Question

What type of electron orbital (i.e., s, p, d, or \(\mathrm{f}\) ) is designated by an electron with quantum numbers (a) \(\mathrm{n}=1, \ell=0, \mathrm{~m}_{\ell}=0 ?\) (b) \(\mathrm{n}=3, \ell=2, \mathrm{~m}_{\ell}=-1 ?\) (c) \(\mathrm{n}=4, \ell=3, \mathrm{~m}_{\ell}=3\) ?

Short answer

a) n=1, ℓ=0, mℓ=0 b) n=3, ℓ=2, mℓ=-1 c) n=4, ℓ=3, mℓ=3 Answer: a) s orbital b) d orbital c) f orbital

Step-by-step solution

Identify electron orbital type for (a) n=1, ℓ=0, mℓ=0

In this case, we have the angular quantum number, \(\ell=0\). Based on the information given in the analysis, an \(\ell\) value of 0 corresponds to an "s" orbital. Therefore, the electron orbital type is "s".

Identify electron orbital type for (b) n=3, ℓ=2, mℓ=-1

In this case, we have the angular quantum number, \(\ell=2\). Based on the information given in the analysis, an \(\ell\) value of 2 corresponds to a "d" orbital. Therefore, the electron orbital type is "d".

Identify electron orbital type for (c) n=4, ℓ=3, mℓ=3

In this case, we have the angular quantum number, \(\ell=3\). Based on the information given in the analysis, an \(\ell\) value of 3 corresponds to an "f" orbital. Therefore, the electron orbital type is "f".

Key concepts

Quantum Numbers

Quantum numbers are the sets of numerical values that provide solutions to the Schrödinger equation, a fundamental equation in quantum mechanics. These numbers describe the unique quantum state of an electron in an atom, and they include the principal quantum number (), the angular quantum number (), the magnetic quantum number (), and the spin quantum number ().

The principal quantum number, denoted as , specifies the energy level of an electron and its relative distance from the nucleus. The values of start from 1 and increase as the energy levels move further out. The angular quantum number, represented by , defines the shape of an electron's orbital, and its values range from 0 to -1. The magnetic quantum number, , indicates the orientation of the orbital in space and can take on integer values between and . Lastly, the spin quantum number, denoted by , describes the direction of the electron's spin and can either be - or +.

Angular Quantum Number

The angular quantum number, often symbolized by or , is crucial to understanding an electron's position and the shape of its orbital. This number can take on any integer value from 0 to -1 for a given principal quantum level (). The values of are typically associated with the shape of electron orbitals: (spherically shaped orbitals), 1 (dumbbell-shaped orbitals), 2 (cloverleaf-shaped orbitals), and 3 (complex shaped orbitals).

From the angular quantum number, we can also deduce the number of sublevels and orbitals in a given energy level; for example, for =0 or the s orbital, there is 1 sublevel and 1 orbital. When =1 (p orbital), there are 3 orbitals; for =2 (d orbital), 5 orbitals; and =3 (f orbital) corresponds to 7 orbitals. The allowed values of the magnetic quantum number, ranging from - to +, are determined based on the value of , showcasing the orientation of these orbitals within the sublevel.

s p d f Orbitals

The s, p, d, and f orbitals refer to the shapes of areas in space where there's a higher probability of finding an electron. 's' orbitals are the simplest, with a spherical shape and a single sublevel in each energy level (n). In contrast, 'p' orbitals are shaped like dumbbells with three orientations and thus three sublevels, typically starting from the second energy level (n=2).

'd' orbitals are more complex with a cloverleaf shape and five orientations, present from the third energy level (n=3). Lastly, the 'f' orbitals are even more complex in shape, with seven different orientations available from the fourth energy level (n=4). The s orbital can hold up to 2 electrons, p orbitals can hold 6, d orbitals can hold 10, and f orbitals can hold up to 14, reflecting the number of orientations multiplied by the two possible spins of an electron.Understanding these orbital shapes and their orientations is fundamental to visualizing how electrons are distributed around an atom and how they contribute to the atom's chemical properties and bonding behavior.