What is the maximum number of orbitals that can be...

What is the maximum number of orbitals that can be identified by each of the following sets of quantum numbers? When "none" is the correct answer, explain your reasoning.
(c) $n=7, \ell=5$
(a) $n=3, \ell=0, m_{\ell}=+1$
(b) $n=5, \ell=1$
(d) $n=4, \ell=2, m_{\ell}=-2$

Key Concepts

Principal Quantum Number

The principal quantum number (n) specifies the main energy level of an electron and is a positive integer starting at 1. It sets the size and energy of orbitals and determines the upper limit for the values of the angular momentum quantum number (?), which must be less than n.

Angular Momentum Quantum Number

The angular momentum quantum number (?) determines the shape of the orbital and can take on integer values from 0 up to n?1. Its value categorizes the type of orbital (s, p, d, f, etc.) and determines the number of possible magnetic quantum numbers through the expression 2? + 1, which represents the number of orbitals in the subshell.

Magnetic Quantum Number

The magnetic quantum number (m?) describes the orientation of an orbital in space relative to an external magnetic field. It can have integer values ranging from -? to +?. This quantum number uniquely identifies the individual orbitals within a subshell based on their spatial orientation.

Quantum Number Restrictions

Quantum number restrictions refer to the rules governing the allowed values of n, ?, and m?. These rules ensure that the values are consistent for valid orbitals, such as requiring that ? is less than n and that m? is an integer within the range of -? to +?. Violating these restrictions means that the set of quantum numbers does not correspond to any physical orbital.

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