A radioactive detector records an initial disintegration rate of 6400\text{ counts per min… — JAMB UTME

A radioactive detector records an initial disintegration rate of $6400\text{ counts per minute}$ from a freshly prepared isotope. If the half-life of the isotope is $5\text{ hours}$ , determine the total time, in hours, required for the count rate to decrease to $400\text{ counts per minute}$ .

Answer: 20 hours

Answer

The total time required for the disintegration rate to decrease to

400\text{ counts per minute}

20\text{ hours}

The fraction of activity remaining is

\frac{400}{6400} = \frac{1}{16}

. Expressing this as a power of one-half,

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

, shows that

4

half-lives have elapsed. Multiplying

4

half-lives by

5\text{ hours}

per half-life yields a total duration of

20\text{ hours}

Step-by-Step Solution

Calculate the ratio of remaining activity to initial activity

\frac{N}{N_0} = \frac{400}{6400} = \frac{1}{16}

To find the fraction of the original radioactive substance that remains undecayed.

Determine the number of elapsed half-lives

n = 4

Since \left(\frac{1}{2}\right)^n = \frac{1}{16} = \left(\frac{1}{2}\right)^4, four complete half-lives have passed.

Compute the total elapsed time

t = 4 \times 5 = 20\text{ hours}

Total time equals the number of half-lives multiplied by the duration of one half-life.

Key Concept

Radioactive Decay Law and Half-life

Estimated Time:1m 30s

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