A building has become accidentally contaminated with radioactivity. The longest-lived material in the building is strontium-90. (The atomic mass of
90
38
Sr
is 89.9077 u.) If the building initially contained 6.9 kg of this substance and the safe level is less than 10.5 counts/min, how long will the building be unsafe?
To determine how long the building will be unsafe, we need to consider the radioactive decay of strontium-90.
Strontium-90 undergoes radioactive decay with a half-life of 28.8 years. The half-life is the time it takes for half of a given quantity of a radioactive substance to decay.
Here's how we can calculate the time it takes for the strontium-90 to decay to a safe level:
1. Calculate the initial number of strontium-90 atoms in the building:
- Convert the mass of strontium-90 (6.9 kg) to moles by dividing by the molar mass of strontium-90.
- The molar mass of strontium-90 is 89.9077 g/mol.
- Use the formula: moles = mass / molar mass.
- Calculate the moles: moles = 6.9 kg / 89.9077 g/mol.
2. Calculate the initial number of strontium-90 atoms using Avogadro's number (6.022 x 10^23 atoms/mol):
- Multiply the moles calculated in step 1 by Avogadro's number.
- The formula is: number of atoms = moles x Avogadro's number.
3. Determine how many half-lives are needed to decay to a safe level:
- A safe level is less than 10.5 counts/min.
- Assuming the decay is continuous, we can use the formula: time = half-life x log(initial count rate / safe count rate).
- Rearrange the formula to solve for half-lives: number of half-lives = log(initial count rate / safe count rate) / log(1/2).
4. Calculate the time it takes for the strontium-90 to decay to a safe level:
- Multiply the number of half-lives calculated in step 3 by the half-life of strontium-90.
By following these steps, we can determine how long the building will be unsafe.