A and B are two radioactive substances. The half-life of A is same as the average life
of B. The decay constant of B is 3.33 per day. (assume ln 2 = 0.70)

To understand the relationship between the half-lives and decay constants of radioactive substances A and B, we need to delve into some fundamental concepts of radioactivity. Let's break this down step by step.

Radioactive Decay Basics

Radioactive decay is a process by which unstable atomic nuclei lose energy by emitting radiation. This decay can be characterized by two key parameters: the half-life and the decay constant.

Half-Life and Decay Constant

The half-life (T_1/2) of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. The decay constant (λ) is a probability rate at which a single atom will decay per unit time. The relationship between these two is given by the formula:

• T_1/2 = ln(2) / λ

Here, ln(2) is approximately 0.693, but for this problem, we will use the provided value of ln(2) = 0.70 for our calculations.

Analyzing Substance B

We know that the decay constant of substance B is given as 3.33 per day. Using the formula for half-life, we can calculate the half-life of B:

• T_1/2 = ln(2) / λ
• T_1/2 = 0.70 / 3.33
• T_1/2 ≈ 0.21 days

This means that substance B has a half-life of approximately 0.21 days.

Relating Substance A to B

The problem states that the half-life of substance A is the same as the average life of substance B. The average life (or mean life) of a radioactive substance is related to its decay constant by the formula:

• Average Life = 1 / λ

Now, let's calculate the average life of substance B:

• Average Life = 1 / 3.33
• Average Life ≈ 0.30 days

Since the half-life of substance A is the same as the average life of substance B, we conclude that:

• T_1/2 (A) = Average Life (B) ≈ 0.30 days

Summary of Findings

In summary, we have determined the following:

• The half-life of substance B is approximately 0.21 days.
• The average life of substance B is approximately 0.30 days.
• The half-life of substance A is also approximately 0.30 days.

This analysis illustrates the interconnectedness of half-life and decay constants in radioactive decay, providing a clearer understanding of how these concepts apply to different substances. If you have any further questions or need clarification on any point, feel free to ask!