Ch 43: Nuclear Physics

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 43: Nuclear Physics

Problem 36

Chapter 43, Problem 36

It has become popular for some people to have yearly whole-body scans (CT scans, formerly called CAT scans) using x rays, just to see if they detect anything suspicious. A number of medical people have recently questioned the advisability of such scans, due in part to the radiation they impart. Typically, one such scan gives a dose of $12$ mSv, applied to the whole body. By contrast, a chest x ray typically administers $0.20$ mSv to only $5.0$ kg of tissue. How many chest x rays would deliver the same total amount of energy to the body of a $75$ -kg person as one whole-body scan?

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding how many chest x-rays would deliver the same total energy to a 75-kg person as one whole-body CT scan. The key is to compare the energy imparted by the two procedures.

Step 2: Recall the relationship between dose (in mSv), energy, and mass. The dose (D) is related to the energy (E) absorbed and the mass (m) of the tissue by the formula:

D = \frac{E}{m}

. Rearranging, the energy absorbed is given by:

E = D \times m

Step 3: Calculate the energy imparted by the whole-body CT scan. The dose is 12 mSv (or 0.012 J/kg in SI units), and the mass of the whole body is 75 kg. Using the formula

E = D \times m

, substitute the values to find the total energy imparted by the CT scan.

Step 4: Calculate the energy imparted by a single chest x-ray. The dose is 0.20 mSv (or 0.0002 J/kg in SI units), and the mass of the tissue exposed is 5.0 kg. Again, use the formula

E = D \times m

to find the energy imparted by one chest x-ray.

Step 5: Determine the number of chest x-rays needed. Divide the total energy imparted by the CT scan (from Step 3) by the energy imparted by one chest x-ray (from Step 4). This will give the number of chest x-rays required to deliver the same total energy as the CT scan.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radiation Dose Measurement

Radiation dose is measured in sieverts (Sv), with millisieverts (mSv) being a common unit for medical imaging. It quantifies the biological effect of ionizing radiation on human tissue. Understanding this measurement is crucial for comparing the radiation exposure from different imaging techniques, such as whole-body scans and chest x-rays.

Energy Absorption in Tissue

When radiation is administered to the body, it interacts with tissues, leading to energy absorption. The amount of energy absorbed depends on the radiation dose and the mass of the tissue exposed. In this context, calculating how many chest x-rays would equal the energy absorbed from a whole-body scan requires knowledge of both the dose and the mass of the tissues involved.

Proportionality in Radiation Exposure

Proportionality in radiation exposure refers to the relationship between the dose of radiation and the mass of tissue it affects. For example, if a chest x-ray delivers a specific dose to a smaller mass of tissue, one must calculate how many such doses are needed to equal the total dose delivered to a larger mass, such as the entire body in a whole-body scan. This concept is essential for determining equivalent radiation exposure across different imaging methods.

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Related Practice

Textbook Question

In a diagnostic x-ray procedure, $5.00 \times 10^{10}$ photons are absorbed by tissue with a mass of $0.600$ kg. The x-ray wavelength is $0.0200$ nm.

(a) What is the total energy absorbed by the tissue?

(b) What is the equivalent dose in rem?

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Textbook Question

Measurements on a certain isotope tell you that the decay rate decreases from $8318$ decays/min to $3091$ decays/min in $4.00$ days. What is the half-life of this isotope?

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Textbook Question

Calculate the energy released in the fusion reaction: $_{2}^{3} H e +_{1}^{2} H \to_{2}^{4} H e +_{1}^{1} H$

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Textbook Question

The unstable isotope $to the 40th power K$ is used for dating rock samples. Its half-life is $1.28 \times 10^{9}$ y.

(a) How many decays occur per second in a sample containing $1.63 \times 10^{- 6}$ g of $to the 40th power K$ ?

(b) What is the activity of the sample in curies?

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Textbook Question

A $67$ -kg person accidentally ingests $0.35$ Ci of tritium.

(a) Assume that the tritium spreads uniformly throughout the body and that each decay leads on the average to the absorption of $5.0$ keV of energy from the electrons emitted in the decay. The half-life of tritium is $12.3$ y, and the RBE of the electrons is $1.0$ . Calculate the absorbed dose in rad and the equivalent dose in rem during one week.

(b) The $β^{-}$ decay of tritium releases more than $5.0$ keV of energy. Why is the average energy absorbed less than the total energy released in the decay?

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Textbook Question

At an archeological site, a sample from timbers containing $500$ g of carbon provides $2690$ decays/min. What is the age of the sample?

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