Question

You want to find the half-life of an element. At 12 AM on the first day you find that the element has decayed 50000 times after 1 min. At 12 AM on the second day you find that the element has decayed 45000 times in 1 minute. What is the half-life of the element?

Answer 1

Answer on Question #82132 - Physics - Atomic and Nuclear Physics

Question: You want to find the half-life of an element. At 12 AM on the first day you find that the element has decayed 50000 times after 1 min. At 12 AM on the second day you find that the element has decayed 45000 times in 1 minute. What is the half-life of the element?

Answer:

Solution of the problem is based on the utilization of the law of radioactive decay which states

$$N(t) = N_0 2^{-\frac{t}{T}},$$

where $N_0$ is the number of the initially existing nuclei, $N(t)$ is the number of the not decayed nuclei by the time $t$, $T$ is the half-life period.

Hence, the number of already decayed nuclei can be calculated as:

$$N_{dec}(t) = N_0 - N(t) = N_0 \left(1 - 2^{-\frac{t}{T}}\right).$$

For the first observation we have:

$$t_1 = \Delta t = 1 \text{min}, \quad \Delta N_1 = 50000,$$ $$\Delta N_1 = N_0 \left(1 - 2^{-\frac{\Delta t}{T}}\right).$$

For the second observation we have:

$$t_2 = t_0 + \Delta t, \quad t_0 = 24h, \quad \Delta N_2 = 45000,$$ $$\Delta N_2 = N(t_0) \left(1 - 2^{-\frac{\Delta t}{T}}\right) = N_0 2^{-\frac{t}{T}} \left(1 - 2^{-\frac{\Delta t}{T}}\right),$$

where we utilize $N(t_0)$ as the number of still existing nuclei after 24 hours.

Dividing (4) by (6), we obtain:

$$\frac{\Delta N_1}{\Delta N_2} = \frac{N_0 \left(1 - 2^{-\frac{\Delta t}{T}}\right)}{N_0 2^{-\frac{t}{T}} \left(1 - 2^{-\frac{\Delta t}{T}}\right)} = 2^{\frac{t}{T}}.$$

By solving (7) in respect to $T$, we obtain:

$$T = \frac{t_0}{\log_2 \frac{\Delta N_1}{\Delta N_2}} = \frac{24h}{\log_2 \frac{50000}{45000}} \approx 158h.$$

So, the half-life of the element is around 158 hours.

Answer provided by https://www.AssignmentExpert.com