Question

The half-life of a radioactive substance is $30$ minutes. The time (in minutes) taken between $40\%$ decay and $85\%$ decay of the same radioactive substance is

• A. 15
• B. 30
• C. 45
• D. 60

Answer 1

The Correct Option is D

Half-Life Calculation

The concept of half-life refers to the time it takes for half of the initial amount of a substance (such as a radioactive isotope) to decay. Let's break down the steps for calculating the time based on the given data.

Given Information: 

• N₁ = 0.6 N₀: The quantity of the substance at some point after decay (0.6 times the original amount).
• N₂ = 0.15 N₀: The quantity of the substance after a further period of time (0.15 times the original amount).

Step 1: Ratio of N₂ and N₁

We can calculate the ratio between \(N\) and \(N\) as follows:

\(\frac{N_2}{N_1} = \left(\frac{1}{2}\right)^2\)

This shows that the ratio is equal to 1/4, meaning that two half-lives have passed. Here’s why:

• In each half-life, the amount of the substance reduces by half. The formula above shows that \(N\) is 1/4 of the original quantity, which is equivalent to two half-lives.

Step 2: Calculating Time Taken

Now that we know two half-lives have passed, we can calculate the total time taken:

\(2t\)

So, the time taken for this process to occur is 60 minutes.

Summary:

Given the decrease from \(0.6 N\) to \(0.15 N\), we find that two half-lives have passed. Since each half-life is 30 minutes, the total time taken is 60 minutes.