \) min\(^{-1}\) with the given options: \( (a) 2.46 \times 10^{2} \text{ min}^{-1} \), \( (b) 4.62 \times 10^{-2} \text{ min}^{-1} \), \( (c) 3 \times 10^{-5} \text{ min}^{-1} \), \( (d) 3 \times 10^{-4} \text{ min}^{-1} \). The closest match is option \( (b) \), \( k = 4.62 \times 10^{-2} \text{ min}^{-1} \).

Key concepts

Half-life

The concept of half-life is like the ticking of a clock in the world of chemistry. It is the time it takes for half of a substance to react or decay in a given process. In the context of a first-order reaction, half-life becomes especially important due to its independence from concentration. In essence, regardless of how much reactant you start with, the time it takes to fall to half its original amount remains constant. This key characteristic makes it a crucial parameter in studying chemical reactions.

• For first-order reactions, the half-life is constant and independent of the initial concentration of the reactant.
• The half-life can help predict how long a substance will last or how quickly it will be used up in a reaction.
• Its mathematical relationship with the rate constant is given by the formula: \[ t_{1/2} = \frac{0.693}{k} \]

Understanding the half-life allows scientists to forecast the behavior of a reaction over time, making it a pivotal concept in fields such as pharmacology and environmental science.

Rate constant

In chemical kinetics, the rate constant is a vital element that tells us how quickly a reaction proceeds. It is unique to each particular reaction and provides deep insights into the reaction dynamics. For a first-order reaction, the rate constant has units of time inverse, usually expressed as \( \text{min}^{-1} \) or \( \text{s}^{-1} \). This signifies the proportional relationship between the rate of reaction and the concentration of reactants.

• The rate constant, \( k \), represents the rate of the reaction under specific conditions.
• It varies with temperature and can be influenced by catalysts, changing the speed of the reaction.
• In first-order reactions, the mathematical linkage with half-life is via the formula: \[ k = \frac{0.693}{t_{1/2}} \]

Calculating the rate constant gives a clear picture of how fast the reactants are being converted into products, essential for chemical manufacturing and laboratory studies.

Chemical kinetics

Chemical kinetics is the study of reaction rates and the steps through which reactions occur. It delves into the mechanics of how rapidly chemical transformations take place and what influences these dynamics. This field helps chemists understand reaction speed, which factors affect it, and how to control the process for desired outputs.

• It explains the cause of varying speeds of different reactions.
• Kinetic studies involve determining rate laws, which reveal how concentration affects the rate.
• Factors such as temperature, concentration, and catalysts are pivotal in kinetic analysis.

Studying chemical kinetics broadens our understanding of natural processes and enables the advancement of technologies such as drug development, where knowing how fast a drug reacts in the body is vital.