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

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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.

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).

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.

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.

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

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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