In a diagnostic x-ray procedure, 5.00×10105.00\$$times10^{10}$$ photons are absorbed by tissue with a mass of 0.6000.600 kg. The x-ray wavelength is 0.0200 0.0200 nm.(a) What is the total energy absorbed by the tissue?(b) What is the equivalent dose in rem?

Ch 43: Nuclear Physics

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 43: Nuclear Physics

Problem 39

Chapter 43, Problem 39

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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Step 1: Calculate the energy of a single photon using the formula $ E = \frac{hc}{\lambda} $, where $ h $ is Planck's constant ($ 6.626 \times 10^{-34} \ \text{J·s} $), $ c $ is the speed of light ($ 3.00 \times 10^8 \ \text{m/s} $), and $ \lambda $ is the wavelength of the x-ray ($ 0.0200 \ \text{nm} = 0.0200 \times 10^{-9} \ \text{m} $). Substitute the values to find the energy of one photon.

Step 2: Multiply the energy of a single photon by the total number of photons absorbed ($ 5.00 \times 10^{10} $) to calculate the total energy absorbed by the tissue. Use the formula $ E_{\text{total}} = E_{\text{photon}} \times N $, where $ N $ is the number of photons.

Step 3: To calculate the equivalent dose in rem, first determine the absorbed dose in grays (Gy). Use the formula $ D = \frac{E_{\text{total}}}{m} $, where $ E_{\text{total}} $ is the total energy absorbed (in joules) and $ m $ is the mass of the tissue ($ 0.600 \ \text{kg} $). This gives the absorbed dose in joules per kilogram, which is equivalent to grays (1 Gy = 1 J/kg).

Step 4: Convert the absorbed dose in grays to the equivalent dose in sieverts (Sv) by multiplying by the radiation weighting factor ($ W_R $). For x-rays, $ W_R = 1 $, so the equivalent dose in sieverts is the same as the absorbed dose in grays.

Step 5: Convert the equivalent dose from sieverts to rem by using the conversion factor $ 1 \ \text{Sv} = 100 \ \text{rem} $. Multiply the equivalent dose in sieverts by 100 to find the equivalent dose in rem.

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Key Concepts

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

Photon Energy

The energy of a photon is determined by its wavelength and can be calculated using the formula E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. In this context, the x-ray wavelength of 0.0200 nm is crucial for determining the energy of each photon absorbed by the tissue.

Total Energy Absorbed

The total energy absorbed by the tissue can be found by multiplying the energy of a single photon by the total number of photons absorbed. This is expressed mathematically as Total Energy = Number of Photons × Energy per Photon. This calculation is essential for understanding the impact of the x-ray exposure on the tissue.

Equivalent Dose

Equivalent dose is a measure of the biological effect of radiation on human tissue, expressed in rem or sieverts. It accounts for the type of radiation and its impact on different tissues. To calculate the equivalent dose, the absorbed dose (in gray) is multiplied by a quality factor that reflects the relative biological effectiveness of the radiation type, which is important for assessing health risks.

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In a diagnostic x-ray procedure, 5.00×10105.00\(\times\)10^{10} photons are absorbed by tissue with a mass of 0.6000.600 kg. The x-ray wavelength is 0.0200 0.0200 nm.(a) What is the total energy absorbed by the tissue?(b) What is the equivalent dose in rem?

Verified video answer for a similar problem:

Key Concepts

Photon Energy

Total Energy Absorbed

Equivalent Dose

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?