Understanding the Total Number of Orbitals Associated with Principal Quantum Number n3

Quantum mechanics plays a crucial role in understanding atomic structure, particularly the arrangement of electrons in an atom. One key parameter in this context is the principal quantum number (n), which defines the principal energy level or shell of an electron. When n3, the atom is in its third principal energy level, and a series of sub-shells and orbitals are associated with it. This article delves into the details of these sub-shells and how to determine the total number of orbitals associated with n3.

What is the Principal Quantum Number (n)?

The principal quantum number (n) is one of the four quantum numbers used to describe the state of an electron in an atom. It provides information about the main energy level or shell of the electron. For n3, the atom is in its third principal energy level.

Sub-Shells and Orbitals

Each value of n corresponds to a principal energy level and is accompanied by a series of sub-shells, denoted by the azimuthal quantum number (l). The values of l range from 0 to n-1. Each value of l corresponds to a different sub-shell type, such as s, p, d, and f. Within each sub-shell, there are additional orbitals, defined by the magnetic quantum number (ml).

Sub-Shells in n3

For n3, the possible values of l are 0, 1, and 2. This corresponds to the following sub-shells:

s sub-shell (l0) p sub-shell (l1) d sub-shell (l2)

Each sub-shell has a specific number of orbitals, which can be determined from the formula 2l 1:

s sub-shell (l0): 2(0) 1 1 orbital (3s) p sub-shell (l1): 2(1) 1 3 orbitals (3px, 3py, 3pz) d sub-shell (l2): 2(2) 1 5 orbitals (3dxy, 3dxz, 3dyz, 3dx2-y2, 3dz2)

Thus, the total number of orbitals in n3 is 1 3 5 9.

Electrons and Orbitals

Each orbital can hold a maximum of two electrons with opposite spins. Therefore, the total number of electrons associated with n3 can be calculated as 2 times the number of orbitals:

2 * 9 18 electrons

This means that when n3, the maximum number of electrons in the third principal energy level is 18.

Summary

In summary, for n3, there are 9 orbitals in total, which include 1 orbital in the s sub-shell (3s), 3 orbitals in the p sub-shell (3px, 3py, 3pz), and 5 orbitals in the d sub-shell (3dxy, 3dxz, 3dyz, 3dx2-y2, 3dz2). The maximum number of electrons associated with n3 is 18.

Related Concepts

Quantum Number n3: The principal quantum number that defines the third principal energy level of an electron.

Orbitals: Regions in an atom where electrons are most likely to be found, as defined by quantum numbers l and ml.

Electrons: Negatively charged particles that orbit the nucleus of an atom and determine its chemical properties.