Question

What is the key distinction between sublevel energies in oneelectron species, such as the \(\mathrm{H}\) atom, and those in many-electron species, such as the \(\mathrm{C}\) atom? What factors lead to this distinction? Would you expect the pattern of sublevel energies in \(\mathrm{Be}^{3+}\) to be more like that in \(\mathrm{H}\) or that in \(\mathrm{C}\) ? Explain.

Short answer

In one-electron species like H, sublevel energies depend only on principal quantum number. In many-electron species like C, electron repulsion and shielding cause the energies to differ. Be^{3+} resembles H due to having a single electron.

Step-by-step solution

Understanding sublevel energies in one-electron species

For a one-electron species like the hydrogen atom, the energy levels depend only on the principal quantum number (). This means that all sublevels within a principal shell (e.g., 2s, 2p) have the same energy.

Understanding sublevel energies in many-electron species

In many-electron atoms, electron-electron repulsion causes the sublevel energies (e.g., 2s, 2p) within the same principal level to split and differ in energy. This leads to the sublevels having different energies.

Factors causing the distinction

The main factors that cause the distinction in the energy sublevels of many-electron atoms are electron shielding and electron-electron repulsion. These interactions are not present in one-electron species.

Comparison with Be^{3+}

The Be^{3+} ion is similar to a one-electron species like H because it has only one electron, leading to sublevel energies that depend mainly on the principal quantum number. Hence, the pattern of sublevel energies in Be^{3+} is more like that in H than in C.

Conclusion

Sublevel energies in a one-electron species like H depend only on the principal quantum number, whereas in many-electron species like C, electron-electron repulsion and electron shielding cause these energies to differ. The energy pattern of Be^{3+} is expected to resemble H more due to similar electron configurations.

Key concepts

one-electron species

In one-electron species, such as the hydrogen atom \((\mathrm{H})\), there is only one electron interacting with the nucleus. The electron's energy levels depend solely on the principal quantum number, denoted by \(n\). This means all sublevels within the same principal shell, like 2s and 2p, have the same energy. There are no other electrons to cause interactions, so the energy levels are not split.

many-electron species

In many-electron species, such as the carbon atom \((\mathrm{C})\), multiple electrons interact with each other as well as with the nucleus. These interactions cause sublevel energies within the same principal level to split and differ. For example, in \(\mathrm{C}\), the 2s and 2p sublevels have different energies despite being in the same principal quantum number shell. The presence of multiple electrons changes the energy landscape considerably.

electron shielding

Electron shielding occurs in many-electron species, where inner electrons partially block the positive charge of the nucleus from the outer electrons. This reduces the effective nuclear charge felt by the outermost electrons. Due to shielding:

• Electrons in inner shells shield outer shell electrons from the nucleus, reducing their attraction.
• Electrons in the same shell can also shield each other to a lesser extent.

Shielding affects the energy levels, making some sublevels lower in energy than others within the same principal shell.

electron-electron repulsion

In many-electron atoms, electrons not only interact with the nucleus but also repel each other due to like charges. This electron-electron repulsion further splits the sublevel energies. The electrons in the same shell and subshell push away from each other, which shifts their energy levels.
For example:

• In \(\mathrm{C}\), the repulsion between 2s and 2p electrons leads to different energies for these sublevels.
• Repulsion contributes to the splitting of sublevel energies beyond what is observed in one-electron species.

principal quantum number

The principal quantum number, represented by \(n\), defines the main energy level of an electron in an atom. For one-electron species, the energy levels are determined solely by this number.
Key ideas include:

• The higher the \(n\) value, the higher the energy level and the further the electron is from the nucleus.
• For many-electron species, sublevel energies depend on \(n\) as well as other factors like electron shielding and repulsion.

For example, in hydrogen \(\mathrm{H}\), all sublevels within a shell \((n=2, \text{e.g.,} 2s, 2p)\) have the same energy. However, in carbon \((\mathrm{C})\), the sublevels will have different energies despite having the same \(n\) value.