Question

The half-life of a certain radioactive isotope is 32 hours. What fraction of the sample would remain after 16 huors?

0.50
0.25
0.62
0.71

Answer 1

Answer on Question 59428, Physics, Atomic and Nuclear Physics

Question:

The half-life of a certain radioactive isotope is 32 hours. What fraction of the sample would remain after 16 hours?

a) 0.50

b) 0.25

c) 0.62

d) 0.71

Solution:

Let's use the famous equation for the radioactive decay:

$$N = N _ {0} e ^ {- \lambda t},$$

here, $N_0$ is the amount of a certain radioactive isotope at time $t = 0$, $N$ is the amount of the certain radioactive isotope in any given time interval $\Delta t$, $\lambda = 0.693 / T_{1/2}$ is the radioactive decay constant, $T_{1/2} = 32$ hours is the half-life of the radioactive isotope, $t = 16$ hours is the elapsed time.

Then, from this equation, we can calculate the fraction of a sample $(N / N_0)$ remaining after 16 hours:

$$\frac {N}{N _ {0}} = e ^ {- \lambda t} = e ^ {- \left(\frac {0 . 6 9 3}{3 2}\right) \cdot 1 6} = e ^ {- 0. 3 4 6 5} = 0. 7 1.$$

Thus, the correct answer is d).

Answer:

d) 0.71

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