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Chapter 38: Problem 13

Which model of the hydrogen atom-the Bohr model or the quantum mechanical model-predicts that the electron spends more time near the nucleus?

Short Answer

Expert verified
Answer: The quantum mechanical model predicts that the electron spends more time near the nucleus due to its probability distribution nature, which places the highest probability densities closer to the nucleus.

Step by step solution

01

Introduce the Bohr model

The Bohr model is an early model of the hydrogen atom, proposed by Niels Bohr in 1913. In this model, the electron orbits the nucleus in discrete circular orbits, similar to planets orbiting around the sun. The electron can only occupy specific energy levels, and it must stay in these fixed orbits unless it gains or loses energy to jump to another orbit.
02

Introduce the quantum mechanical model

The quantum mechanical model, developed in the 1920s, is a more accurate and advanced description of the hydrogen atom. In this model, electrons do not have fixed orbits; instead, their position is described by probability distributions, represented by wave functions. The wave functions, also known as orbitals, represent the areas where the electron is most likely to be found, with no definite path or position around the nucleus.
03

Compare electron distribution in both models

In the Bohr model, the electron spends an equal amount of time in each part of its orbit, meaning that the electron is on average the same distance from the nucleus at all points along the orbit. In quantum mechanical model, however, the probability distribution of the electron's position varies around the nucleus, with the highest probability densities typically found closer to the nucleus. For example, in the ground state of the hydrogen atom (1s orbital), the highest probability density is located at the nucleus itself.
04

Determine which model predicts that the electron spends more time near the nucleus

Based on our comparison of the Bohr model and the quantum mechanical model, it is clear that the quantum mechanical model predicts that the electron spends more time near the nucleus. This is due to its probability distribution nature, which places the highest probability densities closer to the nucleus, unlike the fixed orbits in the Bohr model.

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Most popular questions from this chapter

Consider a muonic hydrogen atom, in which an electron is replaced by a muon of mass \(105.66 \mathrm{MeV} / \mathrm{c}^{2}\) that orbits a proton. What are the first three energy levels of the muon in this type of atom?

An electron in a hydrogen atom is in the ground state (1s). Calculate the probability of finding the electron within a Bohr radius \(\left(a_{0}=0.05295 \mathrm{nm}\right)\) of the proton. The ground state wave function for hydrogen is: \(\psi_{1 s}(r)=A_{1 s} e^{-r / a_{0}}=e^{-r / a_{0}} / \sqrt{\pi a_{0}^{3}}\).

A beam of electrons is incident upon a gas of hydrogen atoms. What minimum speed must the electrons have to cause the emission of light from the \(n=3\) to \(n=2\) transition of hydrogen?

Electrons with the same value of quantum number \(n\) are said to occupy the same electron shell \(K, L, M, N,\) etc. Calculate the maximum allowed number of electrons for the a) \(K\) shell, b) \(L\) shell, and c) \(M\) shell.

Given that the hydrogen atom has an infinite number of energy levels, why can't a hydrogen atom in the ground state absorb all possible wavelengths of light?
See all solutions

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