Question

The radioactive 60co isotope is used in nuclear medicine to treat certain types of cancer. Calculate the wavelength and frequency of an emitted gamma particle having the energy of 1.29 × 1011 j/mol. Enter your answer in scientific notation.

Answer 1

1. Frequency:
`3.23\cdot 10^(20) Hz`

The energy given is the energy per mole of particles:

`E=1.29\cdot 10^(11) J/mol`

1 mole contains a number of Avogadro of particles,
`N_A`, equal to

`N_A=6.022\cdot 10^(23)` particles

So, by setting the following proportion, we can calculate the energy of a single photon:

`1.29 \cdot 10^(11) J/mol : 6.022 \cdot 10^(23) ph/mol = E_1 : 1 ph\\E_1 = ((1.29\cdot 10^(11) J/mol)(1 ph))/(6.022\cdot 10^(23) ph/mol)=2.14\cdot 10^(-13) J`

This is the energy of a single photon; now we can calculate its frequency by using the formula:

`E_1 = hf`

where

`h=6.63\cdot 10^(-34) Js` is the Planck's constant

f is the photon frequency

Solving for f, we find

`f=(E_1)/(h)=(2.14\cdot 10^(-13) J)/(6.63\cdot 10^(-34) Js)=3.23\cdot 10^(20) Hz`

2. Wavelength:
`9.29\cdot 10^(-13) m`

The wavelength of the photon is given by the equation:

`\lambda=(c)/(f)`

where

`c=3\cdot 10^8 m/s`

is the speed of the photon (the speed of light). Substituting,

`\lambda=(3 \cdot 10^8 m/s)/(3.23\cdot 10^(20) Hz)=9.29\cdot 10^(-13) m`