Question

A first-order reaction is \(50 \%\) completed in 30 minutes at \(27^{\circ} \mathrm{C}\). Its rate constant is (a) \(2.31 \times 10^{-2} \mathrm{~min}^{-1}\) (b) \(3.21 \times 10^{-2} \mathrm{~min}^{-1}\) (c) \(4.75 \times 10^{-2} \mathrm{~min}^{1}\) (d) \(1.33 \times 10^{-3} \mathrm{~min}^{-1}\)

Short answer

The rate constant is \(2.31 \times 10^{-2} \mathrm{~min}^{-1}\).

Step-by-step solution

Identify the Relevant Formula

For a first-order reaction, the relationship between the rate constant \( k \), the time \( t \), and the percentage completion is given by the formula \( t_{1/2} = \frac{0.693}{k} \), where \( t_{1/2} \) is the half-life of the reaction.

Determine Half-life

The exercise states that the reaction is 50% complete in 30 minutes. Since 50% completion corresponds to one half-life, we have \( t_{1/2} = 30 \) minutes.

Calculate the Rate Constant

Using the formula for the half-life of a first-order reaction, \( t_{1/2} = \frac{0.693}{k} \), substitute the given half-life:\[ 30 = \frac{0.693}{k} \]Re-arrange and solve for \( k \):\[ k = \frac{0.693}{30} \approx 0.0231 \text{ min}^{-1} \]

Match Calculated Value to Options

The calculated rate constant \( k \approx 0.0231 \text{ min}^{-1} \) matches option (a) \(2.31 \times 10^{-2} \text{ min}^{-1}\).

Key concepts

Understanding Rate Constant

The rate constant, often represented by the letter 'k', is an essential part of understanding how fast or slow a chemical reaction proceeds. For a first-order reaction, the rate constant provides a quantitative measure of the reaction speed. It describes how the concentration of a reactant decreases over time. A higher rate constant value means the reaction occurs faster. In mathematical terms, the rate of a first-order reaction can be expressed as:\[ \text{Rate} = k \times [A] \]where

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

In the exercise mentioned, we determine the rate constant by using the half-life formula, which is specific for first-order reactions. To put it simply, the rate constant tells us how quickly the reaction reaches its half-life.

The Concept of Half-Life

The concept of half-life is central to understanding first-order reactions. Half-life refers to the time it takes for half of the reactant to be consumed in the reaction. For first-order reactions, the half-life is independent of the initial concentration of the reactant. This unique feature means that regardless of how much reactant you start with, the time it takes to reach half is constant.For first-order reactions, the half-life \( t_{1/2} \) can be calculated using:\[ t_{1/2} = \frac{0.693}{k} \]Here, 0.693 is a constant that originates from natural logarithms and remains the same for all first-order reactions.Understanding half-life helps chemists predict how long a reaction will take to reach a certain level of completion. In the given exercise, it was stated that half of the reaction completed in 30 minutes. This provided us with the half-life needed to calculate the rate constant.

Basics of Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the rates of chemical processes. It answers the question of how fast a reaction takes place and what factors influence this speed. Various aspects of chemical kinetics include understanding different types of reactions (like first-order reactions) and the influence of conditions such as temperature and concentration. Key components of chemical kinetics include:

• Reaction rates: The speed at which reactants are converted into products.
• Rate constants: The proportionality factor in the rate equation that provides information on the reaction speed.
• Reaction order: Indicates the power to which the concentration of a reactant is raised in the rate equation.

In our scenario, the nature of the first-order reaction simplifies the calculations, because the rate depends solely on the concentration of one reactant. Understanding these principles allows chemists to control reactions better and predict outcomes, which is crucial in many industrial and scientific applications.