A radioactive sample has a half-life of 5\text{ days}. If the mass of the sample that has … — JAMB UTME

A radioactive sample has a half-life of $5\text{ days}$ . If the mass of the sample that has decayed after $20\text{ days}$ is $45\text{ g}$ , what was the initial mass of the sample?

$48\text{ g}$ Answer
B
$720\text{ g}$
C
$60\text{ g}$
D
$180\text{ g}$

Answer

The initial mass of the sample was

48\text{ g}

The elapsed time corresponds to 4 half-lives (

20 / 5 = 4

). The remaining mass fraction is

(1/2)^4 = 1/16

, which means

15/16

of the original mass has decayed. Setting

15/16

of the initial mass equal to

45\text{ g}

gives an initial mass of

48\text{ g}

Step-by-Step Solution

Calculate the number of half-lives elapsed

n = \frac{t}{T_{1/2}} = \frac{20\text{ days}}{5\text{ days}} = 4\text{ half-lives}

Determining how many half-life cycles occurred within the total elapsed time.

Determine the fraction of sample decayed

Fraction remaining =

(1/2)^4 = 1/16

; Fraction decayed =

1 - 1/16 = 15/16

The decayed fraction is the complement of the remaining fraction.

Calculate the initial mass

M_{\text{decayed}} = \frac{15}{16} M_0 = 45\text{ g} \implies M_0 = 45 \times \frac{16}{15} = 48\text{ g}

Relating the given decayed mass to the total initial mass.

Key Concept

Radioactive Decay Law and Half-life

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