(a) Show that the number of electron states in a subshell is 4 ℓ+2 . [Hint: First, how many stat

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(a) Show that the number of electron states in a subshell is...

(a) Show that the number of electron states in a subshell is $4 \ell+2 .$ [Hint: First, how many states are in each orbital? Second, how many orbitals are in each subshell?] (b) By summing the number of states in each of the subshells, show that the number of states in a shell is $2 n^{2}$. [Hint: Use the following theorem: the sum of the first $n$ odd integers, from 1 to $2 n-1$, is $n^{2} .$ That comes from regrouping the sum in pairs, starting by adding the largest to the smallest:

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Key Concepts

Electron Capacity of an Entire Shell

The overall shell (defined by the principal quantum number n) includes all subshells with angular momentum quantum numbers l from 0 to n?1. Summing the electron states for each subshell and applying sum formulas (such as the sum of the first n odd integers) results in the well-known formula for the maximum number of electrons in a shell, which is 2n².

Counting Electron States in a Subshell

In a subshell, the total number of electron states is found by multiplying the number of orbitals (2l + 1) by the two possible spin states per orbital. This multiplication yields 2(2l + 1) or 4l + 2 states within that subshell.

Orbital Angular Momentum

This concept refers to the quantum number l, which determines the shape of an electron's orbital and the number of distinct spatial orientations an orbital can have. In any subshell characterized by l, the magnetic quantum number m takes integer values from -l to l, resulting in 2l + 1 possible orientations or orbitals.

Electron Spin

Electron spin is a fundamental property of electrons, represented by a quantum number that can take one of two values (commonly +1/2 or -1/2). Because of this, each orbital defined by the magnetic quantum number can accommodate two electrons, one for each spin state.

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