A laboratory rat is exposed to an alpha-radiation source whose activity is \(14.3 \mathrm{mCi}\). (a) What is the activity of the radiation in disintegrations per second? In becquerels? (b) The rat has a mass of \(385 \mathrm{~g}\) and is exposed to the radiation for \(14.0 \mathrm{~s}\), absorbing \(35 \%\) of the emitted alpha particles, each having an energy of \(9.12 \times 10^{-13} \mathrm{~J} .\) Calculate the absorbed dose in millirads and grays. (c) If the RBE of the radiation is 9.5, calculate the effective absorbed dose in mrem and Sv.

To convert from millicuries (mCi) to disintegrations per second (dps), you need to know the conversion factor between the two units. For every Ci (curie), there are \(3.7 \times 10^4\) dps. Therefore, to convert mCi to dps, we can use the following relationship: 1 mCi = \(1 \times 10^{-3}\) Ci = \(1 \times 10^{-3} \times 3.7 \times 10^4\) dps Now, we can convert the given activity in mCi to dps: \(14.3 \mathrm{mCi} \times \frac{1 \times 10^{-3} \times 3.7 \times 10^4 \mathrm{dps}}{1 \mathrm{mCi}} = 5291 \mathrm{dps}\) In Becquerels (Bq), since 1 Bq = 1 disintegration per second (dps), we have: Activity in Bq = \(5291 \mathrm{Bq}\)

First, find the number of alpha particles absorbed by the rat: Absorbed alpha particles = Total emitted alpha particles * Fraction absorbed Total emitted alpha particles = Activity in dps * Time of exposure = \(5291 \mathrm{dps} \times 14.0 \mathrm{s} = 74174\) Absorbed alpha particles = \(74174 \times 0.35 = 25960.9\) Now, we can calculate the energy absorbed by the rat using the energy per alpha particle: Energy absorbed = Absorbed alpha particles * Energy per alpha particle = \(25960.9 \times 9.12 \times 10^{-13} \mathrm{J} \approx 2.367 \times 10^{-9} \mathrm{J}\) To find the absorbed dose, divide the energy absorbed by the mass of the rat: Absorbed dose (in J/kg) = \(\frac{2.367 \times 10^{-9} \mathrm{J}}{385 \times 10^{-3} \mathrm{kg}} = 6.141 \times 10^{-6} \mathrm{J/kg}\) Convert the absorbed dose from J/kg to millirads and grays: Absorbed dose (in millirads) = \(6.141 \times 10^{-6} \mathrm{J/kg} \times 100 \frac{\mathrm{millirads}}{1 \mathrm{J/kg}} = 0.6141 \mathrm{millirads}\) Absorbed dose (in grays) = \(6.141 \times 10^{-6} \mathrm{J/kg} \times 1 \frac{\mathrm{Gy}}{1 \mathrm{J/kg}} = 6.141 \times 10^{-6} \mathrm{Gy}\)

To find the effective absorbed dose, multiply the absorbed dose in grays by the given RBE: Effective absorbed dose (in Gy) = Absorbed dose (in Gy) * RBE = \(6.141 \times 10^{-6} \mathrm{Gy} \times 9.5 = 5.8339 \times 10^{-5} \mathrm{Gy}\) Now, convert the effective absorbed dose from Gy to Sv: Effective absorbed dose (in Sv) = Effective absorbed dose (in Gy) * \(\frac{1 \mathrm{Sv}}{1 \mathrm{Gy}}\) = \(5.8339 \times 10^{-5} \mathrm{Sv}\) Convert the effective absorbed dose from Sv to mrem: Effective absorbed dose (in mrem) = Effective absorbed dose (in Sv) * \(\frac{100 \mathrm{mrem}}{1 \times 10^{-3} \mathrm{Sv}}\) = \(5.8339 \times 10^{-2} \mathrm{mrem}\)