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Chapter 24: Problem 65

A small region of a cancer patient's brain is exposed for 24.0 min to 475 Bq of radioactivity from \({ }^{60}\) Co for treatment of a tumor. If the brain mass exposed is \(1.858 \mathrm{~g}\) and each \(\beta^{-}\) particle emitted has an energy of \(5.05 \times 10^{-14} \mathrm{~J},\) what is the dose in rads?

Short Answer

Expert verified
1.86 \times 10^{-3} rads

Step by step solution

01

- Calculate Total Energy Emitted

To determine the total energy emitted, we first need to calculate the number of decays. Use the formula: \[ \text{Number of decays} = \text{Activity} \times \text{Time} \] Given \[ \text{Activity} = 475 \text{ Bq} = 475 \text{ decays/second} \] and \[ \text{Time} = 24.0 \text{ minutes} = 24.0 \times 60 \text{ seconds} = 1440 \text{ seconds} \] Thus, \[ \text{Number of decays} = 475 \text{ decays/second} \times 1440 \text{ seconds} = 684000 \text{ decays} \]
02

- Calculate Total Energy Released

Each decay releases an energy of \[ 5.05 \times 10^{-14} \text{ J} \] So, the total energy released is: \[ \text{Total Energy} = \text{Number of decays} \times \text{Energy per decay} \] \[ \text{Total Energy} = 684000 \text{ decays} \times 5.05 \times 10^{-14} \text{ J/decay} \] \[ \text{Total Energy} = 3.4542 \times 10^{-8} \text{ J} \]
03

- Convert to Grays (Gy)

The dose in grays can be found using the formula: \[ \text{Dose (Gy)} = \frac{\text{Total Energy (J)}}{\text{Mass (kg)}} \] Given the mass, convert from grams to kilograms: \[ \text{Mass} = 1.858 \text{ g} = 1.858 \times 10^{-3} \text{ kg} \] Hence, \[ \text{Dose (Gy)} = \frac{3.4542 \times 10^{-8} \text{ J}}{1.858 \times 10^{-3} \text{ kg}} = 1.86 \times 10^{-5} \text{ Gy} \]
04

- Convert to Rads

Now, convert grays to rads, knowing that: \[ 1 \text{ Gy} = 100 \text{ rads} \] Therefore, \[ \text{Dose (rads)} = 1.86 \times 10^{-5} \text{ Gy} \times 100 \text{ rads/Gy} \] \[ \text{Dose (rads)} = 1.86 \times 10^{-3} \text{ rads} \]

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Key Concepts

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

radioactivity
Radioactivity is the process by which unstable atomic nuclei lose energy by emitting radiation. This decay happens over time, transforming the original nucleus into a more stable state. There are different types of radioactive decay, including alpha, beta, and gamma decay. The rate at which these decays happen is measured in units called Becquerels (Bq), where one Bq equals one decay per second. Understanding radioactivity is crucial in fields such as medicine, where it is used in radiation therapy to treat cancers.
beta particle emission
Beta particles are high-energy, high-speed electrons or positrons emitted during radioactive decay processes like beta decay. In the specific case of our exercise involving \textsuperscript{60}Co, beta particles are emitted which have significant energy. Each \textbeta^{-} particle emitted in our exercise has an energy of \( 5.05 \times 10^{-14}\ \mathrm{J} \). Calculating the number of such emissions helps determine the total energy released, which is crucial for calculating the radiation dose.
radiation dose in rads
The radiation dose measures the amount of energy deposited by ionizing radiation in a given mass of tissue. There are different units to measure this dose. The gray (Gy) is a standard unit that represents one joule of radiation energy deposited per kilogram of tissue. However, in the exercise, it converts to rads, where 1 Gy equals 100 rads. Understanding this conversion is essential to comprehend the magnitude of radiation exposure.
radiation therapy
Radiation therapy is a treatment method that uses high doses of radiation to kill or damage cancer cells. It works by damaging the DNA inside these cells, causing them to die or stop dividing. The effectiveness of radiation therapy depends on delivering the precise amount of radiation to the targeted area. In the exercise, the patient's brain mass exposed to radioactivity is treated with specific calculations to determine the dose required for effective treatment, ensuring minimal damage to surrounding healthy tissue.
energy conversion
Energy conversion in the context of the exercise involves determining the total energy released during radioactive decay and then converting that energy into a form that can be used to measure the radiation dose. The total energy released by beta particles must be calculated first, followed by converting this energy into grays and finally into rads. This process involves several steps: starting from calculating the activity, determining the number of decays, finding the total energy, and converting it to the relevant radiation dose units. These conversions ensure that the calculated radiation dose accurately reflects the energy absorbed by the tissue, which is vital for effective cancer treatment.

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Most popular questions from this chapter

Uranium and radium are found in many rocky soils throughout the world. Both undergo radioactive decay, and one of the products is radon-222, the heaviest noble gas \(\left(t_{1 / 2}=\right.\) 3.82 days). Inhalation of this gas contributes to many lung cancers. According to the Environmental Protection Agency, the level of radioactivity from radon in homes should not exceed \(4.0 \mathrm{pCi} / \mathrm{L}\) of air. (a) What is the safe level of radon in Bq/L of air? (b) A home has a radon measurement of \(41.5 \mathrm{pCi} / \mathrm{L}\). The owner vents the basement in such a way that no more radon enters the living area. What is the activity of the radon remaining in the room air (in Bq/L) after 9.5 days? (c) How many more days does it take to reach the EPA recommended level?

Write balanced nuclear equations for the following: (a) Formation of \({ }_{22}^{48} \mathrm{Ti}\) through positron emission (b) Formation of silver- 107 through electron capture (c) Formation of polonium- 206 through \(\alpha\) decay

If \(1.00 \times 10^{-12} \mathrm{~mol}\) of \({ }^{135} \mathrm{Cs}\) emits \(1.39 \times 10^{3} \beta^{-}\) particles in \(1.00 \mathrm{yr},\) what is the decay constant?

Write balanced nuclear equations for the following: (a) \(\beta^{-}\) decay of sodium- 26 (b) \(\beta^{-}\) decay of francium- 223 (c) Alpha decay of \({ }^{212} \mathrm{Bi}\) i

Why might it be difficult to use only a nuclide's \(N / Z\) ratio to predict whether it will decay by \(\beta^{+}\) emission or by e \(^{-}\) capture? What other factor is important?
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