Definition and explanation of Azimuthal Quantum Numbers
Azimuthal Quantum Number (𝒍) Subsidiary Quantum Number
According to Sommerfeld, to facilitate the revolution of electrons within an atom, each principal energy level is subdivided into a specific number of subshells (sub-energy levels). The quantum number that designates the specific subshell in which an electron is revolving within a principal energy level is called the Azimuthal Quantum Number (or Subsidiary Quantum Number), denoted by 𝒍.
Key Characteristics:
- It is generally represented by the lowercase bold-italic letter 𝒍.
- The value of 𝒍 depends strictly on the value of the principal quantum number (n).
- For any given value of n, there are exactly n number of possible values for 𝒍.
- The permitted values of 𝒍 range sequentially from 0 to (n − 1).
| For n = 1: Total number of 𝒍 values = 1; which is 𝒍 = 0 | ∴ 1st energy level contains 1 subshell. |
| For n = 2: Total number of 𝒍 values = 2; which are 𝒍 = 0, 1 | ∴ 2nd energy level contains 2 subshells. |
| For n = 3: Total number of 𝒍 values = 3; which are 𝒍 = 0, 1, 2 | ∴ 3rd energy level contains 3 subshells. |
| For n = 4: Total number of 𝒍 values = 4; which are 𝒍 = 0, 1, 2, 3 | ∴ 4th energy level contains 4 subshells. |
Depending on whether 𝒍 = 0, 1, 2, or 3, the corresponding subshells are designated alphabetically as s (sharp), p (principal), d (diffuse), and f (fundamental).
To denote the exact state of an electron in an atom, the principal shell designation (n = 1, 2, 3, …) is written first, followed immediately to its right by the subshell symbol (s, p, d, f).
Mapping of Shells and Subshells:
| Shell Designation | K | L | M | N | ||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Principal Quantum Number (n) | 1 | 2 | 3 | 4 | ||||||
| Azimuthal Quantum Number (𝒍) | 0 | 0 | 1 | 0 | 1 | 2 | 0 | 1 | 2 | 3 |
| Subshell / Orbital Notation | 1s | 2s | 2p | 3s | 3p | 3d | 4s | 4p | 4d | 4f |
These subshells (s, p, d, f) signify specific energy states within a primary quantum shell. For a given principal quantum shell, the relative magnitudes of energy follow the sequential order:
Example: For the 2s subshell, n = 2 and 𝒍 = 0. Here, (n − 1) = 1. Since 𝒍 < (n − 1), it was historically classified as elliptical. Conversely, for the 2p subshell, n = 2 and 𝒍 = 1. Since 𝒍 = (n − 1), its path was treated as circular. However, modern quantum mechanics recognizes actual wave-mechanical electron cloud probabilities as detailed below.
Subshell Geometries and Quantum Configurations:
| Principal (n) | Azimuthal (𝒍) & Subshell | Mathematical Evaluation (Sommerfeld Context) | Modern Quantum Shape |
|---|---|---|---|
| n = 1 | 𝒍 = 0 (1s) | Given n = 1, (n − 1) = 0. Since 𝒍 = (n − 1) | Spherical |
| n = 2 | 𝒍 = 0 (2s) | Given n = 2, (n − 1) = 1. Since 𝒍 < (n − 1) | Spherical |
| 𝒍 = 1 (2p) | Given n = 2, (n − 1) = 1. Since 𝒍 = (n − 1) | Dumbbell-shaped | |
| n = 3 | 𝒍 = 0 (3s) | Given n = 3, (n − 1) = 2. Since 𝒍 < (n − 1) | Spherical |
| 𝒍 = 1 (3p) | Given n = 3, (n − 1) = 2. Since 𝒍 < (n − 1) | Dumbbell-shaped | |
| 𝒍 = 2 (3d) | Given n = 3, (n − 1) = 2. Since 𝒍 = (n − 1) | Complex (Cloverleaf) | |
| n = 4 | 𝒍 = 0 (4s) | Given n = 4, (n − 1) = 3. Since 𝒍 < (n − 1) | Spherical |
