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Chapter 3: Problem 66

How does de Broglie's hypothesis that electrons behave like waves explain the stability of the electron orbits in a hydrogen atom?

Short Answer

Expert verified
In summary, the stability of electron orbits in a hydrogen atom can be explained through de Broglie's hypothesis, which proposes that electrons possess both wave-like and particle-like properties. According to this wave-particle duality, the electron's wave-like behavior results in standing waves around the nucleus, which can only exist in specific quantized orbits that allow for constructive interference. These quantized orbits correspond to the observed energy levels in hydrogen atoms and help explain the stability of electron orbits in a hydrogen atom.

Step by step solution

01

1. Understand de Broglie's Hypothesis

First, we need to understand de Broglie's hypothesis which states that every particle, particularly electrons, has both wave-like and particle-like properties. This idea is known as wave-particle duality. De Broglie proposed that the wavelength of an electron (or any other particle) is inversely proportional to its momentum, as described by the equation: $$\lambda = \frac{h}{p}$$ where \(\lambda\) is the wavelength, \(h\) is Planck's constant, and \(p\) is the particle's momentum. This equation links the wave-like and particle-like properties of electrons, enabling us to treat them like waves when necessary.
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2. Wave behavior around the hydrogen atom

As electrons are in motion around the nucleus of the hydrogen atom, their wave properties become significant. According to de Broglie's hypothesis, the electron's wave-like behavior results in standing waves around the nucleus to maintain stable orbits. A standing wave is formed when a wave interferes constructively with its reflection or another wave. In the case of the hydrogen atom, this means that the electron wave must constructively interfere with itself for the electron to maintain a stable orbit. In other words, the circumference of the electron's orbit must be an integer multiple of its wavelength.
03

3. Quantization of electron orbits

For an electron's standing wave to create a stable orbit, the following condition must be met: $$n\lambda = 2\pi r$$ where \(n\) is an integer (1, 2, 3, ...), \(\lambda\) is the electron's wavelength, and \(r\) is the radius of the electron's orbit. This equation shows that only specific values of \(r\) (i.e., quantized orbits) will lead to constructive interference in the standing wave. If the electron's orbit does not meet this condition, the electron wave will destructively interfere with itself, and the orbit will not be stable.
04

4. Stable electron orbits in hydrogen atom

De Broglie's hypothesis explains the stability of electron orbits in a hydrogen atom through the wave-particle duality concept. When the electron's wave-like properties are considered, it becomes clear that the electron can only exist in specific, quantized orbits that allow for constructive interference of the wave. These stable orbits correspond to the quantized energy levels that are observed in hydrogen atoms and are the basis of the Bohr model for atomic structure. The wave-particle duality of electrons and the resulting quantization of orbits help explain the stability of electron orbits in a hydrogen atom and provide an important foundation for our understanding of atomic structure.

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