If a radioactive element with a half-life of 30 min undergoes bet… — Solution | JEE Physics

MCQEasyJEE 2023Radioactive Decay & Half-Life

If a radioactive element with a half-life of $30 \, \text{min}$ undergoes beta decay, the fraction of the radioactive element that remains undecayed after $90 \, \text{min}$ is:

A
$\frac{1}{8}$
B
$\frac{1}{16}$
C
$\frac{1}{4}$
D
$\frac{1}{2}$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: Half-life $T_{1/2} = 30 \, \text{min}$ and elapsed time $t = 90 \, \text{min}$ .

Find: The fraction of the radioactive element that remains undecayed after $90 \, \text{min}$ .

For radioactive decay,

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

where $n$ is the number of half-lives.

First, calculate the number of half-lives:

n = \frac{t}{T_{1/2}} = \frac{90}{30} = 3

Now substitute into the decay formula:

\frac{N}{N_0} = \left(\frac{1}{2}\right)^3 = \frac{1}{8}

Therefore, the fraction remaining undecayed is $\frac{1}{8}$ . The correct option is A.

Common mistakes

Using the total time directly in the decay formula without first finding the number of half-lives. This is wrong because the exponent must be the count of half-life intervals. First compute $n = \frac{t}{T_{1/2}}$ , then use $\left(\frac{1}{2}\right)^n$ .
Assuming that after $90 \, \text{min}$ the remaining fraction is $\frac{1}{2}$ because one half-life means half remains. This is wrong because $90 \, \text{min}$ corresponds to three half-lives, not one. The quantity halves repeatedly.
Confusing decayed fraction with undecayed fraction. This is wrong because the question asks for the fraction that remains. After three half-lives, remaining fraction is $\frac{1}{8}$ , while decayed fraction would be $1 - \frac{1}{8} = \frac{7}{8}$ .

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