Question

For a given value of the principal quantum number, \(n\) how do the energies of the \(s, p, d\), and \(f\) subshells vary for (a) hydrogen, (b) a many-electron atom?

Short answer

For a given principal quantum number \(n\), the energies of the \(s\), \(p\), \(d\), and \(f\) subshells are identical in a hydrogen atom, as the energy depends solely on \(n\) and is given by the formula \(E_n = \frac{-13.6 \, \text{eV}}{n^2}\). However, in many-electron atoms, the subshell energy depends on both \(n\) and the azimuthal quantum number \(l\), leading to different energies for subshells even when \(n\) is the same. Here, the energies of subshells typically follow the values of \((n + l)\), with lower totals indicating lower energy.

Step-by-step solution

Recalling the energy-level diagram for hydrogen atom

The energy-level diagram for hydrogen atom is determined by the principal quantum number (n). In a hydrogen atom, the energy of an electron depends only on the value of n as the formula for energy is given by: \[E_n = \frac{-13.6 \,\text{eV}}{n^2}\] For a given value of n, all the subshells (s, p, d, and f) have the same energy. We can represent the energy-level diagram for hydrogen atom as follows: \[1s = 2s = 2p = 3s = 3p = 3d = 4s = \cdots\]

Recalling the energy-level diagram for many-electron atoms

For many-electron atoms, the energy of an electron depends not only on the value of n, but also on the energy subshell (l). The energy ordering of subshells is usually given by the values of (n + l). This energy ordering can be memorized using the diagram below: n\l | s p d f ----|----------- 1 | 1 2 | 2 3 3 | 4 5 6 4 | 7 8 9 10 We can see that for many-electron atoms, the energies of the subshells vary according to the values of (n + l), where subshells with lower values have lower energy.

Comparing subshell energies for hydrogen and many-electron atoms

Now, let's compare the energies of s, p, d, and f subshells for both hydrogen and many-electron atoms. (a) In a hydrogen atom (one-electron atom), the energy of a subshell depends only on n, so all subshells with the same value of n have the same energy. Therefore, for hydrogen atom: \[1s = 2s = 2p = 3s = 3p = 3d = 4s = \cdots\] (b) For many-electron atom, the subshell energy depends on both n and l. Therefore the subshells do not have the same energies even when n is the same. For many-electron atoms, the energies of subshells vary according to the values of (n + l), where subshells with lower values have lower energy. In conclusion, for a given value of the principal quantum number n, the energies of s, p, d, and f subshells are the same for hydrogen atom, while for many-electron atoms, the energies of these subshells vary depending on their respective values of the quantum numbers (n + l).

Key concepts

energy levels

In the realm of quantum mechanics, energy levels refer to specific energies that electrons can have within an atom. These are defined by a principal quantum number, denoted as \( n \). The principal quantum number is like an address, giving a general idea of how far an electron is from the nucleus.
But energy levels are not just about distance, they encapsulate the energy required to occupy each level. These energy states become extremely significant when discussing the arrangement of electrons in atoms.
For instance, in the hydrogen atom, a simple one-electron system, the energy of an electron solely depends on the principal quantum number \( n \). This results in all subshells (\( s, p, d, f \)) having the same energy for a given \( n \).

• Principal Quantum Number (\( n \)): Defines the size and energy of the orbital.
• Same energy across subshells for hydrogen when \( n \) is constant.

Conversely, for more complex, many-electron atoms, electrons experience various repulsions and attractions, making energy level calculations more complex.

subshell ordering

Subshell ordering refers to the sequence in which electrons fill the various subshells within an atom. In hydrogen, all subshells at a given energy level have identical energies. However, in many-electron atoms, this simplicity vanishes.
The concept of subshell ordering is determined by both the principal quantum number \( n \) and the azimuthal quantum number \( l \). Collectively, \( n+l \) gives a method to rank subshell energies from lowest to highest.
In this system, subshells with lower \( n+l \) values are filled first. If two subshells share the same \( n+l \) value, the one with the lower \( n \) fills first, providing a nuanced and systematic order to how electrons occupy subshells:

• Subshell Energy: Determined by \( n+l \).
• Lower \( n+l \) values mean lower energy, thus filled first.
• Same \( n+l \): lower \( n \) fills first.

This ordering dictates the structure of the periodic table and the chemical properties of elements.

hydrogen atom

The hydrogen atom, consisting of just one proton and one electron, is the simplest atom. Its simple nature makes it a perfect subject to study atomic structures and quantum levels.
In hydrogen, energy levels are described by a straightforward formula \( E_n = \frac{-13.6 \,\text{eV}}{n^2} \). This means that for each principal quantum number \( n \), every subshell - whether \( s, p, d, \) or \( f \) - holds the same energy. This is because \( l \), which influences energy in more complex atoms, does not affect energy in hydrogen. This results in an evenly spaced energy ladder:

• Energy Levels: All subshells have the same energy for a specific \( n \).
• Formula: Describes energy purely in terms of \( n \), not \( l \).

This energy uniformity simplifies the electronic configuration of hydrogen considerably.

many-electron atoms

In many-electron atoms, complexities arise thanks to the interactions between multiple electrons. These interactions affect the energies of electron subshells, changing them significantly compared to the single-electron hydrogen atom.
Energy levels in these atoms depend on both \( n \) and \( l \), meaning subshells at the same principal energy level can have varying energies. This differentiation is achieved through the \( n+l \) rule, dictating the chemical behavior and stability of atoms:

• Electron Interactions: Cause variations in energy levels.
• Energy Dependence: Varies with both \( n \) and \( l \).
• \( n+l \) Rule: Guides subshell filling order.
  

This complexity provides a richer understanding of elements and underlies the behavior of matter, influencing chemical reactions and properties.