Question

For the following pairs of orbitals, indicate which is higher in energy in a many-electron atom. (a) \(3 \mathrm{~s}\) or \(2 \mathrm{p}\) (b) 4 s or \(4 \mathrm{~d}\) (c) \(4 \mathrm{f}\) or \(6 \mathrm{~s}\) (d) 1 s or \(2 \mathrm{~s}\)

Short answer

Answer: (a) 3s, (b) 4d, (c) 4f, (d) 2s

Step-by-step solution

Identify the principal and azimuthal quantum numbers of each orbital

We have the following orbitals: (a) 3s: Principal quantum number (n) = 3, azimuthal quantum number (l) = 0 2p: Principal quantum number (n) = 2, azimuthal quantum number (l) = 1 (b) 4s: Principal quantum number (n) = 4, azimuthal quantum number (l) = 0 4d: Principal quantum number (n) = 4, azimuthal quantum number (l) = 2 (c) 4f: Principal quantum number (n) = 4, azimuthal quantum number (l) = 3 6s: Principal quantum number (n) = 6, azimuthal quantum number (l) = 0 (d) 1s: Principal quantum number (n) = 1, azimuthal quantum number (l) = 0 2s: Principal quantum number (n) = 2, azimuthal quantum number (l) = 0

Apply the n + l rule to compare the energies of the orbitals

Now we'll apply the n + l rule to determine which orbital in each pair has higher energy: (a) 3s: n + l = 3 + 0 = 3 2p: n + l = 2 + 1 = 3 Since both orbitals have the same n + l value, we compare their n values. The 2p orbital has a lower n value, so it has lower energy. Therefore, the 3s orbital has higher energy. (b) 4s: n + l = 4 + 0 = 4 4d: n + l = 4 + 2 = 6 The 4s orbital has a lower n + l value, so it has lower energy. Therefore, the 4d orbital has higher energy. (c) 4f: n + l = 4 + 3 = 7 6s: n + l = 6 + 0 = 6 The 6s orbital has a lower n + l value, so it has lower energy. Therefore, the 4f orbital has higher energy. (d) 1s: n + l = 1 + 0 = 1 2s: n + l = 2 + 0 = 2 The 1s orbital has a lower n + l value, so it has lower energy. Therefore, the 2s orbital has higher energy.

Report the higher-energy orbital for each pair

Based on our comparisons, the higher-energy orbitals in the given pairs are as follows: (a) 3s (b) 4d (c) 4f (d) 2s

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), is one of the key concepts in quantum mechanics, particularly in the study of atomic structure. It is a vital number that determines the overall size and energy level of an electron orbital in an atom. The value of \( n \) is always a positive integer, namely 1, 2, 3, and so forth. Here’s how it influences the electron orbital:

• It determines the "shell" or energy level in which the electron is located. Higher \( n \) values correspond to orbitals that are farther from the nucleus and have higher energy.
• As the value of the principal quantum number increases, the electron is likely to be found at a greater average distance from the nucleus, resulting in larger atomic radii.
• The principal quantum number is crucial in quantifying the energy levels of electrons in multi-electron atoms, where the complexity increases due to electron-electron interactions.

In comparing orbitals, knowing the \( n \) value is a starting point for comparing their energy levels, largely because electrons further from the nucleus are less tightly bound and often have greater energy.

Azimuthal Quantum Number

The azimuthal quantum number, designated as \( l \), adds more depth to our understanding of an atom’s electron configuration and its energy level. Also referred to as the angular momentum quantum number, \( l \) determines the shape of the electron’s orbital and reflects subshells within a given principal energy level. Here's how \( l \) is defined and what it affects:

• The azimuthal quantum number \( l \) can take on any integer value from 0 to \( n-1 \), where \( n \) is the principal quantum number. This means if \( n = 3 \), \( l \) can be 0, 1, or 2.
• Each value of \( l \) corresponds to a unique subshell, commonly denoted by letters: \( l = 0 \) is an "s" subshell, \( l = 1 \) is a "p" subshell, \( l = 2 \) is a "d" subshell, and \( l = 3 \) is an "f" subshell.
• It also impacts the energy levels within a principal quantum number. Generally, within the same \( n \), a higher \( l \) value implies a higher energy subshell due to increased angular momentum.

Understanding the azimuthal quantum number is essential when comparing orbitals' energy levels, such as in the exercises where one analyzes the impact of \( l \) on energy.

n + l Rule

The \( n + l \) rule is a useful guideline in quantum chemistry to predict the relative energy levels of different orbitals within atoms, particularly when dealing with multi-electron systems. This rule helps in determining which orbitals will be filled first with electrons as atoms build up their electron configurations. Here's how it works:

• Calculate the sum of the principal quantum number \( n \) and the azimuthal quantum number \( l \) for each orbital. This total, \( n + l \), helps in ranking them by energy level.
• The lower the value of \( n + l \), the lower the energy of the orbital. Orbital filling follows this sequence from lower to higher \( n + l \) values.
• If two orbitals have the same \( n + l \) value, the one with the lower \( n \) will have lower energy.

This rule is especially valuable when analyzing different orbitals of a multi-electron atom, as it was applied in analyzing exercises for energy comparisons. It intuitively explains why, for example, 3s is of higher energy than 2p due to closer proximity of electrons and available space in higher shells.