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Chapter 8: Problem 39

Which must possess a greater velocity to produce matter waves of the same wavelength (such as \(1 \mathrm{nm}\) ), protons or electrons? Explain your reasoning.

Short Answer

Expert verified
Electrons must possess a greater velocity compared to protons to produce matter waves of the same wavelength, as electrons have a smaller mass than protons.

Step by step solution

01

Recall the formula between de Broglie wavelength and momentum

The fundamental principle underlying this exercise is the de Broglie's wavelength formula, \(\lambda = \frac{h}{p}\), where \(\lambda\) is the de Broglie wavelength, \(h\) is the Planck's constant and \(p\) is the momentum of the particle.
02

Understand the relationship between momentum and velocity

Momentum can be determined using the formula \(p = mv\), where m is the mass of the particle and v is its velocity. Thus, the velocity can be found using the formula:\(v = \frac{p}{m}\).
03

Determine the velocity needed for each particle to have the same wavelength

The same wavelength can be achieved by having the same momentum, according to the de Broglie's wavelength formula. However, having the same momentum means having a different velocity, as the electrons and the protons have different masses. Now, since the mass of a proton is much larger than the mass of an electron, the smaller mass of the electron implies that it must move faster than the proton.

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Most popular questions from this chapter

An atom in which just one of the outer-shell electrons is excited to a very high quantum level \(n\) is called a "high Rydberg" atom. In some ways, all these atoms resemble a Bohr hydrogen atom with its electron in a high-numbered orbit. Explain why you might expect this to be the case.

Calculate the wavelength of the electromagnetic radiation required to excite a proton from the ground state to the level with \(n=4\) in a one-dimensional box 50. pm long.

High-pressure sodium vapor lamps are used in street lighting. The two brightest lines in the sodium spectrum are at 589.00 and \(589.59 \mathrm{nm}\). What is the difference in energy per photon of the radiations corresponding to these two lines?

What electron transition in a hydrogen atom, ending in the orbit \(n=3,\) will produce light of wavelength \(1090 \mathrm{nm} ?\)

Without doing detailed calculations, indicate which of the following electromagnetic radiations has the greatest energy per photon and which has the least: (a) \(662 \mathrm{nm}\) (b) \(2.1 \times 10^{-5} \mathrm{cm} ;\) (c) \(3.58 \mu \mathrm{m} ;\) (d) \(4.1 \times 10^{-6} \mathrm{m}\).
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