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Chapter 10: Problem 23

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
Answer: The electron has a longer de Broglie wavelength when both particles are moving at the same speed.

Step by step solution

01

de Broglie wavelength formula

Recall that the de Broglie wavelength (λ) of a particle is given by: λ = h / p where h is Planck's constant, and p is the momentum of the particle.
02

Find the momentum

For both the electron and the proton to have the same speed (v), their momentum (p) must be calculated: p = mv where m is the mass of the particle and v is the speed.
03

Substitute momentum into the de Broglie wavelength formula

Now, substitute the formula for momentum (p = mv) into the de Broglie wavelength formula (λ = h / p): λ = h / (mv)
04

Compare masses

Since both the electron and proton are moving at the same speed, we can compare their wavelengths by looking at their masses. Note that the mass of an electron (m_e) is smaller than the mass of a proton (m_p): m_e < m_p
05

Determine which has a longer de Broglie wavelength

We can now determine which particle has a longer de Broglie wavelength by observing the relationship between mass and wavelength in the formula λ = h / (mv). Since the electron mass (m_e) is smaller than the proton mass (m_p), when the speed (v) is the same for both particles, the denominator will be smaller for the electron than for the proton. Thus, the wavelength of the electron will be larger: λ_e > λ_p The electron has a longer de Broglie wavelength when both particles are moving at the same speed.

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