The maximum number of electrons that can occupy orbitals with an azimuthal quantum number of 3 is 14.
Understanding Azimuthal Quantum Number (l)
The azimuthal quantum number, denoted by $l$, is one of the four quantum numbers that describe the unique quantum state of an electron in an atom. It determines the shape of an atomic orbital and defines the subshell an electron occupies. The value of $l$ can range from 0 up to n-1, where n is the principal quantum number.
- l = 0: Corresponds to the s-subshell, which contains 1 orbital (spherical shape).
- l = 1: Corresponds to the p-subshell, which contains 3 orbitals (dumbbell shape).
- l = 2: Corresponds to the d-subshell, which contains 5 orbitals (more complex shapes).
- l = 3: Corresponds to the f-subshell, which contains 7 orbitals (even more complex shapes).
Electron Capacity of the f-Subshell (l=3)
For an azimuthal quantum number of $l=3$, we are referring to the f-subshell. The number of orbitals within a given subshell is determined by the formula $2l+1$.
Let's apply this to $l=3$:
- Number of orbitals = $2(3) + 1 = 6 + 1 = 7$ orbitals.
Each atomic orbital, according to the Pauli Exclusion Principle, can hold a maximum of two electrons, provided they have opposite spins.
Therefore, to find the maximum number of electrons in the f-subshell:
- Maximum electrons = Number of orbitals $\times$ Electrons per orbital
- Maximum electrons = $7 \times 2 = 14$ electrons.
Summary of Subshell Capacities
The following table summarizes the relationship between the azimuthal quantum number, subshell designation, number of orbitals, and maximum electron capacity:
| Azimuthal Quantum Number (l) | Subshell Designation | Number of Orbitals ($2l+1$) | Maximum Electrons |
|---|---|---|---|
| 0 | s | 1 | 2 |
| 1 | p | 3 | 6 |
| 2 | d | 5 | 10 |
| 3 | f | 7 | 14 |
This capacity is crucial for understanding electron configurations and the organization of the periodic table, where elements with f-electrons (like the Lanthanides and Actinides) are often found in separate blocks.
