A sample of a radioactive isotope has a half-life of 12\text{ days}. If the initial mass o… — JAMB UTME

A sample of a radioactive isotope has a half-life of $12\text{ days}$ . If the initial mass of the sample is $48\text{ g}$ , what mass of the isotope remains undecayed after $36\text{ days}$ ?

$6\text{ g}$ Answer
B
$16\text{ g}$
C
$12\text{ g}$
D
$24\text{ g}$

Answer

The mass of the isotope remaining undecayed after

36\text{ days}

6\text{ g}

Radioactive decay follows exponential kinetics. The total elapsed time of

36\text{ days}

represents

3

half-lives of

12\text{ days}

each. Halving the initial

48\text{ g}

sample three consecutive times (

48 \rightarrow 24 \rightarrow 12 \rightarrow 6

) yields

6\text{ g}

Step-by-Step Solution

Calculate the number of half-lives (

n

) that have elapsed.

n = \frac{\text{Total elapsed time}}{\text{Half-life}} = \frac{36\text{ days}}{12\text{ days}} = 3\text{ half-lives}

Determining the number of elapsed half-lives is necessary to apply exponential halving.

Apply the exponential decay formula

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

N = 48\text{ g} \times \left(\frac{1}{2}\right)^3 = 48\text{ g} \times \frac{1}{8} = 6\text{ g}

Radioactive decay follows exponential kinetics, where the remaining mass decreases by half for each elapsed half-life.

Key Concept

Radioactive Half-Life and Exponential Decay Kinetics

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