Question

A photon passes near a nucleus and creates an electron and a positron, each with a total energy of \(8.0 \mathrm{MeV} .\) What was the wavelength of the photon?

Short answer

Answer: The wavelength of the photon is approximately 7.731 nm.

Step-by-step solution

Express the total energy of both particles#create_latexcode_in_later_steps

Given that the total energy of each created particle (electron and positron) is 8.0 MeV, we can write the total energy as: E_total = E_electron + E_positron = 2 * 8.0 MeV

Convert the energy from MeV to Joules#create_latexcode_in_later_steps

Since we will use the energy and wavelength relationship of a photon, we have to convert the given energy in MeV to Joules. The conversion factor is 1 MeV = 1.602 x 10^{-13} J. E_total = 2 * 8.0 MeV * (1.602 x 10^{-13} J/MeV) = 2.5632 x 10^{-12} J

Use the energy-wavelength relationship for photons#create_latexcode_in_later_steps

The energy (E) of a photon is related to its wavelength (λ) using the equation: E = h * c / λ, where h is the Planck's constant (6.626 x 10^{-34} Js) and c is the speed of light (3.0 x 10^8 m/s). We need to solve for the wavelength λ.

Rearrange the equation and solve for the wavelength#create_latexcode_in_later_steps

Rearrange the equation to solve for λ: λ = h * c / E Now, plug in the values for E (2.5632 x 10^{-12} J), h (6.626 x 10^{-34} Js), and c (3.0 x 10^8 m/s) into the equation: λ = (6.626 x 10^{-34} Js) * (3.0 x 10^8 m/s) / (2.5632 x 10^{-12} J)

Calculate the result and express it in nanometers (nm)#create_latexcode_in_later_steps

Performing the calculation, we get: λ = 7.731 x 10^{-12} m To express the result in nanometers (nm), we need to multiply by 10^9: λ = 7.731 x 10^{-12} m * 10^9 nm/m = 7.731 nm So, the wavelength of the photon is approximately 7.731 nm.