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Chapter 43: Problem 63

A \(^6$$^0\)Co source with activity 2.6 \(\times\) 10\(^-$$^4\) Ci is embedded in a tumor that has mass 0.200 kg. The source emits \(\gamma\) photons with average energy 1.25 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 g rays is 0.70? (d) What exposure time is required for an equivalent dose of 200 rem?

Short Answer

Expert verified
(a) 9.65 脳 10鈦烩伔 J/s, (b) 4.825 脳 10鈦烩伌 rad/s, (c) 3.378 脳 10鈦烩伌 rem/s, (d) 164.44 hours.

Step by step solution

01

Calculate the Energy Emitted per Second

First, let's find the number of disintegrations per second, which is the activity in Becquerels (1 Ci = 3.7 脳 10鹿鈦� Bq). So, for the activity of 2.6 脳 10鈦烩伌 Ci, we get \( A = 2.6 \times 10^{-4} \times 3.7 \times 10^{10} \) \( = 9.62 \times 10^6 \text{ disintegrations per second} \). Each disintegration emits a photon with 1.25 MeV, converting it to joules gives \( 1.25 \times 1.602 \times 10^{-13} \text{ J} \). Thus, energy emitted per second is \( 9.62 \times 10^6 \times 1.25 \times 1.602 \times 10^{-13} \approx 1.93 \times 10^{-6} \text{ J/s} \).
02

Determine Energy Absorbed by Tumor per Second

Since half of the photons are absorbed by the tumor, the energy absorbed per second is half the energy emitted per second. Therefore, \( E_{\text{absorbed}} = \frac{1.93 \times 10^{-6}}{2} = 9.65 \times 10^{-7} \text{ J/s} \).
03

Calculate Absorbed Dose in rad

The absorbed dose (D) in rad is given by \( D = \frac{E}{m} \times 100 \), where E is the energy absorbed in joules and m is the mass in kg. Here, \( D = \frac{9.65 \times 10^{-7}}{0.200} \times 100 = 4.825 \times 10^{-4} \text{ rad/s} \).
04

Compute Equivalent Dose in rem

The equivalent dose (H) in rem is calculated by multiplying absorbed dose with the relative biological effectiveness (RBE). Given RBE is 0.70, we find \( H = 4.825 \times 10^{-4} \times 0.70 = 3.378 \times 10^{-4} \text{ rem/s} \).
05

