Final answer:
To find the mass of Radium-226 causing a 2.50 mSv/year dose, one must convert the dose to joules, relate it to the alpha particle energy, and use the decay constant and Avogadro's number to find the mass. However, without the half-life value or decay constant, the exact calculation cannot be completed. It is not surprising that a small mass would have a measurable effect due to the high energy of radioactive decays.
Step-by-step explanation:
To calculate the mass of Radium-226 in an 80 kg man's body who receives a dose of 2.50 mSv/year from it, we must convert the dose in millisieverts to joules, relate it to the emitted energy of the alpha particles in joules, and then find the number of decays per year. We can then utilize the decay constant and Avogadro's number to find the actual mass.
First, convert the radiation dose to joules (1 Sv = 1 J/kg), so 2.50 mSv = 2.50 × 10⁻³ Sv and for an 80 kg man, it would be 2.50 × 10⁻³ Sv × 80 kg = 0.2 J. To find the number of decays, we need the energy of an alpha particle in joules (1 eV = 1.602 × 10⁻ J). 4.80 MeV is 4.80 × 10⁶ eV or 4.80 × 10⁶ × 1.602 × 10⁻ J per decay. By dividing the total annual energy absorption (0.2 J) by the energy per decay, we get the number of decays per year.
Using the decay constant (λ) for Radium-226, which is related to its half-life (λ = ln(2) / half-life) and Avogadro's number, we can find the mass of Radium. This is based on the equation N = mass / (molar mass) × Avogadro's number, where N is the number of decay events. Though this problem doesn't provide the decay constant (λ), you would generally use it to find the activity (A = λN) and solve for the mass.
For the exact calculation, additional steps, including the specific decay constant of Radium-226, would be required. However, without the half-life or decay constant provided, we cannot complete the numeric calculation. Still, the mass would indeed be extremely small and still produce a measurable dose of radiation. This is not surprising because radiation can cause significant biological effects, even at low mass of radioactive material, due to the high energy carried by each decay event.