Question

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.

Short answer

(a) To convert the activity to disintegrations per second and becquerels: $$14.3\ \mathrm{mCi} \times \frac{1\ \mathrm{Ci}}{1000\ \mathrm{mCi}} \times 3.7 \times 10^{10}\ \mathrm{Bq} = 5.3 \times 10^{8}\ \mathrm{Bq}$$ (b) Absorbed energy: $$E_{absorbed} = 5.3 \times 10^{8}\ \mathrm{disintegrations/s} \times 9.12 \times 10^{-13}\ \mathrm{J/disintegration} \times 14 \ \mathrm{s} \times 0.35 = 2.368 \times 10^{-3}\ \mathrm{J}$$ Absorbed dose: $$D = \frac{2.368 \times 10^{-3}\ \mathrm{J}}{385\ \mathrm{g}} = 6.144 \times 10^{-3}\ \mathrm{J/g}$$ Convert to millirads and grays: $$6.144 \times 10^{-3}\ \mathrm{J/g} \times 100 \mathrm{mrad/J} = 614.4\ \mathrm{mrad}$$ $$6.144 \times 10^{-3}\ \mathrm{J/g} \times 1000\ \mathrm{g/kg} \times 0.01\ \mathrm{Gy/J} = 0.06144\ \mathrm{Gy}$$ (c) Effective absorbed dose: $$D_{eff} = 614.4\ \mathrm{mrad} \times 9.5 = 5,837.6\ \mathrm{mrem}$$ $$D_{eff} = 0.06144\ \mathrm{Gy} \times 9.5 = 0.58368\ \mathrm{Sv}$$

Step-by-step solution

(a) Convert the activity to disintegrations per second and becquerels

To convert the activity from \(\mathrm{mCi}\) to disintegrations per second and becquerels, use the conversion factors \(1\mathrm{Ci} = 3.7 \times 10^{10}\ \mathrm{Bq}\) and \(1\mathrm{Bq} = 1\ \mathrm{disintegration/second}\). Given, the activity is \(14.3\ \mathrm{mCi}\). To convert to disintegrations per second and becquerels: $$14.3\ \mathrm{mCi} \times \frac{1\ \mathrm{Ci}}{1000\ \mathrm{mCi}} \times 3.7 \times 10^{10}\ \mathrm{Bq}$$

(b) Calculate the absorbed energy

Firstly, we need to calculate the energy absorbed. The absorbed energy can be found using the formula: $$E_{absorbed} = N \times E_{\alpha} \times T \times \%$$ Here, \(N\) represents the activity of the radiation in disintegrations per second, \(E_{\alpha}\) is the energy per alpha particle, \(T\) is the exposure time in seconds, and \(\%\) is the absorbed percentage of alpha particles.

(b) Calculate the absorbed dose in millirads and grays

To find the absorbed dose, divide the absorbed energy by the mass of the rat and use the conversion factors \(1 \mathrm{rad} = 0.01\ \mathrm{Gy}\) and \(1\ \mathrm{Gy} = 1\ \mathrm{J/kg}\). The formula is: $$D = \frac{E_{absorbed}}{m}$$ Here, \(D\) represents the absorbed dose, \(E_{absorbed}\) is the absorbed energy we calculated above, and \(m\) is the mass of the rat.

(c) Calculate the effective absorbed dose in mrem and Sv

To find the effective absorbed dose, multiply the absorbed dose by the RBE of the radiation. Then, convert the dose from rads to mrem and from grays to Sv. The formula for effective absorbed dose is: $$D_{eff} = D \times RBE$$ Here, \(D_{eff}\) represents the effective absorbed dose, \(D\) is the absorbed dose we calculated earlier, and \(RBE\) is the Relative Biological Effectiveness. Using the conversion factors \(1 \mathrm{mrad} = 0.001\ \mathrm{rad}\) and \(1 \mathrm{mSv} = 0.001\ \mathrm{Sv}\), we can convert the effective absorbed dose to mrem and Sv.