Radioactive Substance Half-Life: Understanding Decay

The concept of half-life is fundamental in understanding the decay of radioactive substances. It represents the time required for half of the radioactive nuclei in a sample to undergo decay.

Half-Life Definition

• The half-life ($\(T_{1/2}\)$) of a radioactive substance is a constant value that indicates how quickly it decays.
• After one half-life, the amount of the original radioactive substance reduces to 50%.
• After two half-lives, it reduces to 25%, and so on. This is an exponential decay process.

Calculating Number of Half-Lives

To determine the remaining amount of a radioactive substance after a certain period, we first need to find out how many half-lives have passed during that time. The question states that the half-life ($\(T_{1/2}\)$) is 10 years and the total time ($\(t\)$) elapsed is 20 years.

The number of half-lives ($\(n\)$) is calculated using the formula:

\[n = \frac{\text{Total time }(t)}{\text{Half-life }(T_{1/2})}\]

Substituting the given values:

\[n = \frac{20 \text{ years}}{10 \text{ years}} = 2 \text{ half-lives}\]

This means that the substance has undergone decay through two half-life periods.

Remaining Mass of Radioactive Substance

The initial mass of the radioactive substance is given as 1 g. We need to find the remaining amount after 2 half-lives. The general formula to calculate the remaining amount ($\(N\)$) after 'n' half-lives from an initial amount ($\(N_0\)$) is:

\[N = N_0 \left(\frac{1}{2}\right)^n\]

Here, \(N_0 = 1 \text{ g}\) and \(n = 2\).

Substituting these values into the formula:

\[N = 1 \text{ g} \times \left(\frac{1}{2}\right)^2\]

\[N = 1 \text{ g} \times \left(\frac{1}{4}\right)\]

\[N = 0.25 \text{ g}\]

Alternatively, we can track the decay step-by-step:

Therefore, after 20 years, the remaining amount of the radioactive substance is 0.25 g.