Textbook Question
The series of emission lines of the hydrogen atom for which nf = 3 is called the Paschen series. (b) Calculate the wavelengths of the first three lines in the Paschen series—those for which ni = 4, 5, and 6.
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Ch.6 - Electronic Structure of Atoms
Chapter 6, Problem 94a
Bohr's model can be used for hydrogen-like ions—ions that have only one electron, such as He+ and Li2+. (a) Why is the Bohr model applicable to He+ ions but not to neutral He atoms?
Verified step by step guidance1
Understand the Bohr model: The Bohr model is specifically designed for systems where a single electron orbits a nucleus. It simplifies the electron-nucleus interaction by considering only one electron moving in a fixed orbit around the nucleus.
Recognize the electron configuration of He+ and neutral He: Helium (He) in its neutral state has two electrons. When it loses one electron, becoming He+, it has only one electron remaining.
Consider electron-electron interaction: In neutral helium, the presence of two electrons means that electron-electron repulsion must be considered, which complicates the system beyond what the Bohr model can accurately handle.
Apply the Bohr model to He+: Since He+ has only one electron, the system is simplified to a single electron orbiting the nucleus, similar to the hydrogen atom. This fits the conditions under which the Bohr model provides a good approximation.
Conclude why Bohr model is not applicable to neutral He: The Bohr model does not account for interactions between multiple electrons, which is necessary for accurately describing the behavior of neutral helium with its two electrons.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Bohr's Model of the Atom
Bohr's model describes the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits. It quantizes the angular momentum of electrons, allowing only certain stable orbits. This model is particularly effective for hydrogen-like ions, where a single electron is influenced by the nucleus's charge.
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Bohr Model of the Atom
Hydrogen-like Ions
Hydrogen-like ions are ions that contain only one electron, similar to hydrogen. Examples include He<sup>+</sup> and Li<sup>2+</sup>. The simplicity of having a single electron allows for the application of Bohr's model, as the electron's behavior can be accurately predicted by the model's equations, unlike in multi-electron systems.
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Electron-Electron Interactions
In neutral helium (He), there are two electrons that repel each other due to their negative charges. This electron-electron interaction complicates the energy levels and orbital shapes, making the Bohr model inadequate. The model does not account for these interactions, which are significant in multi-electron atoms, leading to deviations from the predictions made for hydrogen-like ions.
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Related Practice
Textbook Question
Determine whether each of the following sets of quantum numbers for the hydrogen atom are valid. If a set is not valid, indicate which of the quantum numbers has a value that is not valid: (a) n = 3, l = 3, ml = 2, ms = +1/2 (b) n = 4, l = 3, ml = -3, ms = +1/2 (c) n = 3, l = 1, ml = 2, ms = +1/2 (d) n = 5, l = 0, ml = 0, ms = 0 (e) n = 2, l = 1, ml = 1, ms = -1/2