A sample of a radioactive nuclide has a half-life of 4\text{ hours}. What is the ratio of … — JAMB UTME

A sample of a radioactive nuclide has a half-life of $4\text{ hours}$ . What is the ratio of the number of nuclei that have decayed to the number of undecayed nuclei remaining after a total elapsed time of $16\text{ hours}$ ?

$15 : 1$ Answer
B
$1 : 15$
C
$15 : 16$
D
$7 : 1$

Answer

The ratio of decayed nuclei to remaining undecayed nuclei is

15 : 1

In an elapsed time of

16\text{ hours}

(which is

4

half-lives), the remaining undecayed portion of the sample is

(1/2)^4 = 1/16

of the original amount. Consequently,

1 - 1/16 = 15/16

of the nuclei have decayed. The ratio of decayed nuclei to remaining nuclei is

(15/16) : (1/16) = 15 : 1

Step-by-Step Solution

Determine the number of half-lives (

n

) that have elapsed.

n = \frac{t}{T_{1/2}} = \frac{16\text{ hours}}{4\text{ hours}} = 4\text{ half-lives}

The number of half-lives is the total time divided by the half-life duration.

Calculate the fraction of original nuclei remaining undecayed (

N/N_0

\frac{N}{N_0} = \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^4 = \frac{1}{16}

After

n

half-lives, the remaining fraction is

(1/2)^n

Find the fraction of original nuclei that have decayed (

N_{\text{decayed}}/N_0

\frac{N_{\text{decayed}}}{N_0} = 1 - \frac{N}{N_0} = 1 - \frac{1}{16} = \frac{15}{16}.

Decayed fraction equals total initial fraction (1) minus the undecayed fraction.

Compute the ratio of decayed nuclei to remaining undecayed nuclei.

\text{Ratio} = \frac{N_{\text{decayed}}}{N} = \frac{15/16}{1/16} = 15 : 1.

The question asks for the ratio of decayed nuclei to remaining undecayed nuclei.

Key Concept

Radioactive Decay Law and Half-life calculations involving ratios of decayed versus remaining quantities.

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