A sample of a radioactive substance has a decay constant of 0.0154\text{ h}^{-1}. What is … — JAMB UTME

A sample of a radioactive substance has a decay constant of $0.0154\text{ h}^{-1}$ . What is the elapsed time, in hours, required for $87.5\%$ of the original sample to decay? (Take $\ln 2 = 0.693$ )

Answer: 135 hours

Answer

135 hours

First, calculate the half-life of the substance using the relation

T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{0.0154\text{ h}^{-1}} = 45\text{ hours}

. Since

87.5\%

of the sample has decayed, the remaining fraction of the sample is

100\% - 87.5\% = 12.5\% = \frac{1}{8}

. Expressing

\frac{1}{8}

as a power of

\frac{1}{2}

gives

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

, indicating that

3

half-lives have passed. The total elapsed time is therefore

3 \times 45\text{ hours} = 135\text{ hours}

Step-by-Step Solution

Calculate the half-life from the given decay constant

Half-life

T_{1/2} = 45\text{ hours}

The decay constant

\lambda

and half-life

T_{1/2}

are related by the formula

T_{1/2} = \frac{\ln 2}{\lambda}

Find the remaining percentage and fraction of the sample

Remaining fraction is

12.5\%

\frac{1}{8}

Radioactive decay equations use the undecayed remaining amount, which is

100\% - 87.5\% = 12.5\%

Calculate the number of half-lives elapsed

Number of half-lives

n = 3

Since

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

, solving for

n

gives

n = 3

Compute the total elapsed time

Total elapsed time

t = 135\text{ hours}

Total elapsed time is the product of the number of half-lives and the half-life duration (

3 \times 45\text{ hours}

)

Key Concept

Radioactive Decay Law and Half-life Relationship

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