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Chapter 7: Problem 138

The radioactive \(\mathrm{Co}-60\) isotope is used in nuclear medicine to treat certain types of cancer. Calculate the wavelength and frequency of an emitted gamma photon having the energy of \(1.29 \times 10^{11} \mathrm{~J} / \mathrm{mol} .\)

Short Answer

Expert verified
The frequency of the emitted gamma photon is approximately \(3.23 \times 10^{20}\) Hz and the wavelength is approximately \(9.28 \times 10^{-14}\) m.

Step by step solution

01

Convert given energy from Joules/mol to Joules/photon

Given energy of gamma photon is \(1.29 \times 10^{11}\) J/mol. To convert from J/mol to J/photon, we use the Avogadro's number which is \(6.022 \times 10^{23}\) photons/mol. So, the energy per photon would be \( \frac{1.29 \times 10^{11}}{6.022 \times 10^{23}} \) J/photon, which is approximately \(2.14 \times 10^{-13}\) J/photon.
02

Calculate frequency using Planck-Einstein relation

Using Planck's constant \(h = 6.626 \times 10^{-34}\) Js, we can find the frequency using Planck-Einstein relation \(E = hv\). Therefore, frequency \(v = \frac{E}{h} = \frac{2.14 \times 10^{-13}}{6.626 \times 10^{-34}}\) s^-1, which calculates to approximately \(3.23 \times 10^{20}\) Hz.
03

Calculate wavelength using wave equation

Finally, using speed of light \(c = 299792458\) m/s, we can find the wavelength using the wave equation \(v = c/λ\). Therefore, wavelength \(λ = \frac{c}{v} = \frac{299792458}{3.23 \times 10^{20}}\) m , which calculates to approximately \(9.28 \times 10^{-14}\) m.

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