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Maximum Electrons in Shell n = 3

electrons (2n²)

Principal quantum number (n)	3
Azimuthal range (l)	0 to 2
Number of subshells	3
Number of orbitals (n²)	9
Magnetic quantum range (mₗ)	−2 to +2 per subshell

What Is the Quantum Numbers Calculator?

This calculator takes a principal quantum number n and instantly works out the set of allowed quantum numbers and capacities for that electron shell. Quantum numbers describe the unique "address" of an electron in an atom: the energy level ($n$), the subshell shape ($l$), the orbital orientation ($m_\ell$), and spin. Given $n$, the rules of quantum mechanics fix the allowed values of $l$ and $m_\ell$ as well as how many subshells, orbitals, and electrons the shell can contain.

How to Use It

Enter a whole-number principal quantum number $n$ (1 for the first shell, 2 for the second, and so on). The calculator returns the maximum number of electrons ($2n^2$), the range of azimuthal values $l$ (0 to $n-1$), the number of subshells ($n$), the number of orbitals ($n^2$), and the magnetic quantum number range ($-l$ to $+l$) for each subshell.

The Formula Explained

For a given shell $n$: the azimuthal quantum number l can be any integer from 0 up to $n-1$, giving exactly $n$ subshells (s, p, d, f...). For each $l$, the magnetic quantum number mₗ takes the $2l+1$ integer values from $-l$ to $+l$. Summing $2l+1$ over all $l$ from 0 to $n-1$ gives $n^2$ orbitals. Since each orbital holds two electrons (spin $\pm\tfrac{1}{2}$), the maximum electron count is $2n^2$.

$$\begin{gathered} \text{Orbitals} = \text{n}^{2}, \quad \text{Max Electrons} = 2\,\text{n}^{2} \\[1.5em] \text{where}\quad \left\{ \begin{aligned} \text{Subshells} &= \text{n} \\ \ell_{\max} &= \text{n} - 1 \end{aligned} \right. \end{gathered}$$

Tree diagram showing principal quantum number n branching into l values, each l branching into m sub-l orbital boxes — How the principal quantum number n determines the allowed l and mₗ values and the orbitals in each subshell.

Worked Example

Take $n = 3$. Then $l = 0, 1, 2$ (three subshells: 3s, 3p, 3d). The orbital count is $n^2 = 9$ (one s, three p, five d). The maximum electrons is $$2n^2 = 2 \times 9 = 18.$$ The $m_\ell$ values reach from $-2$ to $+2$ for the d subshell.

Bar chart of shells n=1 to 4 with bars showing maximum electrons 2, 8, 18, 32 — Maximum electrons per shell follow 2n²: 2, 8, 18, 32 for n = 1 to 4.

FAQ

Why is the maximum 2n²? There are $n^2$ orbitals in shell $n$, and each orbital holds 2 electrons with opposite spins, so $2 \times n^2 = 2n^2$.

What does l = 0,1,2,3 correspond to? These are the s, p, d, and f subshells respectively.

How many mₗ values does a subshell have? Exactly $2l+1$, ranging from $-l$ to $+l$ in integer steps.

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