Question

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

Short answer

Question: Which orbital has lower energy in a many-electron atom: (a) \(3 \mathrm{~d}\) or \(4 \mathrm{~s}\) (b) \(4 \mathrm{f}\) or \(3 \mathrm{~d}\) (c) \(2 \mathrm{~s}\) or \(2 \mathrm{p}\) (d) \(4 \mathrm{f}\) or \(4 \mathrm{~d}\) Answer: (a) \(4 \mathrm{~s}\) (b) \(3 \mathrm{~d}\) (c) \(2 \mathrm{~s}\) (d) \(4 \mathrm{~d}\)

Step-by-step solution

(a) \(3 \mathrm{~d}\) or \(4 \mathrm{~s}\) comparison

For the \(3 \mathrm{~d}\) orbital, n=3 and l=2 (as d orbitals have l=2). So, n+l=5. For the \(4 \mathrm{~s}\) orbital, n=4 and l=0 (as s orbitals have l=0). So, n+l=4. As n+l is smaller for the \(4 \mathrm{~s}\) orbital, it has lower energy in a many-electron atom.

(b) \(4 \mathrm{f}\) or \(3 \mathrm{~d}\) comparison

For the \(4 \mathrm{f}\) orbital, n=4 and l=3 (as f orbitals have l=3). So, n+l=7. For the \(3 \mathrm{~d}\) orbital, n=3 and l=2 (as d orbitals have l=2). So, n+l=5. As n+l is smaller for the \(3 \mathrm{~d}\) orbital, it has lower energy in a many-electron atom.

(c) \(2 \mathrm{~s}\) or \(2 \mathrm{p}\) comparison

For the \(2 \mathrm{~s}\) orbital, n=2 and l=0 (as s orbitals have l=0). So, n+l=2. For the \(2 \mathrm{p}\) orbital, n=2 and l=1 (as p orbitals have l=1). So, n+l=3. As n+l is smaller for the \(2 \mathrm{~s}\) orbital, it has lower energy in a many-electron atom.

(d) \(4 \mathrm{f}\) or \(4 \mathrm{~d}\) comparison

For the \(4 \mathrm{f}\) orbital, n=4 and l=3 (as f orbitals have l=3). So, n+l=7. For the \(4 \mathrm{~d}\) orbital, n=4 and l=2 (as d orbitals have l=2). So, n+l=6. As n+l is smaller for the \(4 \mathrm{~d}\) orbital, it has lower energy in a many-electron atom.

Key concepts

Many-electron Atom

A many-electron atom is one that has more than one electron, such as helium (with two electrons), oxygen (with eight), or uranium (with ninety-two). Unlike the hydrogen atom, which has a single electron, these atoms have multiple electrons interacting with each other and with the positive nucleus. These interactions make predicting the behavior of electrons in such atoms more complex.

In a many-electron atom, each electron feels not only the attraction of the nucleus but also the repulsive forces from other electrons. This leads to a shielding effect, where inner electrons shield the outer electrons from the full charge of the nucleus. As a result, outer electrons do not feel as strong of an attractive force and therefore have higher energy levels compared to what would be seen in a single-electron atom with the same nuclear charge.

The situation becomes even more complicated due to electron-electron repulsions, which cause additional energy splitting and shifts in the energy levels of the orbitals. It is a principle of quantum mechanics that electrons arrange themselves in a way that minimizes the overall energy of the atom. Therefore, they occupy the available orbitals in a specific order, influenced by the orbital energies.

Quantum Numbers

Quantum numbers are the sets of numerical values that give a complete description of the energy state and spatial distribution of an electron in an atom. The four quantum numbers—principal (n), azimuthal (l), magnetic (m_l), and spin (m_s)—determine the size, shape, orientation, and spin direction of an electron's probability cloud, or orbital.

The principal quantum number (n) indicates the energy level of the electron and is always a positive integer. As n increases, the electron is farther from the nucleus, thereby increasing its energy. The azimuthal quantum number (l) defines the shape of the orbital and can range from 0 to (n-1). The magnetic quantum number (m_l) reflects the orientation of the orbital in space and can have integer values between -l and +l, including zero. Lastly, the spin quantum number (m_s) indicates the direction of the electron's spin, which can either be +1/2 or -1/2. These quantum numbers are essential in predicting an electron’s position and energy within an atom and are used in models and equations to understand atomic structure.

Orbital Energy Levels

Orbital energy levels are like rungs on a ladder that electrons can occupy within an atom, with each level corresponding to a specific amount of energy. The energy levels are defined by the principal quantum number (n), but within each principal energy level, there are also sublevels, which are determined by the azimuthal quantum number (l). The sublevels are classified as s, p, d, or f, corresponding to l values of 0, 1, 2, or 3, respectively.

In a many-electron atom, the energy of an orbital is not solely defined by the principal quantum number but also by the azimuthal quantum number due to electron shielding and repulsion. This ordering of sublevels by increasing energy in many-electron atoms tends to follow the order of increasing (n + l). When two orbitals have the same (n + l) value, the one with the lower n value is lower in energy. For example, the difference in the energy level between a 2s and 2p orbital comes from the different l values.

Understanding how orbital energy levels are filled according to the Aufbau principle, which states that electrons occupy the lowest energy orbitals available, is foundational in chemistry. This knowledge helps to explain how electrons are distributed in atoms and how this relates to the atom’s chemical properties.