Understanding the Half-Life of a First-Order Reaction and Time to Complete 99% of Reactant

In chemical kinetics, understanding the behavior of a first-order reaction is crucial for predicting its rates and dynamic properties. This article delves into two specific aspects of a first-order reaction: the half-life and the time required for 99% of the reactant to be converted. The given reaction constant, k, is 2.10×10-3s-1, and we will explore how to calculate the half-life and the time required for the reactant to be reduced to 1%.

Half-Life of a First-Order Reaction

The half-life, denoted as t1/2, of a first-order reaction can be determined using the formula:

t1/2 0.693}{k}

Given k 2.0 × 10-3s-1, we can substitute the value into the formula to find the half-life:

t1/2 0.693}{2.0 × 10-3} ≈ 346.5 seconds

Time to Complete 99% of Reactant

In a first-order reaction, the relationship between the concentration of the reactant and time can be described using the equation:

ln([A]? / [A]) kt

Where:

[A]? is the initial concentration of the reactant, [A] is the concentration at time t, k is the rate constant, t is the time.

To determine the time when 99% of the reactant has reacted, we need to find the time when [A] 0.01[A]?. Therefore:

ln([A]? / [A]) ln(100) ≈ 4.605

Substituting the given values into the equation:

4.605 2.0 × 10-3 × t

Solving for t:

t 4.605 / 2.0 × 10-3 ≈ 2302.5 seconds

Summary

Half-life: Approximately 346.5 seconds

Time for the 99% of the reactant to remain: Approximately 2302.5 seconds

Conclusion

By understanding the half-life and the time required for 99% of the reactant to be converted, chemists and engineers can better predict and optimize the behavior of first-order reactions in various applications, such as pharmaceuticals, environmental science, and industrial processes. The given values and calculations provide a clear example of how to apply these concepts.

For further reading on this topic, consider exploring the mathematical foundations of first-order reactions and their applications in real-world scenarios.