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Chapter 7: Problem 36

How does de Broglie's hypothesis account for the fact that the energies of the electron in a hydrogen atom are quantized?

Short Answer

Expert verified
De Broglie's hypothesis, by introducing the idea of wave-particle duality, allows for the behavior of electrons in a hydrogen atom to be modeled as standing waves in certain permissible orbits. This leads to the concept of energy quantization, as the electron can only have the specific energy levels associated with these orbits, and changes in energy level require specific amounts of energy to be absorbed or emitted.

Step by step solution

01

Understand de Broglie's Hypothesis

De Broglie's hypothesis suggests that every particle has a dual nature. It can exhibit either wave-like properties or particle-like properties based on the situation. This idea of wave-particle duality was groundbreaking in quantum physics, and is the underlying theory behind quantum mechanics. In the context of electron behavior, this is why electrons exhibit wave-like properties when revolving around the nucleus.
02

Establish a Standing Wave in the Electron's Orbit

Now, consider the electron moving around the nucleus in a circular orbit. According to de Broglie, it behaves like a standing wave in this orbital path. A standing wave condition is one where the wave appears stationary, i.e., it does not seem to move in any particular direction. For a standing wave condition to exist, the circumference of the orbit needs to be an integral multiple of the wavelength (lambda) of the electron. So, 2 \pi r = n \lambda, the de Broglie relation, where 'r' is the radius of the orbit and 'n' is a natural number known as the quantum number.
03

Determine Energy Quantization from the Wave Condition

The wave condition reveals that only certain specific orbits are permissible, which are those where the circumference is an integral multiple of the wavelength. This leads to the concept of energy quantization - the electron can only occupy these certain orbits and therefore can only have certain specific energy levels. This results in the energies of the electron being quantized, because jumping from one orbit to another involves absorbing or emitting specific 'quanta' or amounts of energy, that correspond to the energy difference between those orbits.

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Most popular questions from this chapter

An electron in a hydrogen atom is excited from the ground state to the \(n=4\) state. Comment on the correctness of the following statements (true or false). (a) \(n=4\) is the first excited state. (b) It takes more energy to ionize (remove) the electron from \(n=4\) than from the ground state. (c) The electron is farther from the nucleus (on average) in \(n=4\) than from the ground state. (d) The wavelength of light emitted when the electron drops from \(n=4\) to \(n=1\) is longer than that from \(n=4\) to \(n=2\). (e) The wavelength the atom absorbs in going from \(n=1\) to \(n=4\) is the same as that emitted as it goes from \(n=4\) to \(n=1\).

What is the noble gas core? How does it simplify the writing of electron configurations?

Why do the \(3 s, 3 p,\) and \(3 d\) orbitals have the same energy in a hydrogen atom but different energies in a many-electron atom?

Thermal neutrons are neutrons that move at speeds comparable to those of air molecules at room temperature. These neutrons are most effective in initiating a nuclear chain reaction among \({ }^{235} \mathrm{U}\) isotopes. Calculate the wavelength (in \(\mathrm{nm}\) ) associated with a beam of neutrons moving at \(7.00 \times 10^{2} \mathrm{~m} / \mathrm{s}\). (Mass of a neutron \(\left.=1.675 \times 10^{-27} \mathrm{~kg} .\right)\)

List the hydrogen orbitals in increasing order of energy.
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