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Chapter 20: Problem 73

A 14-m by \(10-\mathrm{m}\) by \(3.0-\mathrm{m}\) basement had a high radon content. On the day the basement was sealed off from its surroundings so that no exchange of air could take place, the partial pressure of \({ }^{222} \mathrm{Rn}\) was \(1.2 \times 10^{-6} \mathrm{mmHg} .\) Calculate the number of \({ }^{222} \mathrm{Rn}\) isotopes \(\left(t_{1}=3.8 \mathrm{~d}\right)\) at the beginning and end of 31 days. Assume STP conditions.

Short Answer

Expert verified
To get the initial and final number of radon atoms, one must apply the ideal gas law to find the initial number of atoms and then use the radioactive decay formula (considering the number of half-lives passed) to find the remaining atoms after 31 days.

Step by step solution

01

Use the Ideal Gas Law to find the initial number of radon atoms

The ideal gas law is given by \(PV=nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature. Here, we know the temperature is at standard conditions (0 degree Celsius or 273 K), R = 0.0821 L.atm/K.mol, and we're given P in mmHg, so it needs to be converted to atmosphere (1 atm = 760 mmHg). The volume of the basement is given as \(14m \times 10m \times 3m = 420 m^3\), which is \(420 \times 10^3 L\). We solve for n, the number of moles, and then find the number of atoms by multiplying by Avogadro's number (\(6.022 \times 10^{23} / mol\)).
02

Calculate the number of half-lives elapsed

31 days have passed, so one must calculate how many half-lives this time period corresponds to. This is done by dividing the elapsed time by the half-life of radon: \(N = t/t_{1/2}\).
03

Calculate the remaining number of radon atoms

After finding the number of half-lives, the remaining number of radon atoms are found using the equation \(N_{final} = N_{initial} * (1/2)^N\), where N_{final} is the number of atoms remaining after N half-lives, N_{initial} is the starting number of atoms, and N is the number of half-lives.

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