Bohr Model Calculator
Calculate orbital radius, energy levels, electron velocity, and photon wavelengths for hydrogen-like atoms using the Bohr model of the atom.
Quantum Number Calculator
Analyze and validate sets of quantum numbers (n, l, mₗ, mₛ), explore subshell capacities, orbital nodes, and generate electron configurations using the Aufbau principle.
Quantum Number Calculator
Shell number (n = 1, 2, 3, …)
Must be 0 to 1 (s=0, p=1, d=2, f=3)
Must be −l to +l (−1 to +1)
Orbital Notation⧉
2p
n=2, l=1 (p), mₗ=0, mₛ=+½
Validation⧉
✅ Valid set
All four quantum numbers satisfy the selection rules
Max Electrons (subshell)⧉
6
2p holds up to 2(2l+1) = 6 electrons in 3 orbital(s)
Max Electrons (shell)⧉
8
Shell n=2 holds 2n² = 8 electrons total
Nodes⧉
1 total (0 radial + 1 angular)
Radial nodes = n−l−1 = 0. Angular nodes = l = 1. Total = n−1 = 1
Orbital Angular Momentum⧉
1.4142 ħ
L = √(l(l+1))·ħ = 0.0000 J·s
Orbital Orientations (mₗ values for l=1)
mₗ = -1
mₗ = 0
mₗ = +1
Subshell Capacities for Shell n=2
2s
2 e⁻
2p
6 e⁻
Subshell Details for n=2
| Subshell | l | Orbitals | Max e⁻ | Radial Nodes | Angular Nodes |
|---|---|---|---|---|---|
| 2s | 0 | 1 | 2 | 1 | 0 |
| 2p | 1 | 3 | 6 | 0 | 1 |
Planning notes, formulas, and examples
About the Quantum Number Calculator
Every electron in an atom is uniquely described by four quantum numbers — a result known as the Pauli exclusion principle. The principal quantum number n (1, 2, 3, …) determines the electron's shell and overall energy. The angular momentum quantum number l (0 to n−1) defines the subshell shape: l = 0 is an s orbital (spherical), l = 1 is p (dumbbell), l = 2 is d (cloverleaf), and l = 3 is f. The magnetic quantum number mₗ (−l to +l) specifies the orbital's orientation in space, and the spin quantum number mₛ (+½ or −½) distinguishes the two electrons that can occupy the same orbital.
Together, these four numbers encode the full quantum state of an electron and determine properties like orbital shape, angular momentum, number of nodes, and maximum occupancy. This calculator validates any set of quantum numbers, computes orbital properties (capacity, nodes, angular momentum), and shows the subshell structure of any shell. The electron configuration mode builds the ground-state configuration for any element up to Z = 118 using the Aufbau principle.
Whether you're checking homework answers, preparing for an exam, or exploring how quantum numbers determine atomic structure, it gives clear visual feedback with orbital diagrams, capacity bar charts, and configuration tables.
When This Page Helps
Quantum numbers are fundamental to understanding atomic structure, chemical bonding, and the periodic table. This calculator helps students validate quantum number sets, visualize subshell capacities, and generate electron configurations — all common tasks in general chemistry and introductory physics courses. The instant validation catches common mistakes before they propagate through multi-step problems.
How to Use the Inputs
- Enter the principal quantum number n (1–7 for common elements).
- Enter the angular momentum quantum number l (0 to n−1).
- Enter the magnetic quantum number mₗ (−l to +l).
- Select the spin quantum number mₛ (+½ or −½).
- Check the validation result to confirm the quantum numbers are allowed.
- Switch to "Electron Configuration" mode and enter an atomic number to see the full Aufbau configuration.
- Use the presets for common orbitals like 1s, 2p, 3d, and 4f.
