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Chapter 27: Problem 55

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

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

Step by step solution

01

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
02

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
03

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.
04

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)
05

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.

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Most popular questions from this chapter

A photon of energy \(240.0 \mathrm{keV}\) is scattered by a free electron. If the recoil electron has a kinetic energy of \(190.0 \mathrm{keV},\) what is the wavelength of the scattered photon?
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