Final answer:
To find the decay rate of Calcium-47, you calculate the decay constant using its half-life and use it along with the sample's mass to determine the number of decaying nuclei per second, which gives you the activity in becquerels.
Step-by-step explanation:
To calculate the decay rate of Calcium-47, which is a β− emitter with a half-life of 4.50 days, we will use the formula that relates the decay constant (λ) to the half-life (t₁/₂): λ = ln(2)/t₁/₂. First, we find the decay constant by substituting the half-life value into the equation.
λ = ln(2)/4.50 days = 0.1541 days⁻¹
Next, to find the number of decays per second (activity in becquerels), we will use the formula A = λN, where N is the number of unstable nuclei in the sample. To calculate N, we need the mass of the isotope and its molar mass (which is approximately 47 g/mol for Calcium-47).
N = (mass of isotope in grams / molar mass) × Avogadro's number
N = (2.34g / 47 g/mol) × 6.022×10¹¹ nuclei/mol
This gives us the number of nuclei. We then calculate the activity:
A = λN = 0.1541 days⁻¹ × N nuclei
This calculation will yield the decay rate of Calcium-47 in the bone sample in becquerels.