A radioactive isotope has a half-life of 4\text{ hours}. If a sample initially contains 80… — JAMB UTME

A radioactive isotope has a half-life of $4\text{ hours}$ . If a sample initially contains $80\text{ g}$ of the isotope, what mass of the isotope, in grams, will remain undecayed after $12\text{ hours}$ ?

Answer: 10 g

Answer

The mass of the radioactive isotope remaining undecayed after

12\text{ hours}

10\text{ g}

After

3

half-lives (

12\text{ hours}

total elapsed time with a half-life of

4\text{ hours}

), the fraction of the initial sample remaining is

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

. Multiplying this fraction by the initial mass of

80\text{ g}

gives

10\text{ g}

Step-by-Step Solution

Determine the number of elapsed half-lives (

n

)

n = \frac{12\text{ hours}}{4\text{ hours}} = 3

half-lives

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

Calculate the mass remaining after

3

half-lives

N = 80 \times \left(\frac{1}{2}\right)^3 = 80 \times \frac{1}{8} = 10\text{ g}

The remaining mass halves during each half-life interval according to the exponential decay rule

N = N_0 (1/2)^n

Key Concept

Radioactive Decay Law and Half-life

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