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Chapter 10: Problem 21

An electron and a proton are moving with the same speed. Which has the longer de Broglie wavelength? (You may want to look ahead at some useful information in the table in the back inside cover.)

Short Answer

Expert verified
Question: Compare the de Broglie wavelength of an electron and a proton moving with the same speed. Answer: The de Broglie wavelength of the electron is longer than that of the proton when they are moving with the same speed.

Step by step solution

01

Write down the formula for de Broglie wavelength

The formula for calculating the de Broglie wavelength of a particle is given by λ = h / p, where λ is the de Broglie wavelength, h is Planck's constant, and p is the momentum of the particle.
02

Write down the formula for momentum

The formula for calculating the momentum of a particle is given by p = mv, where m is the mass of the particle and v is its speed.
03

Find the momentum of the electron and proton

Since both the electron and the proton have the same speed, we can write down their momenta as: p_electron = m_electron * v p_proton = m_proton * v
04

Write down the formula for de Broglie wavelength in terms of mass and speed

Since λ = h / p and p = mv, we can write the formula for de Broglie wavelength in terms of mass and speed as, λ = h / (m * v).
05

Calculate the de Broglie wavelength of the electron and proton

Now, we can write down the de Broglie wavelengths of the electron and the proton as: λ_electron = h / (m_electron * v) λ_proton = h / (m_proton * v)
06

Compare the wavelengths

To compare the wavelengths, we can form the following ratio: (λ_electron / λ_proton) = ((h / (m_electron * v)) / (h / (m_proton * v))) By simplifying the above equation, we get: (λ_electron / λ_proton) = (m_proton / m_electron) Since the mass of the proton (m_proton) is much larger than the mass of the electron (m_electron), the ratio (λ_electron / λ_proton) is greater than 1. Therefore, the de Broglie wavelength of the electron is longer than that of the proton when they are moving with the same speed.

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