Chapter 11.4, Problem 66E

To determine

To find the time for the radioactive decay of radon gas.

Answer to Problem 66E

  $\text{0}\text{.000948 days}$ .

Explanation of Solution

Given:

The initial amount radon gas is $38g$ , and final amount radon gas is $6.8g$

Calculation:

Formula is

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

Here $N$ is the final amount of radon gas, $N_0$ is the initial amount of radon gas, and $t$ is the number of half-life.

Substitute the initial amount radon gas $N_0=38g$ , and final amount radon gas $N=6.8g$

In the equation 1

Rearrange the equation (1) for half life (t).

  $\begin{array}{l}N=N_0{\left(\frac{1}{2}\right)}^t\\ \frac{N}{N_0}={\left(\frac{1}{2}\right)}^t\\ \log \frac{N}{N_0}=\log {\left(\frac{1}{2}\right)}^t\\ \log \frac{N}{N_0}=t\log \left(\frac{1}{2}\right)\\ t=\frac{\log \frac{N}{N_0}}{\log \left(\frac{1}{2}\right)}\\ t=\frac{\log \frac{6.8}{38}}{-0.3010}\\ t=0.0002482\end{array}$

Therefore the time taken to decay the radon gas is

  $\begin{array}{l}0.0002482\times 3.82\text{ days}\\ \text{=0}\text{.000948 days}\end{array}$