A sample of iodine-131 (^{131}\text{I}), a radioactive isotope used in medical diagnosis, … — JAMB UTME

A sample of iodine-131 ( $^{131}\text{I}$ ), a radioactive isotope used in medical diagnosis, has a half-life of $8\text{ days}$ . If only $2.5\text{ g}$ of the sample remains active after an elapsed time of $24\text{ days}$ , what was the initial mass of the sample?

A
$7.5\text{ g}$
B
$10\text{ g}$
$20\text{ g}$ Answer
D
$40\text{ g}$

Answer

The initial mass of the iodine-131 sample was

20\text{ g}

Over an elapsed time of

24\text{ days}

with a half-life of

8\text{ days}

, exactly

3

half-lives pass (

24 \div 8 = 3

). Since the remaining mass is

2.5\text{ g}

, working backward requires doubling the mass three times:

2.5\text{ g} \rightarrow 5.0\text{ g} \rightarrow 10.0\text{ g} \rightarrow 20.0\text{ g}

, giving an initial mass of

20\text{ g}

Step-by-Step Solution

Determine the number of half-lives (

n

) that have elapsed.

n = \frac{\text{Total Time}}{\text{Half-life}} = \frac{24\text{ days}}{8\text{ days}} = 3\text{ half-lives}

Dividing total elapsed time by the half-life period gives the total count of half-life cycles.

Apply the radioactive decay relationship to calculate initial mass (

N_0

N = N_0 \left(\frac{1}{2}\right)^n \implies 2.5\text{ g} = N_0 \left(\frac{1}{2}\right)^3 = \frac{N_0}{8}

Radioactive decay follows an exponential model where remaining amount is reduced by half each cycle.

Solve for initial mass (

N_0

N_0 = 2.5\text{ g} \times 8 = 20\text{ g}

Multiplying the remaining mass by

2^3

reverses the exponential decay process.

Key Concept

Radioactive Half-Life and Exponential Decay Calculations

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