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Chapter 43: Problem 37

In a diagnostic \(x\) -ray procedure, \(5.00 \times 10^{10}\) photons are absorbed by tissue with a mass of \(0.600 \mathrm{~kg}\). The x-ray wavelength is \(0.0200 \mathrm{nm}\). (a) What is the total energy absorbed by the tissue? (b) What is the equivalent dose in rem?

Short Answer

Expert verified
The total energy absorbed by the tissue is calculated in step 2, and the equivalent dose in rem is calculated in step 3. Fill in these values based on the calculations you made in those steps.

Step by step solution

01

Calculate the Energy of a Single X-ray Photon

The energy \(E\) of one photon can be calculated using the Planck's equation \(E = hf\), where \(h\) is the Planck's constant and \(f\) is the frequency of radiation. However, since we are given the wavelength, we can use the relation \(c = \lambda f\), where \(c\) is the speed of light and \(\lambda\) is the wavelength, to express the frequency in terms of wavelength. This leads to \(E = \frac{hc}{\lambda}\). Substituting \(h = 6.63 \times 10^{-34} J.s\), \(\lambda = 0.0200 \times 10^{-9} m\), and \(c = 3.00 \times 10^{8} m/s\), we can calculate \(E\).
02

Calculate the Total Energy Absorbed by the Tissue

The total energy absorbed by the tissue is simply the energy of one photon times the total number of photons. Having determined the energy of one photon in the previous step, we now multiply it by the total number of photons given in the problem, \(5.00 \times 10^{10}\), to get the total energy absorbed.
03

Calculate the Equivalent Dose in Rem

Considering that 1 rem is equivalent to \(1.00 \times 10^{-2} J/kg\), the dose in rems is calculated by dividing the total energy absorbed (found in step 2) by the mass of the tissue, and then converting from Joules/kilogram to rem. The mass of the tissue is given as \(0.600 kg\). Dividing the total energy by this mass, and then multiplying it by 100 gives us the dose in rem units.

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Key Concepts

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

Energy Absorption
When tissue absorbs X-ray radiation, it takes in the photon's energy. This energy transfer is crucial because it directly measures how much energy the body tissues receive during exposure to X-rays. In our case, the tissue absorbs energy from thenumber of X-ray photons specified.
To calculate the total energy absorbed, we follow these steps:
  • First, find the energy of a single photon using the energy formula for electromagnetic radiation: \[ E = \frac{hc}{\lambda} \]where \( h \) is Planck's constant \( (6.63 \times 10^{-34} \; J\cdot s) \), \( c \) is the speed of light \( (3.00 \times 10^{8} \; m/s) \), and \( \lambda \) is the wavelength (converted to meters).
  • Next, multiply the energy of one photon by the total number of photons absorbed (\(5.00 \times 10^{10}\)) to get the total energy absorbed by the tissue.
By understanding these steps, one can comprehend how much energy different tissues absorb from X-ray radiation. Keep this understanding handy, as it's essential in areas such as medical diagnostics and radiation therapy.
Photon Energy Calculation
Photon energy is vital in determining the impact of radiation on biological tissues. Each photon carries a certain amount of energy depending on its wavelength.
Using Planck's formula, the energy of a photon is given by:\[ E = \frac{hc}{\lambda} \]Here is a breakdown:
  • \(h\) (Planck's constant) helps relate the energy to the frequency of the photon. The value is always \(6.63 \times 10^{-34} \; J\cdot s\).
  • \(c\) (speed of light) is \(3.00 \times 10^{8} \; m/s\).
  • \(\lambda\) is the wavelength, often measured in nanometers (\(nm\)), but needs to be converted to meters for calculations.
Substituting the values into the formula yields the energy of one X-ray photon. Understanding these calculations provides insights into the frequency and energy relationship of electromagnetic waves, including X-ray.
Equivalent Dose in Rem
To assess the biological effect of radiation, we convert the absorbed energy into a dose unit called "rem." The equivalent dose in rem takes into account both the energy and the type of radiation interacting with the tissue.
Calculating the dose in rem involves:
  • Finding the absorbed energy per kilogram of tissue, called the absorbed dose. This calculation divides the total energy absorbed by the mass of the tissue.
  • The absorbed dose (in Javaflocker) is then converted to rem by multiplying by the factor \(100\) because \(1 rem = 1.00 \times 10^{-2} J/kg\).
This conversion is crucial for assessing potential biological damage due to radiation exposure. In medical settings, understanding rem helps in balancing the benefits and risks of radiobiological procedures. Proper calculation ensures patient safety while optimizing diagnostic effectiveness.

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