Problem 71 - University Physics with Modern Physics

Question

\(\mathrm{A}^{60} \mathrm{Co}\) source with activity \(2.6 \times 10^{-4} \mathrm{Ci}\) is embedded in a tumor that has mass 0.200 \(\mathrm{kg} .\) The source emits \(\gamma\) photons with average energy 1.25 \(\mathrm{MeV} .\) Half the photons are absorbed in the tumor, and half escape. (a) What energy is delivered to the tumor per second? (b) What absorbed dose (in rad) is delivered per second? (c) What equivalent dose (in rem) is delivered per second if the RBE for these \(\gamma\) rays is 0.70\(?\) (d) What exposure time is required for an equivalent dose of 200 rem?

Step-by-Step Solution

Answer

(a) 9.625 J/s (b) 48.125 rad/s (c) 33.6875 rem/s (d) 5.94 s

Step 1: Calculate Photons Emitted per Second

The activity of the source is given as \(2.6 \times 10^{-4}\,\mathrm{Ci}\). To find the number of photons emitted per second, convert this activity into becquerels (Bq), where \(1\,\mathrm{Ci} = 3.7 \times 10^{10}\,\mathrm{Bq}\): \[\text{Activity in Bq} = 2.6 \times 10^{-4}\,\mathrm{Ci} \times 3.7 \times 10^{10}\,\mathrm{Bq/Ci} = 9.62 \times 10^6\,\mathrm{Bq}\]This means \(9.62 \times 10^6\) photons are emitted per second.

Step 2: Calculate Energy Delivered to the Tumor per Second

Since half of the emitted photons are absorbed by the tumor, the number of absorbed photons is:\[\text{Absorbed Photons} = \frac{9.62 \times 10^6}{2} = 4.81 \times 10^6\]Each photon has an energy of \(1.25\,\mathrm{MeV}\), so the total energy absorbed per second is:\[\text{Energy} = 4.81 \times 10^6 \times 1.25\,\mathrm{MeV} \times 1.6 \times 10^{-13}\,\mathrm{J/MeV} = 9.625\,\mathrm{J/s}\]

Step 3: Calculate Absorbed Dose per Second

The absorbed dose in rads is given by the energy absorbed per unit mass. Since the tumor has a mass of \(0.200\,\mathrm{kg}\), the absorbed dose per second is:\[\text{Dose (rad)} = \frac{9.625}{0.200} \times 100 = 48.125\,\mathrm{rad/s}\]

Step 4: Calculate Equivalent Dose per Second

To find the equivalent dose in rems, we use the relative biological effectiveness (RBE), given as 0.70 for these \(\gamma\) rays. Thus, the equivalent dose per second is:\[\text{Equivalent Dose (rem/s)} = 48.125 \times 0.70 = 33.6875\,\mathrm{rem/s}\]

Step 5: Calculate Exposure Time for 200 rem Equivalent Dose

To determine the time required for an equivalent dose of 200 rem, use the equivalent dose rate:\[\text{Time (s)} = \frac{200\,\mathrm{rem}}{33.6875\,\mathrm{rem/s}} \approx 5.94\,\mathrm{s}\]

Key Concepts

Radiation Absorption

Gamma radiation is a form of high-energy electromagnetic radiation. When it comes to radiation absorption, it's all about the interaction of these gamma photons with matter, such as a tumor in the body. The effectiveness of absorption depends on several factors, including the type of material and the energy of the radiation. In our case, a cobalt-60 (\(^{60} \text{Co}\)) source emits gamma photons into a tumor. Half of these photons get absorbed by the tumor, delivering energy directly to the tissue. This absorption process is critical, as it determines how much of the emitted energy actually contributes to treating the tumor. The essence of radiation therapy lies in maximizing the absorption of high-energy photons in the affected area to maximize therapeutic effects, while minimizing exposure to surrounding healthy tissues.

Energy Delivery

The concept of energy delivery focuses on how much energy is imparted to the tumor by the trapped gamma photons. This is a key factor because more energy delivery means more effective destruction of tumor cells. From the problem, we know that half of the emitted gamma photons from the cobalt-60 source are absorbed by the tumor. Each photon carries 1.25 mega-electron volts (MeV) of energy. Therefore, calculating the energy delivered involves determining the number of absorbed photons and multiplying it by the energy per photon. To convert this to joules, the energy in MeV is multiplied by a factor of 1.6 x 10^-13 joules per MeV. This gives us the total energy delivered to the tumor per second, a figure crucial for understanding the therapy's impact.

Absorbed Dose

The absorbed dose is a measure of how much energy from the gamma rays is deposited in the tumor tissue. It is typically measured in rads, where 1 rad equals 100 ergs of energy absorbed per gram of tissue. In the problem, we calculate the absorbed dose using the mass of the tumor, given as 0.200 kg (which is 200 grams), and the energy delivered per second. The absorbed dose provides a direct indication of the potential for biological effect on the tissue due to radiation. High absorbed doses mean more energy is absorbed by the tumor, potentially resulting in more effective treatment. Understanding this concept is important for assessing treatment doses and predicting therapeutic outcomes.

Equivalent Dose

The equivalent dose enhances the absorbed dose by accounting for the type of radiation and its biological impact. While absorbed dose gives a basic measure of energy deposition, equivalent dose adjusts this value using a dimensionless factor called the relative biological effectiveness (RBE). The RBE varies with different types of radiation due to their varying impacts on biological cells. For gamma rays, the RBE used in the problem is 0.70. Therefore, the equivalent dose, often measured in rems, is calculated by multiplying the absorbed dose by the RBE. This result helps in assessing the potential biological damage to the tissue. Equivalent dose is particularly useful in safety guidelines and ensuring therapeutic procedures align with dose limits to avoid harmful effects.