Question

Bohr's model can be used for hydrogen-like ions-ions that have only one electron, such as \(\mathrm{He}^{+}\)and \(\mathrm{Li}^{2+}\). (a) Why is the Bohr model applicable to \(\mathrm{He}^{+}\)ions but not to neutral He atoms? (b) The ground-state energies of \(\mathrm{H}, \mathrm{He}^{+}\), and \(\mathrm{Li}^{2+}\) are tabulated as follows:

Short answer

(a) The Bohr model is applicable to \(He^+\) ions but not to neutral He atoms because, for single-electron systems like hydrogen-like ions, the electron primarily interacts with the nucleus, and the Bohr model works well in predicting their energy levels. In neutral helium atoms, there are two electrons, and the Bohr model does not account for the significant electron-electron interactions that occur in multi-electron atoms. (b) Ionization energies can be calculated from the ground-state energies provided: For hydrogen, ionization energy, \(I_H = -E_{H}\) For helium, ionization energy, \(I_{He} = -E_{He^+}\) For lithium, ionization energy, \(I_{Li} = -E_{Li^{2+}}\) By substituting the tabulated ground-state energies, we can calculate the ionization energies for hydrogen, helium, and lithium atoms using the formulas above.

Step-by-step solution

(a) Bohr Model Applicability for Hydrogen-like Ions and Neutral Helium

Bohr's model is based on quantization of electron orbits and it works well for atoms with a single electron orbiting the nucleus, such as hydrogen-like ions (e.g., \(\mathrm{He}^{+}\) and \(\mathrm{Li}^{2+}\)). The success of the Bohr model for hydrogen-like ions is due to the simplified nature of these single-electron systems. In these ions, the single electron is primarily interacting with the nucleus and there are no significant electron-electron interactions, which makes it easier to apply the Bohr model. However, neutral helium atoms have two electrons, and the interactions between those electrons become significant and cannot be ignored. The Bohr model does not account for these electron-electron interactions and, as a result, fails to provide accurate predictions for the energy levels and other properties of multi-electron atoms like neutral helium.

(b) Ionization Energies Calculation

To find the ionization energies for \(\mathrm{H}\), \(\mathrm{He}^{+}\), and \(\mathrm{Li}^{2+}\), we need to determine the energy required to remove the outermost electron from each atom. The ionization energies can be calculated from the ground-state energies provided: For hydrogen, ionization energy (\(I_H\)) can be calculated as: \[ I_H = -E_{H} \] For helium, ionization energy (\(I_{He}\)) can be calculated using the energies for \(\mathrm{He}^{+}\) and \(\mathrm{He}\) as follows: \[ I_{He} = -E_{He^+} \] For lithium, ionization energy (\(I_{Li}\)) can be calculated using the energies for \(\mathrm{Li}^{2+}\) and \(\mathrm{Li}^{+}\) as follows: \[ I_{Li} = -E_{Li^{2+}} \] Using the provided tabulated ground-state energies, calculate the ionization energies for hydrogen, helium, and lithium atoms using the above formulas.

Key concepts

Hydrogen-like Ions

Hydrogen-like ions are fascinating objects of study since they have only one electron orbiting the nucleus. This simplicity makes them a favorite target for applying Bohr's model. These ions are derived from larger atoms that have their outer electrons removed. Examples include

• Helium ion \(\mathrm{He}^{+}\)
• Lithium ion \(\mathrm{Li}^{2+}\).

The key characteristic of hydrogen-like ions is their single electron configuration which resembles the hydrogen atom. Bohr's model applies best to these ions because there's only one electron interacting mainly with the nucleus. This reduction to just one primary interaction simplifies many calculations and predictions. The Bohr model breaks accuracy, however, when multiple electrons are present because it does not account for the complex electron-electron interactions.
Such similarity to hydrogen gives scientists an excellent tool to predict various properties, such as energy levels and spectral lines. The elegance of hydrogen-like ions lies in their simplicity, allowing classic physics models like Bohr’s to shine without the complexity of electron clouds interfering.

Ionization Energy

Ionization energy represents the energy required to remove the outermost electron from an atom or an ion. This energy helps us understand how difficult it is to strip an electron away from an atom. Different atoms and ions have different ionization energies, depending mainly on two factors:

• The nuclear charge, which is the number of protons in the nucleus.
• The distance of the electron from the nucleus, meaning how tightly the electron is bound.

For hydrogen-like ions, the ionization energy is a crucial measure because it reveals how strongly the electron is bound to the nucleus. We can calculate the ionization energy using the concept of ground-state energies, which are the energies of electrons in their lowest energy states. For example, in hydrogen \(\mathrm{H}\), helium ion \(\mathrm{He}^{+}\), and lithium ion \(\mathrm{Li}^{2+}\), their ionization energies can be computed by negating their corresponding ground-state energies, illustrating directly how much energy is required for electron removal.
This calculation helps give insight into atomic stability, reactivity, and the ability to form ions, essential aspects of chemistry and quantum physics.

Electron-Electron Interactions

In atoms with more than one electron, such as neutral helium, interactions between electrons significantly affect atomic behavior and energy levels. These interactions occur because electrons carry negative charges, which cause them to repel each other. This repulsion complicates the simple picture Bohr's model provides.
When considering electron-electron interactions, multi-electron systems become far more complex than the hydrogen-like ions. The interactions not only affect energy levels but also influence atomic orbitals and bonding characteristics. In multi-electron atoms, electrons are found in various energy levels or orbitals, which need to be considered when analyzing atomic structure and properties. Understanding these interactions requires advanced models beyond the Bohr model, such as the Schrödinger equation, which can account for

• Multiple electron participations.
• Correlations in electron positions and movements.

Thus, while Bohr's model is a great starting point for simple systems, electron-electron interactions necessitate more complex quantum mechanical treatments to fully grasp atomic structure and energies in larger atoms.