Question

A radioactive element having half-life (30 min) is undergoing beta decay. The fraction of radioactive element remains undecayed after (90 min) will be

• A. (\frac{1}{2})
• B. (\frac{1}{4})
• C. (\frac{1}{8})
• D. [suspicious link removed]

Hint:

Remaining quantity $N = N_0(1/2)^n$, where $n$ is the number of elapsed half-lives.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the fraction of a radioactive substance that remains after a certain amount of time, given its half-life.
Step 2: Key Formula or Approach:
The amount of substance remaining after \(n\) half-lives is given by:
\[ N = N_{0} \left(\frac{1}{2}\right)^n \]
where \(n = \frac{\text{total time}}{\text{half-life}}\) and \(\frac{N}{N_{0}}\) is the undecayed fraction.
Step 3: Detailed Explanation:
Given:
Half-life \(T_{1/2} = 30 \text{ min}\)
Total time \(t = 90 \text{ min}\)
Calculate the number of half-lives (\(n\)):
\[ n = \frac{t}{T_{1/2}} = \frac{90}{30} = 3 \]
The fraction remaining is:
\[ \text{Fraction} = \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^3 \]
\[ \text{Fraction} = \frac{1}{8} \]
Step 4: Final Answer:
The fraction remaining undecayed is \(\frac{1}{8}\).