When uranium disintegrates into lead, one step in the process is the radioactive decay of radium into radon gas. Radon enters through the soil into home basements, where it presents a health hazard if inhaled. In the simplest case of radon detection, a sample of air with volume V is taken. After equilibrium has been established, the radioactive decay D of the radon gas is counted with efficiency E over time t. The radon concentration C present in the sample of air varies directly as the product of D and E and inversely as the product of V and t. For a fixed radon concentration C and time t, find the change in the radioactive decay count D if V is multiplied by 4 and E is reduced by 20%.

Question

Quizplus · Accepted Answer

Given that $C \propto \frac{DE}{Vt}$, for a fixed $C$ and $t$, if $V$ is multiplied by 4 and $E$ is reduced by 20%, we have $E_{new} = 0.8E$ and $V_{new} = 4V$. To keep $C$ constant, $D$ must adjust accordingly. The new relationship becomes $C \propto \frac{D_{new} \times 0.8E}{4V \times t}$. To counteract the effects of quadrupling $V$ and reducing $E$ by 20%, $D_{new}$ must increase by a factor of $5$ (since $4 \times 0.8 = 3.2$, and to maintain the same proportion, $D_{new}$ needs to be $5$ times $D$ to offset the $3.2$ denominator change), which is a $500$% increase.

When Uranium Disintegrates into Lead, One Step in the Process