Radium-226 (atomic mass = 226.025402 amu) decays to radon-222 (a radioactive gas) with a half-life of 1.6⨉10³ years. What volume of radon gas (at 25.0 °C and 1.0 atm) does 25.0 g of radium produce in 5.0 days? (Report your answer to two significant digits.)

Verified step by step guidance

1. Convert the mass of radium-226 to moles using its atomic mass: \( \text{moles of } \text{Ra} = \frac{25.0 \text{ g}}{226.025402 \text{ amu}} \).
2. Determine the fraction of radium that decays in 5.0 days using the half-life formula: \( N_t = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \), where \( t = 5.0 \text{ days} \) and \( t_{1/2} = 1.6 \times 10^3 \text{ years} \).
3. Calculate the moles of radon-222 produced, which is equal to the moles of radium-226 that decayed.
4. Use the ideal gas law \( PV = nRT \) to find the volume of radon gas, where \( P = 1.0 \text{ atm} \), \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \), and \( T = 25.0 \degree C = 298.15 \text{ K} \).
5. Solve for \( V \) in the ideal gas law equation to find the volume of radon gas produced.