Determine Time for Equivalent Dose of 200 rem

To find the time required for a certain equivalent dose, we use \( \, \text{Equivalent Dose} = \, H \times t \) and solve for time, \( t = \frac{200}{3.378 \times 10^{-4}} \approx 5.92 \times 10^5 \text{ seconds} \). Converting seconds to hours, we get \( \frac{5.92 \times 10^5}{3600} \approx 164.44 \text{ hours} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radioactive Decay
Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This radiation can be in the form of alpha particles, beta particles, or gamma rays. Understanding radioactive decay helps us comprehend the behavior of radioactive materials such as Cobalt-60, a radionuclide frequently used in medical treatments and research.
  • During decay, a radioactive substance undergoes disintegration, leading to the emission of radiation.
  • The rate of decay is measured in terms of activity, expressed in units like Becquerels (Bq) or Curies (Ci), which denote disintegrations per second.
  • This decay process decreases the number of radioactive atoms over time, following an exponential decay law.
Radioactive decay is instrumental in applications ranging from cancer treatment to archaeological dating. It is crucial to calculate the emitted energy accurately, especially in medical applications to administer precise doses.
Absorbed Dose
The absorbed dose is a measure of the energy deposited by radiation in a unit mass of tissue. This concept plays a significant role in evaluating the potential damage radiation can cause to living tissues.
  • The absorbed dose is commonly measured in rads or grays, with 1 gray equivalent to 100 rads.
  • It is calculated as the energy absorbed divided by the tissue mass, giving insights into the concentration of radiation energy delivered.
In the context of radiation therapy, knowing the absorbed dose helps clinicians optimize treatment plans to maximize tumor destruction while minimizing damage to surrounding healthy tissue. Thus, precision in calculating the absorbed dose ensures both efficacy and safety in therapeutic procedures.
Equivalent Dose
The equivalent dose takes into account not only the absorbed dose but also the type of radiation and its varying biological effects. This metric is expressed in sieverts (Sv) or rems, where 1 Sv equals 100 rems.
  • To calculate the equivalent dose, multiply the absorbed dose by the relative biological effectiveness (RBE) factor, which denotes the biological impact of the type of radiation.
  • Different radiations have different RBE values; gamma rays have a lower RBE compared to alpha particles.
Using the equivalent dose, professionals can assess the potential biological effects of radiation on humans, guiding safety standards and regulatory limits for radiation exposure. This evaluation ensures human tissue is exposed to levels within manageable risk margins.
Cobalt-60
Cobalt-60 is a synthetic radioactive isotope of cobalt known for its applications in medical and industrial settings. With a half-life of about 5.27 years, it serves as a prominent source of gamma rays.
  • In medical fields, Cobalt-60 is utilized in radiotherapy for treating cancer, leveraging its ability to emit high-energy gamma radiation to target and destroy tumor cells.
  • Industrial uses include sterilization of medical equipment and non-destructive testing of materials.
Its stability and gamma emission make Cobalt-60 invaluable in various fields requiring precise radiation application. Due to its intense radiation, it must be handled with caution, strictly adhering to safety protocols to prevent unwanted exposure.
Relative Biological Effectiveness (RBE)
Relative Biological Effectiveness (RBE) is a measure that compares the biological effectiveness of different types of radiation. RBE is crucial in understanding the varying impacts of different radiation types despite having similar absorbed doses.
  • The RBE reflects how particles like alpha or gamma rays interact with biological tissues, leading to different extents of damage.
  • This factor is significant when calculating equivalent doses, as it allows for translating absorbed energy into potential biological harm.
By evaluating the RBE, scientists and medical professionals can tailor radiation treatments and safety measures more precisely according to the specific radiation type encountered. Thus, RBE is an essential tool for ensuring accurate assessments of radiation risks and effectiveness.

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Most popular questions from this chapter

What nuclide is produced in the following radioactive decays? (a) \(\alpha\) decay of \(^{239}_{94}Pu\); (b) \(\beta$$^-\) decay of \(^{24}_{11}Na\); (c) \(\beta$$^+\) decay of \(^{15}_{8}O\).

The polonium isotope \(^{210}_{84}Po\) has atomic mass 209.982874 u. Other atomic masses are \(^{206}_{82}Pb\), 205.974465 u; \(^{209}_{83}Bi\), 208.980399 u; \(^{210}_{83}Bi\), 209.984120 u; \(^{209}_{84}Po\), 208.982430 u; and \(^{210}_{85}At\), 209.987148 u. (a) Show that the alpha decay of \(^{210}_{84}Po\) is energetically possible, and find the energy of the emitted a particle. (b) Is \(^{210}_{84}Po\) energetically stable with respect to emission of a proton? Why or why not? (c) Is \(^{210}_{84}Po\) energetically stable with respect to emission of a neutron? Why or why not? (d) Is \(^{210}_{84}Po\) energetically stable with respect to \(\beta$$^-\) decay? Why or why not? (e) Is \(^{210}_{84}Po\)energetically stable with respect to \(\beta$$^+\) decay? Why or why not?

The radioactive nuclide \(^1$$^9$$^9\)Pt has a half-life of 30.8 minutes. A sample is prepared that has an initial activity of 7.56 \(\times\) 10\(^1$$^1\) Bq. (a) How many \(^1$$^9$$^9\)Pt nuclei are initially present in the sample? (b) How many are present after 30.8 minutes? What is the activity at this time? (c) Repeat part (b) for a time 92.4 minutes after the sample is first prepared.

A radioactive isotope has a half-life of 43.0 min. At \(t\) = 0 its activity is 0.376 Ci. What is its activity at \(t\) = 2.00 h?

Which reaction produces \(^{131}\)Te in the nuclear reactor? (a) \(^{130}Te + n \rightarrow ^{131}Te\); (b) \(^{130}I + n \rightarrow ^{131}Te\); (c) \(^{132}Te + n \rightarrow ^{131}Te\); (d) \(^{132}I + n \rightarrow ^{131}Te\).
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