| 𝒍 = 1 (4p) | Given n = 4, (n − 1) = 3. Since 𝒍 < (n − 1) | Dumbbell-shaped | |
| 𝒍 = 2 (4d) | Given n = 4, (n − 1) = 3. Since 𝒍 < (n − 1) | Complex | |
| 𝒍 = 3 (4f) | Given n = 4, (n − 1) = 3. Since 𝒍 = (n − 1) | Highly Complex |
Orbitals and Electron Capacities
Under the influence of an external magnetic field, a subshell splits into (2𝒍 + 1) degenerated spatial orientations or orbitals. The total number of individual orbitals within any subshell is given by this formula:
- For s-subshell (𝒍 = 0): Total Orbitals = (2 × 0 + 1) = 1 orbital
- For p-subshell (𝒍 = 1): Total Orbitals = (2 × 1 + 1) = 3 orbitals (px, py, pz)
- For d-subshell (𝒍 = 2): Total Orbitals = (2 × 2 + 1) = 5 orbitals
- For f-subshell (𝒍 = 3): Total Orbitals = (2 × 3 + 1) = 7 orbitals
According to Pauli’s Exclusion Principle, each single orbital can hold a maximum of 2 electrons. Therefore, the maximum electron capacity of any subshell is evaluated using the formula:
- For s-subshell (𝒍 = 0): Maximum capacity = 2(2 × 0 + 1) = 2 electrons
- For p-subshell (𝒍 = 1): Maximum capacity = 2(2 × 1 + 1) = 6 electrons
- For d-subshell (𝒍 = 2): Maximum capacity = 2(2 × 2 + 1) = 10 electrons
- For f-subshell (𝒍 = 3): Maximum capacity = 2(2 × 3 + 1) = 14 electrons
Order of Electron Entry (Aufbau Principle)
During the electronic configuration of an atom in its ground state, electrons progressively occupy subshells in the increasing order of their energy levels. This relative energy is calculated using the (n + 𝒍) rule. The subshell with a lower (n + 𝒍) value possesses lower energy and is filled first.
- For 3d orbital: n = 3, 𝒍 = 2 ⇒ (n + 𝒍) = 3 + 2 = 5
- For 4s orbital: n = 4, 𝒍 = 0 ⇒ (n + 𝒍) = 4 + 0 = 4
Since the (n + 𝒍) value of 4s (4) is smaller than that of 3d (5), the 4s orbital has lower energy. Consequently, electrons enter the 4s orbital before filling 3d.
- For 4p orbital: n = 4, 𝒍 = 1 ⇒ (n + 𝒍) = 4 + 1 = 5
- For 5s orbital: n = 5, 𝒍 = 0 ⇒ (n + 𝒍) = 5 + 0 = 5
When two different subshells share identical (n + 𝒍) values, the subshell with the smaller principal quantum number (n) possesses lower energy and is filled first. Therefore, the 4p orbital is filled before 5s.
Conceptual Proofs (Academic Reasoning)
Question: Why is a 1p orbital not physically possible?
Proof: For the first principal energy shell, n = 1. The permitted values of the azimuthal quantum number range from 0 to (n − 1). Since n = 1, (n − 1) = 1 − 1 = 0. Hence, the only possible value for 𝒍 is 0, which strictly corresponds to the s-subshell (1s). For a p-subshell to exist, 𝒍 must equal 1. Since 𝒍 = 1 cannot exist when n = 1, a 1p orbital is mathematically and physically impossible.
Question: Why is a 2d orbital not physically possible?
Proof: For the second principal shell, n = 2. The permitted values for 𝒍 range from 0 to (n − 1), which gives 𝒍 = 0 (2s subshell) and 𝒍 = 1 (2p subshell). For a d-subshell to exist, the value of 𝒍 must equal 2. However, when n = 2, 𝒍 cannot equal 2 because the maximum value permitted is (n − 1) = 1. Thus, a 2d orbital cannot exist in nature.
Orbital Angular Momentum of an Electron
The orbital angular momentum (L) of an electron revolving in any specific subshell is quantized and determined solely by its azimuthal quantum number (𝒍). The formula is stated as:
