What is the Maximum Number of Electrons in an Atom that Can Have the Following Quantum Numbers?

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What is the maximum number of electrons in an atom that can have the following quantum numbers?

(a) n=2;

(b) n=3 and ℓ =1;

(c) n=3 and and ℓ =1 and mℓ ​=0;

(d) n=3 and and ℓ =1 and mℓ ​=-1 and ms​ = -1/2;

Answer: 

Part (a): 8 electrons;

Part (b): 6 electrons;

Part (c): 2 electrons;

Part (d): 1 electron.

Explanation

Part A:

For a given principal quantum number n, there are a maximum total of 2n^2 electrons. Therefore, an n = 2 shell has a maximum total of 2 (2)^2 = 8 electrons.

Part B:

For a given secondary (angular momentum) quantum number ℓ, there are a maximum total of 2 (2ℓ + 1) electrons. Therefore, an ℓ = 1 subshell has a maximum total of 2 [2 (1) + 1] = 6 electrons.

Part C:

A set of three quantum numbers represents a specific orbital, which contains a maximum of two electrons with opposite spins.

Part D:

A set of four quantum numbers represents an electron in a specific orbital with a specific spin. The Pauli exclusion principle states that no two electrons in an atom may have the same set of four quantum numbers. Therefore, this set of quantum numbers contains a maximum of one electron.

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