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Chapter 6: Problem 82

Find the wavelength and energy of a photon with momentum \(5.00 \times 10^{-29} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\).

Short Answer

Expert verified
The wavelength of the photon is approximately \(1.325 \times 10^{-6}\,\mathrm{m}\) and its energy is approximately \(1.50 \times 10^{-20}\,\mathrm{J}\).

Step by step solution

01

Write down the de Broglie's equation

The de Broglie's equation states the relationship between the momentum of a particle, its wavelength, and the Planck's constant: \(λ = \frac{h}{p}\) Where: \(λ\) is the wavelength of the photon, \(h\) is Planck's constant (\(6.626 \times 10^{-34} \mathrm{J} \cdot \mathrm{s}\)), \(p\) is the momentum of the photon.
02

Find the wavelength

We are given the momentum (\(p = 5.00 \times 10^{-29} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\)). We can now calculate the wavelength: \(λ = \frac{h}{p} = \frac{6.626 \times 10^{-34} \mathrm{J} \cdot \mathrm{s}}{5.00 \times 10^{-29} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}}\) \(λ \approx 1.325 \times 10^{-6} \mathrm{m}\) The wavelength is approximately \(1.325 \times 10^{-6}\,\mathrm{m}\), which is in the infrared region of the electromagnetic spectrum.
03

Write down the formula relating energy, momentum, and the speed of light

The formula for the energy of a photon in terms of its momentum and the speed of light is: \(E = pc\) Where: \(E\) is the energy of the photon, \(p\) is the momentum of the photon, \(c\) is the speed of light (\(3.00 \times 10^8 \mathrm{m} / \mathrm{s}\)).
04

Find the energy of the photon

Now, we can calculate the energy of the photon using the momentum and the speed of light: \(E = pc = (5.00 \times 10^{-29} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}) (3.00 \times 10^8 \mathrm{m} / \mathrm{s})\) \(E \approx 1.50 \times 10^{-20} \mathrm{J}\) The energy of the photon is approximately \(1.50 \times 10^{-20}\,\mathrm{J}\). In conclusion, the wavelength of the photon is \(1.325 \times 10^{-6}\,\mathrm{m}\) and its energy is \(1.50 \times 10^{-20}\,\mathrm{J}\).

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