Question

The rate constant (k) for a first order reaction is equal to \(4.2 \times 10^{-4} \mathrm{~s}^{-1}\). What is the half life for the reaction? a. \(3.7 \times 10^{3} \mathrm{~s}\) b. \(7.1 \times 10^{3} \mathrm{~s}\) c. \(2.71 \times 10^{3} \mathrm{~s}\) d. \(1.7 \times 10^{3} \mathrm{~s}\)

Short answer

The half-life is approximately \(1.7 \times 10^{3} \, \mathrm{s}\). Option d.

Step-by-step solution

Identify the Formula for Half-Life

Since this is a first-order reaction, the half-life (\( t_{1/2} \) ) can be calculated using the formula \( t_{1/2} = \frac{0.693}{k} \), where \( k \) is the rate constant.

Substitute the Known Values

Insert the given rate constant \( k = 4.2 \times 10^{-4} \, \mathrm{s}^{-1} \) into the half-life formula: \( t_{1/2} = \frac{0.693}{4.2 \times 10^{-4}} \).

Perform the Calculation

Calculate the half-life using the values: \( t_{1/2} = \frac{0.693}{4.2 \times 10^{-4}} = 1.65 \times 10^{3} \, \mathrm{s} \).

Round and Compare

Round \( t_{1/2} = 1.65 \times 10^{3} \, \mathrm{s} \) to match the options and verify it with choice d, which is rounded to \( 1.7 \times 10^{3} \, \mathrm{s} \).

Key concepts

First-Order Reaction

In chemical reactions, understanding the order of the reaction is crucial for predicting how the reaction behaves over time. A **first-order reaction** is characterized by a reaction rate that is directly proportional to the concentration of one reactant. Put simply, as the concentration of the reactant decreases, the rate at which the reaction occurs also decreases at a proportional rate. This means that first-order reactions have a constant half-life, regardless of the initial concentration of the reactant.

First-order reactions are represented by the rate law equation:

• Rate = k[A]

Here, *Rate* is the rate of the reaction, *k* is the rate constant, and *[A]* is the concentration of the reactant.

Understanding the behavior of first-order reactions helps in predicting how long it will take for a reactant to deplete or transform into products, making it a foundational concept in chemistry.

Rate Constant

The **rate constant** is a key component in chemical kinetics, as it quantifies the speed at which a reaction progresses. It is specific to each reaction and varies depending on factors such as temperature and the presence of catalysts.

In the context of a first-order reaction, the rate constant is denoted by "k" and has units of reciprocal time, such as \( \text{s}^{-1} \). This signifies that it defines the rate in terms of how quickly the concentration of a reactant decreases within a unit of time.

• The magnitude of the rate constant provides insight into the reaction's speed;
• Larger values of *k* indicate faster reactions, whereas smaller values suggest slower changes.

To utilize the rate constant effectively, one must understand its integration into various formulas, like the half-life formula, to predict reaction characteristics accurately.

Chemical Kinetics

**Chemical kinetics** is the branch of chemistry that deals with understanding the speed of chemical reactions and the factors that influence this speed. It delves into the transformation of reactants to products, focusing on:

• The reaction rate;
• The steps involved in the reaction mechanism;
• The effect of different variables such as concentration, temperature, and pressure.

In chemical kinetics, it is vital to know if a reaction follows zero, first, or second order as this dictates how calculations and predictions are made.

The study of chemical kinetics allows chemists and scientists to design experiments, plan chemical production, and even control processes involved in industrial applications. Deciphering kinetic data aids in optimizing reactions for yield, safety, and efficiency.

Half-Life Formula

The **half-life** of a reaction refers to the time required for the concentration of a reactant to reduce to half its initial value. For a first-order reaction, the half-life is a constant and can be calculated using a specific formula.The formula used for first-order reactions is:

• and is expressed as \( t_{1/2} = \frac{0.693}{k} \).

In this expression, \( t_{1/2} \) represents the half-life, and \( k \) is the rate constant. The constant 0.693 arises from the natural logarithm of 2.This formula highlights the unique nature of first-order processes where the half-life is independent of the starting concentration. Therefore, whether you begin with a lot or just a little of a substance, it invariably takes the same amount of time to reduce by half.

Understanding and applying this formula is critical for laboratory experiments, drug dosages, and environmental assessments, ensuring that calculations remain consistent for varying concentrations.