Radium-226, which undergoes alpha decay, has a half-life of 1600 yr. (a) How many alpha particles are emitted in 5.0 min by a 10.0-mg sample of ₂₂₆Ra?

Verified step by step guidance

1. Step 1: Determine the number of moles of radium-226 in the 10.0 mg sample. Use the molar mass of radium-226, which is approximately 226 g/mol, to convert the mass to moles.
2. Step 2: Calculate the initial number of radium-226 atoms in the sample using Avogadro's number (6.022 x 10^23 atoms/mol).
3. Step 3: Use the half-life formula to find the decay constant (\( \lambda \)) for radium-226. The formula is \( \lambda = \frac{0.693}{\text{half-life}} \). Convert the half-life from years to minutes for consistency with the time period given.
4. Step 4: Apply the first-order decay equation \( N = N_0 e^{-\lambda t} \) to find the number of radium-226 atoms remaining after 5.0 minutes. Here, \( N_0 \) is the initial number of atoms, \( \lambda \) is the decay constant, and \( t \) is the time in minutes.
5. Step 5: Subtract the number of remaining radium-226 atoms from the initial number to find the number of atoms that decayed. Since each decay corresponds to the emission of one alpha particle, this number is also the number of alpha particles emitted.