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

The activity of the alpha-radiation source is \(5291 \mathrm{dps}\) or \(5291 \mathrm{Bq}\). The absorbed dose by the rat is \(0.6141 \mathrm{millirads}\) or \(6.141 \times 10^{-6} \mathrm{Gy}\). The effective absorbed dose is \(5.8339 \times 10^{-5} \mathrm{Sv}\) or \(5.8339 \times 10^{-2} \mathrm{mrem}\).

Step-by-step solution

Part (a): Converting mCi to disintegrations per second and Becquerels

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 = \(5291 \mathrm{Bq}\)

Part (b): Calculating the absorbed dose in millirads and grays

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) = \(6.141 \times 10^{-6} \mathrm{J/kg} \times 1 \frac{\mathrm{Gy}}{1 \mathrm{J/kg}} = 6.141 \times 10^{-6} \mathrm{Gy}\)

Part (c): Calculating the effective absorbed dose in mrem and Sv

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) * \(\frac{100 \mathrm{mrem}}{1 \times 10^{-3} \mathrm{Sv}}\) = \(5.8339 \times 10^{-2} \mathrm{mrem}\)

Key concepts

Alpha Radiation

Alpha radiation is a type of ionizing radiation which consists of alpha particles, essentially helium nuclei made up of two protons and two neutrons. These particles are emitted from the nuclei of unstable atoms during the decay process. Alpha radiation is generally not hazardous when it's outside the body because it doesn't penetrate the skin very easily.
However, if alpha-emitting materials are ingested or inhaled, they can cause significant harm to living cells, as they can deliver a large amount of energy over a short distance. This makes alpha particles particularly potent at causing biological damage once inside the body.

• Alpha particles are highly charged, resulting in a strong interaction with surrounding atoms.
• Due to their large mass, they have low penetration power compared to other types of radiation like beta or gamma radiation.
• The energy from alpha particles can cause ionization, leading to potential cell damage.

Becquerels

The becquerel (Bq) is the international unit used to measure radioactivity. It is defined as one disintegration or decay per second. In simpler terms, this means each Bq represents one nuclear transformation happening in one second.
When measuring the activity of a radioactive source, such as given in an exercise like this, you often begin with units like curies (Ci) or millicuries (mCi). These units are then converted into becquerels for standardization and easier comparison with other data.

• The conversion between millicuries and becquerels is straightforward: 1 mCi equals 37,000 Bq.
• Becquerels help scientists quantify the intensity of the radiation source or its activity level.
• The higher the becquerel value, the more radioactive decays are occurring per second.

Absorbed Dose

The absorbed dose refers to the amount of radiation energy absorbed per unit mass of an object or individual. It is a crucial concept in radiation protection, indicating how much energy from the radiation is deposited into a specific mass of tissue.
The absorbed dose is measured in grays (Gy) or rads (with 1 Gy equating to 100 rads). Understanding the absorbed dose helps ascertain the potential biological impact of radiation on an organism.

• The absorbed dose is calculated by dividing the total energy absorbed by the mass of the tissue: \(\text{Absorbed Dose} = \frac{\text{Energy absorbed}}{\text{Mass}}\).
• The unit gray (Gy) is equal to 1 joule of energy absorbed per kilogram of tissue.
• In practical situations, converting the absorbed dose into millirads or milligrays allows for a more intuitive understanding of dosage at small scales.

Effective Dose

The effective dose is a metric used to estimate the radiation risk or the biological effect of radiation exposure. It considers not only the absorbed dose but also factors like radiation type and tissue sensitivity, thereby providing a more comprehensive indicator of potential harm.
Radiation types, such as alpha radiation, have different biological impacts based on their ability to cause damage to tissues. The Relative Biological Effectiveness (RBE) helps quantify these differences by modifying the absorbed dose to express potential harm more accurately.

• The effective dose is expressed in sieverts (Sv) or rems, with the understanding that 1 Sv is equivalent to 100 rems.
• To determine the effective dose, you multiply the absorbed dose by the RBE. For example, if the RBE is 9.5 and the absorbed dose is given in grays, then the effective dose becomes \( \text{Effective Dose} = \text{Absorbed Dose} \times \text{RBE} \).
• This calculation aids in government policy and safety measures for protecting individuals from different radiation types.