Formula used
Quantum Number Rules:
• n = 1, 2, 3, … (principal — determines shell and energy)
• l = 0, 1, 2, …, n−1 (angular momentum — determines subshell shape)
• mₗ = −l, −l+1, …, 0, …, l−1, l (magnetic — orbital orientation)
• mₛ = +½ or −½ (spin — electron spin direction)
• Orbitals per subshell: 2l + 1
• Max electrons per subshell: 2(2l + 1)
• Max electrons per shell: 2n²
• Orbital angular momentum: L = √(l(l+1)) ħ
• Nodes: radial = n−l−1, angular = l, total = n−1Example Calculation
Result: 3d orbital, 10 max electrons, 2 angular + 0 radial nodes
The set n=3, l=2, mₗ=0, mₛ=+½ describes an electron in a 3d orbital. The 3d subshell has 5 orbitals holding up to 10 electrons, with 2 angular nodes and 0 radial nodes.
Tips & Best Practices
- Remember: l ranges from 0 to n−1, not 1 to n. This is a very common exam mistake.
- Each orbital (unique n, l, mₗ) holds exactly 2 electrons — one spin up, one spin down.
- The letters s, p, d, f come from historical spectroscopic terms: sharp, principal, diffuse, fundamental.
- Exceptions to Aufbau: Cr is [Ar] 3d⁵4s¹ and Cu is [Ar] 3d¹⁰4s¹ due to half/full-shell stability.
- The total number of nodes (n−1) increases with shell number, making higher orbitals more complex.
- Use electron configuration mode to quickly check ground-state configurations for any element.
Quantum Numbers and the Periodic Table
The structure of the periodic table directly reflects the quantum number filling rules. Each row (period) corresponds to filling a new principal shell n. The s-block (groups 1–2) fills l=0, the p-block (groups 13–18) fills l=1, the d-block (transition metals, groups 3–12) fills l=2, and the f-block (lanthanides and actinides) fills l=3. Understanding quantum numbers therefore explains why the periodic table has its characteristic shape and why elements in the same group share similar chemical properties.
Orbital Shapes and Chemistry
The shape of an orbital, determined by l, directly influences chemical bonding. s orbitals are spherically symmetric and form sigma bonds. p orbitals are directional and can form both sigma and pi bonds, explaining the geometry of molecules like water and ammonia. d orbitals enable the rich coordination chemistry of transition metals, while f orbitals are responsible for the unique magnetic and optical properties of rare-earth elements used in magnets, lasers, and phosphors.
Beyond Hydrogen: Electron-Electron Interactions
In hydrogen-like atoms, energy depends only on n. In multi-electron atoms, electron-electron repulsion splits the energy levels — orbitals with the same n but different l have different energies. This splitting determines the Aufbau order and explains exceptions like chromium and copper. Advanced computational methods (Hartree-Fock, density functional theory) solve the multi-electron problem numerically, but quantum numbers remain the fundamental language for describing electronic states.
Sources & Methodology
Last updated:
Frequently Asked Questions
n gives the shell (size/energy), l gives the subshell shape, mₗ gives the orbital orientation, and mₛ gives the electron spin direction. Together they uniquely identify one electron state in an atom.
The Pauli exclusion principle forbids two identical fermions (like electrons) from occupying the same quantum state. This is why each orbital holds at most 2 electrons with opposite spins.
The Aufbau (building-up) principle states that electrons fill orbitals starting from the lowest energy. The filling order roughly follows increasing n+l, with lower n filled first when n+l is equal, giving the sequence 1s, 2s, 2p, 3s, 3p, 4s, 3d, etc.
Nodes are surfaces where the probability of finding the electron is zero. Angular nodes are planes or cones (l of them), radial nodes are spherical surfaces (n−l−1 of them), and the total is always n−1.
In multi-electron atoms, 4s has lower energy than 3d due to electron-electron repulsion and penetration effects. The 4s electron penetrates closer to the nucleus than 3d, giving it a lower effective energy.
No. The rule is l = 0, 1, …, n−1. So for n=1 only l=0 (s) is allowed; for n=2, l can be 0 (s) or 1 (p); and so on.
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