Principal Quantum Number — Revision Notes

⚡ 30-Second Revision

Symbol: — 'n'\n- Allowed Values: Positive integers ($1, 2, 3, \dots$)\n- Primary Role: Determines main energy level (shell) and orbital size.\n- Energy: Higher 'n' = higher energy (less stable). For H-atom: $E_n \propto -1/n^2$.\n- Size: Higher 'n' = larger orbital (further from nucleus). For H-atom: $r_n \propto n^2$.\n- Shell Names: $n=1$ (K), $n=2$ (L), $n=3$ (M), $n=4$ (N).\n- Number of Orbitals in a Shell: $n^2$\n- Maximum Electrons in a Shell: $2n^2$\n- Relation to 'l': $l$ can range from $0$ to $n-1$.

2-Minute Revision

The Principal Quantum Number, 'n', is the most fundamental quantum number, taking positive integer values ($1, 2, 3, \dots$). It primarily dictates the electron's main energy level and the average size of the orbital.

Higher 'n' values mean higher energy (less negative, less stable) and larger orbitals (further from the nucleus). For hydrogenic atoms, energy $E_n = -R_H Z^2/n^2$ and radius $r_n = n^2 a_0/Z$. Each 'n' corresponds to a specific electron shell: K ($n=1$), L ($n=2$), M ($n=3$), etc.

A shell with principal quantum number 'n' contains $n^2$ orbitals and can accommodate a maximum of $2n^2$ electrons. It's crucial to remember that 'n' limits the azimuthal quantum number 'l', where 'l' can only take values from $0$ to $n-1$.

This relationship is frequently tested in NEET for identifying valid sets of quantum numbers. For multi-electron atoms, the (n+l) rule helps determine the relative energy order of orbitals.

5-Minute Revision

The Principal Quantum Number (n) is the first of the four quantum numbers, serving as the primary descriptor of an electron's state. It is always a positive integer ($n=1, 2, 3, \dots$). Its core functions are to define the main energy level (or electron shell) and the average size of the orbital.

\n\nEnergy: As 'n' increases, the electron's energy increases (becomes less negative), meaning it is less tightly bound to the nucleus and therefore less stable. For a hydrogen atom, the energy is given by $E_n = -2.

18 \times 10^{-18} \frac{1}{n^2}$ J. This explains why electrons in higher shells are easier to remove (lower ionization energy). \n\nSize: A higher 'n' value signifies a larger orbital, implying the electron is, on average, further away from the nucleus.

This directly impacts the atomic radius. For hydrogen, the radius of the $n^{th}$ orbit is $r_n = 0.529 \times n^2$ Å. \n\nShells: Each 'n' value corresponds to a specific electron shell, historically designated by letters: K-shell ($n=1$), L-shell ($n=2$), M-shell ($n=3$), N-shell ($n=4$), and so on.

\n\nOrbital and Electron Capacity: For any given shell 'n':\n* The total number of subshells is equal to 'n'.\n* The total number of orbitals is $n^2$. For example, for $n=2$ (L-shell), there are $2^2=4$ orbitals (one 2s, three 2p).

\n* The maximum number of electrons that can be accommodated is $2n^2$. For $n=2$, this is $2 \times 2^2 = 8$ electrons. \n\nRelationship with other Quantum Numbers: 'n' sets the limit for the azimuthal quantum number 'l'.

For a given 'n', 'l' can take integer values from $0$ to $n-1$. This means an $n=1$ shell only has $l=0$ (s subshell), while an $n=3$ shell can have $l=0, 1, 2$ (s, p, d subshells). This interdependence is crucial for determining valid quantum number sets.

\n\nExample: Consider an electron in the 4p orbital. Here, $n=4$ and $l=1$. This is a valid combination because $l=1$ is within the allowed range of $0$ to $n-1$ (i.e., $0, 1, 2, 3$). The electron is in the N-shell, has a specific energy level, and occupies a p-shaped orbital of a certain size.

Prelims Revision Notes

Principal Quantum Number (n) - NEET Revision Notes\n\n1. Definition & Role:\n* Symbol: 'n'\n* Primary Function: Defines the main energy level (shell) and the average size of the electron orbital.\n* Quantization: Energy and size are quantized, meaning they can only take specific discrete values determined by 'n'.\n\n2. Allowed Values:\n* 'n' can only be a positive integer: $n = 1, 2, 3, 4, \dots$\n* Cannot be zero, negative, or fractional.\n\n3. Shell Designations:\n* Each 'n' value corresponds to a specific electron shell:\n * $n=1 \rightarrow$ K-shell\n * $n=2 \rightarrow$ L-shell\n * $n=3 \rightarrow$ M-shell\n * $n=4 \rightarrow$ N-shell\n\n4. Energy Dependence:\n* Higher 'n' = Higher Energy: Electrons in shells with larger 'n' values have higher energy (less negative, less stable) and are less tightly bound to the nucleus.\n* Formula (Hydrogenic Atoms): $E_n = -R_H \frac{Z^2}{n^2}$, where $R_H$ is Rydberg constant ($2.18 \times 10^{-18}$ J) and Z is atomic number.\n* Energy Gaps: The energy difference between successive shells decreases as 'n' increases (e.g., $E_2-E_1 > E_3-E_2$).\n\n5. Orbital Size Dependence:\n* Higher 'n' = Larger Orbital: Orbitals with larger 'n' values are, on average, further from the nucleus and have a larger spatial extent.\n* Formula (Hydrogenic Atoms): $r_n = \frac{n^2 a_0}{Z}$, where $a_0$ is Bohr radius ($0.529$ Å).\n\n6. Orbital and Electron Capacity per Shell:\n* Number of Subshells: For a given 'n', there are 'n' subshells (e.g., for $n=3$, there are 3s, 3p, 3d subshells).\n* Number of Orbitals: Total orbitals in a shell = $n^2$.\n * Example: $n=3 \rightarrow 3^2 = 9$ orbitals (one 3s, three 3p, five 3d).\n* Maximum Electrons: Maximum electrons in a shell = $2n^2$ (due to Pauli's Exclusion Principle, 2 electrons per orbital).\n * Example: $n=3 \rightarrow 2 \times 3^2 = 18$ electrons.\n\n7. Relationship with Azimuthal Quantum Number (l):\n* 'n' limits 'l': For a given 'n', 'l' can take integer values from $0$ to $n-1$.\n* Example: If $n=2$, 'l' can be $0$ (s-subshell) or $1$ (p-subshell). It cannot be $2$.\n\n8. Multi-electron Atoms:\n* Energy of orbitals depends on both 'n' and 'l'.\n* Aufbau Principle / (n+l) Rule: Orbitals with lower (n+l) values fill first. If (n+l) is the same, the orbital with lower 'n' fills first (e.g., 4s fills before 3d because 4+0=4 and 3+2=5).

Vyyuha Quick Recall

Nice Elephants Sing Out Loudly: \nN - Principal Quantum Number\nE - Energy (main determinant)\nS - Size (main determinant)\nO - Orbitals ($n^2$ per shell)\nL - Limits 'l' ($l < n$)