s, p, d and f Orbitals — Revision Notes

⚡ 30-Second Revision

Quantum Numbers — $n$ (energy/size), $l$ (shape, $0 dots n-1$), $m_l$ (orientation, $-l dots +l$), $m_s$ (spin, $pm 1/2$).
Orbital Types — $l=0 implies s$ (spherical), $l=1 implies p$ (dumbbell), $l=2 implies d$ (cloverleaf/complex), $l=3 implies f$ (very complex).
Orbitals per Subshell — s: 1, p: 3, d: 5, f: 7.
Max Electrons per Subshell — s: 2, p: 6, d: 10, f: 14.
Nodes — Angular nodes = $l$. Radial nodes = $n-l-1$. Total nodes = $n-1$.
Energy Order (Multi-electron) — $(n+l)$ rule. Lower $(n+l)$ is lower energy. If $(n+l)$ is same, lower $n$ is lower energy.
Filling Rules — Aufbau, Pauli Exclusion, Hund's Rule.

2-Minute Revision

Atomic orbitals are regions of high electron probability, defined by quantum numbers. The principal quantum number ($n$) dictates energy level and size. The azimuthal quantum number ($l$) determines the orbital's shape: $l=0$ for spherical s-orbitals, $l=1$ for dumbbell-shaped p-orbitals, $l=2$ for complex d-orbitals, and $l=3$ for very complex f-orbitals.

The magnetic quantum number ($m_l$) specifies spatial orientation, leading to 1 s-orbital, 3 p-orbitals, 5 d-orbitals, and 7 f-orbitals for a given $n$ and $l$. Each orbital holds a maximum of two electrons with opposite spins.

Nodal planes (angular nodes, equal to $l$) are regions of zero electron probability. Radial nodes are spherical regions of zero probability, calculated as $n-l-1$. Electron filling in multi-electron atoms follows the Aufbau principle (using the $(n+l)$ rule), Pauli exclusion principle, and Hund's rule, which are crucial for determining electronic configurations and understanding the periodic table.

5-Minute Revision

To master s, p, d, and f orbitals for NEET, focus on these core concepts:

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Quantum Numbers — Understand their roles: $n$ (principal) defines energy shell and size; $l$ (azimuthal) defines subshell and shape (s, p, d, f); $m_l$ (magnetic) defines spatial orientation; $m_s$ (spin) defines electron spin. Remember their allowed values: $n ge 1$; $0 le l le n-1$; $-l le m_l le +l$; $m_s = pm 1/2$. Practice identifying valid sets.

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Orbital Characteristics

* **s-orbitals ($l=0$)**: Spherical, 1 orbital per subshell, max 2 electrons. Example: 1s, 2s, 3s. Radial nodes = $n-1$. * **p-orbitals ($l=1$)**: Dumbbell, 3 orbitals ($p_x, p_y, p_z$) per subshell, max 6 electrons.

Starts from $n=2$. Angular nodes = 1. Radial nodes = $n-2$. Example: 2p, 3p. * **d-orbitals ($l=2$)**: Cloverleaf (4 types) or dumbbell with donut ($d_{z^2}$), 5 orbitals per subshell, max 10 electrons.

Starts from $n=3$. Angular nodes = 2. Radial nodes = $n-3$. Example: 3d, 4d. * **f-orbitals ($l=3$)**: Very complex, 7 orbitals per subshell, max 14 electrons. Starts from $n=4$. Angular nodes = 3. Radial nodes = $n-4$.

Example: 4f.

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Nodes — Crucial for NEET. Angular nodes = $l$. Radial nodes = $n-l-1$. Total nodes = $n-1$. For instance, a 4p orbital ($n=4, l=1$) has 1 angular node and $4-1-1=2$ radial nodes, totaling $4-1=3$ nodes.

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Energy Ordering (Aufbau Principle) — For multi-electron atoms, orbitals fill according to increasing energy, determined by the $(n+l)$ rule. Lower $(n+l)$ means lower energy. If $(n+l)$ is equal, the orbital with lower $n$ is filled first. Example: $4s (n+l=4)$ fills before $3d (n+l=5)$.

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Electron Filling Rules

* Pauli Exclusion Principle: Max 2 electrons per orbital, with opposite spins. * Hund's Rule: For degenerate orbitals, fill singly with parallel spins first before pairing up.

Worked Example: Determine the number of angular and radial nodes for a 5f orbital.

For 5f, $n=5$ and $l=3$.
Angular nodes = $l = 3$.
Radial nodes = $n-l-1 = 5-3-1 = 1$.
Total nodes = $n-1 = 5-1 = 4$. (Check: $3+1=4$).

Prelims Revision Notes

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Quantum Numbers — $n$ (Principal) $ightarrow$ shell, energy, size. $l$ (Azimuthal) $ightarrow$ subshell, shape. $m_l$ (Magnetic) $ightarrow$ orientation. $m_s$ (Spin) $ightarrow$ electron spin.

* $n = 1, 2, 3, dots$ * $l = 0, 1, dots, n-1$ (s, p, d, f) * $m_l = -l, dots, 0, dots, +l$ * $m_s = +1/2, -1/2$

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Orbital Types and Properties

* **s-orbital ($l=0$)**: Spherical. 1 orbital. Max 2 electrons. Min $n=1$. * **p-orbital ($l=1$)**: Dumbbell. 3 orbitals ($p_x, p_y, p_z$). Max 6 electrons. Min $n=2$. * **d-orbital ($l=2$)**: Cloverleaf/dumbbell+ring. 5 orbitals. Max 10 electrons. Min $n=3$. * **f-orbital ($l=3$)**: Complex. 7 orbitals. Max 14 electrons. Min $n=4$.

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Number of Orbitals/Electrons

* Total orbitals in a shell ($n$) = $n^2$. * Max electrons in a shell ($n$) = $2n^2$. * Orbitals in a subshell ($l$) = $2l+1$. * Max electrons in a subshell ($l$) = $2(2l+1)$.

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Nodes

* Angular nodes = $l$. * Radial nodes = $n-l-1$. * Total nodes = $n-1$.

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Energy Order (Multi-electron atoms)

* Follows $(n+l)$ rule. Lower $(n+l)$ means lower energy. * If $(n+l)$ is same, lower $n$ means lower energy. * Order: $1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p$.

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Electron Filling Rules

* Aufbau Principle: Fill lowest energy orbitals first. * Pauli Exclusion Principle: Max 2 electrons per orbital, with opposite spins. * Hund's Rule: For degenerate orbitals, fill singly with parallel spins before pairing.

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Exceptions — Cr ($[Ar]3d^54s^1$), Cu ($[Ar]3d^{10}4s^1$). For ions, remove electrons from highest $n$ first (e.g., $Fe^{2+}$ is $[Ar]3d^6$, not $[Ar]4s^23d^4$).

Vyyuha Quick Recall

To remember the order of filling orbitals (Aufbau principle) using the (n+l) rule, think of: 'Some People Don't Follow' for s, p, d, f. For the energy order, visualize a diagonal rule diagram, or simply remember: 'S-P-S-P-D-P-S-D-P-F-D-P...' (1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p). For nodes: Angular = L ($l$), Radial = No Longer 1 ($n-l-1$). Total = No 1 ($n-1$). This helps quickly recall the formulas.