Determine the activity of 10.0 mg of ²²⁶Ra in units of Bq and Ci. The half-life of ²²⁶Ra is 1600 years.

Verified step by step guidance

1. Step 1: Convert the mass of ²²⁶Ra from milligrams to grams. Since 1 mg = 0.001 g, multiply 10.0 mg by 0.001 to get the mass in grams.
2. Step 2: Calculate the number of moles of ²²⁶Ra using its molar mass. The molar mass of ²²⁶Ra is approximately 226 g/mol. Use the formula: moles = mass (g) / molar mass (g/mol).
3. Step 3: Determine the number of atoms of ²²⁶Ra using Avogadro's number, which is 6.022 x 10²³ atoms/mol. Multiply the number of moles by Avogadro's number to find the total number of atoms.
4. Step 4: Calculate the decay constant (λ) using the half-life formula: λ = ln(2) / half-life. Convert the half-life from years to seconds (1 year = 3.156 x 10⁷ seconds) before using it in the formula.
5. Step 5: Determine the activity (A) in becquerels (Bq) using the formula: A = λN, where N is the number of atoms. To convert the activity from Bq to curies (Ci), use the conversion factor: 1 Ci = 3.7 x 10¹⁰ Bq.

Key Concepts

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This decay occurs at a predictable rate characterized by the half-life, which is the time required for half of the radioactive atoms in a sample to decay. Understanding this concept is crucial for calculating the activity of a radioactive substance.

Half-Life

The half-life of a radioactive isotope is the time it takes for half of the original amount of the isotope to decay into another element or isotope. For ²²⁶Ra, the half-life is 1600 years, meaning that after this period, only half of the original amount will remain. This concept is essential for determining the remaining quantity of a radioactive substance over time.

Activity Measurement Units

Activity is a measure of the rate at which a radioactive substance decays, typically expressed in becquerels (Bq) or curies (Ci). One Bq corresponds to one decay per second, while one Ci is equivalent to 3.7 x 10^10 decays per second. Converting between these units is important for expressing the activity of radioactive materials in a standardized manner.