Question

The half-life of radon is 3.82 days. How long will it take for 60 percent of a
sample of radon to decay?

Answer 1

Answer on Question 60786, Physics, Atomic and Nuclear Physics

Question:

The half-life of radon is 3.82 days. How long will it take for 60 percent of a sample of radon to decay?

Solution:

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

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

here, $N_0$ is the amount of radon at time $t = 0$ (100%), $N$ is the amount of radon when 60% of it is decayed ($N = 40\%$), $\lambda = 0.693 / T_{1/2}$ is the radioactive decay constant, $T_{1/2} = 3.82$ days is the half-life of radon, $t$ is the elapsed time which we are searching for.

Then, we get:

$$\frac {N}{N _ {0}} = e ^ {- \lambda t},$$ $$\ln \left(\frac {N}{N _ {0}}\right) = \ln \left(e ^ {- \lambda t}\right),$$ $$\ln \left(\frac {N}{N _ {0}}\right) = - \frac {0 . 6 9 3}{T _ {1 / 2}} t.$$

From the last formula we can find the time needed for 60 percent of a sample of radon to decay:

$$t = \left[ \frac {\ln \left(\frac {N}{N _ {0}}\right)}{(- 0 . 6 9 3)} \right] T _ {1 / 2} = \left[ \frac {\ln (0 . 4)}{(- 0 . 6 9 3)} \right] \cdot 3. 8 2 d a y s = 5. 0 5 d a y s.$$

Answer:

$t = 5.05$ days.