Answer:
25 minutes
Explanation:
V = Volume = [tex]90\ \text{m}^3[/tex]
[tex]C_0[/tex] = Radon concentration under steady state = 1.5 Bq/L
k = Radon decay rate = [tex]2.09\times 10^{-6}\ \text{s}^{-1}[/tex]
[tex]Q[/tex] = Venting rate = [tex]0.14\ \text{m}^3/\text{s}[/tex]
[tex]C_f[/tex] = Final concentration of radon = 0.15 Bq/L
Theoretical detention time is given by
[tex]\theta=\dfrac{V}{Q}\\\Rightarrow \theta=\dfrac{90}{0.14}\\\Rightarrow \theta=642.86\ \text{s}[/tex]
We have the relation
[tex]C_f=C_0e^{-(\dfrac{1}{\theta}+k)t}\\\Rightarrow t=\dfrac{\ln\dfrac{C_f}{C_0}}{-(\dfrac{1}{\theta}+k)}\\\Rightarrow t=\dfrac{\ln\dfrac{0.15}{1.5}}{-(\dfrac{1}{642.86}+2.09\times 10^{-6})}\\\Rightarrow t=1478.25\ \text{s}=\dfrac{1478.25}{60}=24.63\approx 25\text{minutes}[/tex]
The time taken to reach the acceptable level of concentration is 25 minutes.
