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 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: $$ \begin{array}{l}{\text { Atom or ion } \quad \quad\quad\quad\quad\quad \mathrm{H} \quad\quad\quad\quad\quad\quad \text { He }^{+} \quad\quad\quad\quad\quad\quad\quad \mathrm{Li}^{2+}} \\ {\text { Ground- state }\quad-2.18 \times 10^{-18} \mathrm{J}\quad-8.72 \times 10^{-18} \mathrm{J}\quad-1.96 \times 10^{-17} \mathrm{J}} \\ {\text { energy }}\end{array} $$ By examining these numbers, propose a relationship between the ground-state energy of hydrogen-like systems and the nuclear charge, \(Z .(\mathbf{c})\) Use the relationship you derive in part (b) to predict the ground-state energy of the \(\mathrm{C}^{5+}\) ion.

Short answer

The Bohr model is applicable to He+ ions but not to neutral He atoms because the model only works for systems with a single electron, like He+. The relationship between the ground-state energy (E) and nuclear charge (Z) for hydrogen-like systems is \(E_{ground-state} = \frac{-2.18 \times 10^{-18}\, \mathrm{J}}{Z^2}\). Using this relationship, the ground-state energy of the \(\mathrm{C}^{5+}\) ion is approximately \(-6.05 \times 10^{-18}\, \mathrm{J}\).

Step-by-step solution

(a) Applicability of Bohr Model to Hydrogen-like Ions

Bohr model can be applied to hydrogen-like ions that have only one electron, such as He+ and Li2+. It is essential to understand why this model is applicable to He+ ions but not to neutral He atoms. The Bohr model is based on the assumption that there is only one electron orbiting the nucleus in a quantized orbit. Due to this, the electron does not lose energy continuously, and only energy transitions happen between orbitals when the electron loses its energy. In the case of neutral helium atoms, there are 2 electrons orbiting the nucleus, therefore the Bohr model does not apply because it only works for systems with a single electron. However, for He+ ions, there is only one electron left, which makes it applicable to the Bohr model.

(b) Relationship between Ground-State Energy and Nuclear Charge (Z)

To find the relationship between the ground-state energy (E) of hydrogen-like systems and the nuclear charge (Z), we examine and compare the given ground-state energies of the hydrogen-like systems H, He+, and Li2+. The ground-state energy for hydrogen atom H is given as \[ E_H = -2.18 \times 10^{-18}\, \mathrm{J} \] The ground-state energy for helium ion He+ is given as \[ E_{He^+} = -8.72 \times 10^{-18}\, \mathrm{J} \] The ground-state energy for lithium ion Li2+ is given as \[ E_{Li^{2+}} = -1.96 \times 10^{-17}\, \mathrm{J} \] Now, we know that the nuclear charge of hydrogen (Z_H) is 1, helium (Z_{He}) is 2, and lithium (Z_{Li}) is 3. Let's compare the ratios of energy values to the square of the nuclear charge values of these hydrogen-like systems: \[ \frac{E_{He^+}}{(Z_{He^+})^2} = \frac{-8.72 \times 10^{-18}\, \mathrm{J}}{2^2} = -2.18 \times 10^{-18}\, \mathrm{J}, \] \[ \frac{E_{Li^{2+}}}{(Z_{Li^{2+}})^2} = \frac{-1.96 \times 10^{-17}\, \mathrm{J}}{3^2} = -2.18 \times 10^{-18}\, \mathrm{J}. \] Based on these ratios, we find out that for hydrogen-like systems, the ground-state energy is inversely proportional to the square of the nuclear charge. Therefore, the relationship can be expressed as: \[ E_{ground-state} = \frac{-2.18 \times 10^{-18}\, \mathrm{J}}{Z^2}, \] where Z is the nuclear charge of the system.

(c) Predicting Ground-State Energy of the \(\mathrm{C}^{5+}\) Ion

Based on the relationship we derived in part (b), we can predict the ground-state energy of the \(\mathrm{C}^{5+}\) ion. The nuclear charge of carbon (Z_C) is 6. Use that value of Z in the relationship above: \[ E_{C^{5+}} = \frac{-2.18 \times 10^{-18}\, \mathrm{J}}{6^2} = -6.05 \times 10^{-18}\, \mathrm{J}. \] So, the ground-state energy of the \(\mathrm{C}^{5+}\) ion is approximately \(-6.05 \times 10^{-18}\, \mathrm{J}\).

Key concepts

Hydrogen-like ions

Hydrogen-like ions are a fascinating cornerstone in the study of atomic structure because they offer a simplified scenario where the complexity of electron interactions can be minimized for easier understanding. The term 'hydrogen-like ions' refers to ions that possess only one electron, similar to a hydrogen atom, which makes them critical in exploring the underlying principles of quantum mechanics and the Bohr Model.

The significance of these ions in educational contexts lies in their single-electron setup, enabling a direct application of the Bohr Model. This model describes an electron's quantized orbits around the nucleus, with each orbit corresponding to a specific energy level. The crucial aspect for learners to grasp is that, unlike atoms with multiple electrons where repulsive forces and complex interactions come into play, hydrogen-like ions provide a clean, isolated system that mirrors the simplicity of hydrogen.

Bohr's Model can thereby explain the discrete energy levels represented by these ions, predicting the energy released or absorbed when an electron transitions between levels. This refined approach is invaluable for students because it reinforces the quantum aspect of atomic energy states without the added complexity of electron-electron repulsion. Therefore, studying hydrogen-like ions provides a stepping stone towards understanding more intricate atomic structures.

Ground-state energy

Exploring the exciting territory of ground-state energy helps us understand the lowest energy configuration of an atom or ion. Ground-state energy is a significant concept because it serves as a reference point for all other higher energy levels, akin to the ground floor in a building from which all other floors are accessed.

In the context of the Bohr Model and hydrogen-like ions, ground-state energy is the energy of the single electron when it is in the closest orbit to the nucleus. The importance here for students is that this energy state is the most stable for the electron and represents the minimal potential and kinetic energy combination for that electron in its atomic environment.

By examining the ground-state energies of different hydrogen-like ions, we can discern patterns and predict the energies of others. For example, the exercise shows a direct proportionality between the ground-state energies and the square of their respective nuclear charges. This tangible relationship equips students with a tool to mathematically forecast an ion's inherent energy, which is crucial when studying and comparing the behavior of different atoms and ions in their most relaxed state.

Nuclear charge

In atomic physics, nuclear charge is the total charge of the nucleus, which is due to the number of protons it contains. This is a pivotal concept because the nuclear charge dictates a wide array of atomic properties, particularly the strength by which an electron is attracted to the nucleus.

Understanding nuclear charge is fundamental for students as it directly influences an ion or atom's electron configuration and reactivity. The higher the nuclear charge, the stronger the pull on the surrounding electrons, which affects how these electrons interact with other atoms. This becomes central when applying the Bohr Model to hydrogen-like ions: the ground-state energy and the orbit of an electron are primarily determined by the nuclear charge.

In the exercise provided, the relationship between ground-state energy and nuclear charge of hydrogen-like ions is evident as we observe a mathematical relationship where the energy is inversely proportional to the square of the nuclear charge. By solidifying their understanding of nuclear charge, students can not only predict the energetic behavior of ions but also infer the resulting chemical properties, thereby harnessing a key to unlocking the interaction patterns of different elements in the periodic table.