Question

Within any given shell, how do the energies of the \(s, p, d\), and \(f\) subshells compare? How do the energies of the orbitals belonging to a given subshell compare?

Short answer

Within a shell, subshell energies increase in the order: s < p < d < f. Orbitals within a given subshell have the same energy (degenerate).

Step-by-step solution

Energy Comparison of Subshells within a Shell

Within a given shell, the energy of the subshells increases with the complexity of the orbital shapes. The order of increasing energy is: s < p < d < f. This is based on the number of nodes and the distance from the nucleus. Each letter refers to a different subshell: s (spherical), p (dumbbell-shaped), d (clover-shaped), and f (complex shapes).

Energy Comparison of Orbitals within a Subshell

Within a single subshell, all orbitals have the same energy and are said to be degenerate. This means that, for example, all the 2p orbitals (2p_x, 2p_y, and 2p_z) have the same energy within the same subshell. The degeneracy is broken when a magnetic or an electric field is applied.

Key concepts

Atomic Orbitals

Atomic orbitals are the regions around an atom's nucleus where electrons are most likely to be found. Each orbital can hold a maximum of two electrons with opposite spins. The simplest type of orbital is the spherical s orbital, found in all energy levels from the principal quantum number (n=1) onward. As one moves to higher principal quantum numbers, more complex orbitals come into play, including the p orbitals, which have a dumbbell shape, d orbitals with a clover shape, and f orbitals with even more complex shapes.

Understanding the arrangement and behavior of these orbitals is crucial for studying electronic configurations of elements and the chemical properties that arise as a consequence. Electrons will first fill the lowest energy orbitals available—a principle known as the Aufbau Principle. By visualizing the orbital shapes, students can better predict and explain the behavior of electrons in atoms during chemical reactions.

Orbital Shapes and Energy

The energy levels of orbitals within an atom are not even; they are ordered based on the shape and complexity of the orbital. For a given energy level, or shell, the s orbital is the lowest in energy because it is closest to the nucleus and has the least complex shape. As we move to the p, d, and f orbitals, the energy increases with the complexity of the shape. The more complex the shape, the more nodes they have, which are regions where the probability of finding an electron is zero.

The order of increasing energy for subshells within the same shell is therefore s < p < d < f. This hierarchy of energy influences how electrons populate orbitals and ultimately, how atoms interact with each other to form molecules. By focusing on the energy and shapes of these orbitals, students can predict electron configurations and the potential chemical behavior of elements.

Degenerate Orbitals

Within a given subshell, orbitals have the same energy level and are known as degenerate orbitals. For example, in the 2p subshell, there are three degenerate orbitals: 2p_x, 2p_y, and 2p_z. They are all on the same energy level despite being oriented differently in space. It's important to note that the degeneracy of these orbitals is based on the absence of an external magnetic or electric field. In a uniform field-free environment, such as an isolated atom or a symmetrical molecule, the energy levels remain the same.

When an external field is present, the degeneracy can be broken, causing the energy levels to split. This phenomenon is used in many spectroscopic techniques to study the structure and function of molecules. In the classroom, understanding the concept of degenerate orbitals can be crucial for students when determining electron configurations and the behavior of atoms in various chemical environments.