Question

What type of electron orbital (i.e., s, \(\mathrm{p}, \mathrm{d}\), or \(\mathrm{f}\) ) is designated by (a) \(\mathrm{n}=2, \ell=1, \mathrm{~m}_{\ell}=-1 ?\) (b) \(\mathbf{n}=1, \ell=0, \mathbf{m}_{\ell}=0 ?\) (c) \(\mathrm{n}=5, \ell=2, \mathrm{~m}_{\ell}=2 ?\)

Short answer

Question: Identify the electron orbital type for each of the given sets of quantum numbers: a) n=2, l=1, m_l=-1 b) n=1, l=0, m_l=0 c) n=5, l=2, m_l=2 Answer: a) p-orbital b) s-orbital c) d-orbital

Step-by-step solution

Case (a): n=2, l=1, m_l=-1.

We have \(\ell = 1\), so this quantum number corresponds to a p-orbital. Therefore, the electron orbital type for this case is p.

Case (b): n=1, l=0, m_l=0.

We have \(\ell = 0\), so this quantum number corresponds to an s-orbital. Therefore, the electron orbital type for this case is s.

Case (c): n=5, l=2, m_l=2.

We have \(\ell = 2\), so this quantum number corresponds to a d-orbital. Therefore, the electron orbital type for this case is d.

Key concepts

Electron Orbitals

Electron orbitals are regions around the nucleus where an electron is likely to be found. These regions are defined by quantum numbers, which describe the properties and energies of electrons. The four quantum numbers are:

• The principal quantum number ( "). It indicates the main energy level or shell.
• The angular momentum quantum number ( "). It describes the shape of the orbital. Each shape has a corresponding letter designation: 0 for s, 1 for p, 2 for d, and 3 for f.
• The magnetic quantum number ( "). It specifies the orientation of the orbital in space.
• The spin quantum number (m_s"). It describes the direction of the electron's spin, which can be either +1/2 or -1/2.

These quantum numbers help categorize orbitals into different types, such as s, p, d, and f, based on their shape and spin characteristics. Understanding these categories helps predict how electrons will behave and interact in chemical systems.

s-orbital

The s-orbital is the simplest type of electron orbital. It has a spherical shape, and each energy level can only contain one s-orbital. The s-orbital's spherical shape means that the probability of finding an electron is the same in every direction around the nucleus. Thus, electrons in s-orbitals are equally likely to be found anywhere around the nucleus, at a given distance.

The s-orbitals are associated with the value ". When ", we identify the orbital shape as s. Additionally, since all main energy levels (n"]) have one s-orbital, when n = 1, 2, 3, etc., each level will have a corresponding 1s, 2s, 3s orbital, and so on.
Overall, s-orbitals are unique due to their singular, simple shape and serve as the foundational building block for other complex orbitals.

p-orbital

p-orbitals are more complex than s-orbitals. They have a dumbbell shape and are oriented in three specific directions in space. This shape means there are three p-orbitals per energy level, commonly noted as px, py, and pz. These orbitals increase in size as energy levels increase. Each p-orbital can hold a maximum of two electrons, leading to a total of six electrons in all three p-orbitals for a given energy level.

p-orbitals are associated with the angular momentum quantum number ". A value of 1 corresponds to these p-orbitals. Hence, starting from n = 2, each energy level contains three p-orbitals.
The unique directionality of p-orbitals enables them to form specific bond angles, making them crucial for understanding molecular geometry and bonding. This directional nature allows for diverse molecular structures, particularly in organic chemistry.

d-orbital

The d-orbitals are one level of complexity higher than p-orbitals. They typically resemble cloverleaf shapes, with some variations such as a double dumbbell. The complexity of these shapes allows for five d-orbitals in each energy level, which have unique orientations in 3D space.
As the principal quantum number (n) increases, so does the size and energy of the d-orbitals, and they begin to play a role at the third energy level, starting with 3d-orbitals.
In terms of quantum numbers, d-orbitals are characterized by ". Therefore, these orbitals come into play starting from n = 3. The five d-orbitals can accommodate a total of ten electrons, adding to the chemical and dynamic complexity of elements within the transition metals category. Having a diverse array of shapes, d-orbitals allow for advanced bonding interactions, particularly in metal complexes and coordination chemistry.