SHELL / n
A principal level
Allowed values are n = 1, 2, 3, and so on. A shell contains n² spatial orbitals and has a maximum capacity of 2n² electrons. Capacity is not the same as the occupancy of a particular atom.
2n² = 2, 8, 18, 32 …
Foundations of chemistry / atomic structure
Electron shells.
Beyond the planetary picture.
See atomic orbitals as three-dimensional probability regions, build electron configurations, and connect transitions between energy levels to absorbed and emitted light.
Ready for offline use. Orbital meshes, probability samples, calculations and illustrations are embedded. No remote 3D service or library is needed.
01 / Build the vocabulary
A shell is not an orbit
SUBSHELL / l
A family of orbital shapes
The angular-momentum quantum number l ranges from 0 to n − 1. Values 0, 1, 2 and 3 are called s, p, d and f. They contain 1, 3, 5 and 7 orbitals, respectively.
s: 2 p: 6 d: 10 f: 14
ORBITAL / STATE
Two electrons at most
A spatial orbital can hold at most two electrons, with opposite spin projections. Spin is an intrinsic quantum property, not a picture of an electron physically spinning like a tiny ball.
↑↓ within one orbital
| Principal shell | Allowed subshells | Number of orbitals | Maximum electrons |
|---|---|---|---|
| n = 1 | 1s | 1 | 2 |
| n = 2 | 2s, 2p | 4 | 8 |
| n = 3 | 3s, 3p, 3d | 9 | 18 |
| n = 4 | 4s, 4p, 4d, 4f | 16 | 32 |
OpenStax: quantum numbers and orbitals · Electron configurations.
02 / A mathematical model you can rotate
Three-dimensional orbital explorer
The surfaces are calculated from hydrogen wavefunctions, not assembled from decorative spheres. Select s, p or d orbitals, turn on a cutaway to reveal radial structure, or accumulate independent position samples from the probability distribution.
HYDROGEN 2pz / PROBABILITY SURFACE
Preparing embedded wavefunction surfaces...z axis vertical · orthographic view
Positive wavefunction signNegative wavefunction signAuto-fit is on unless a common physical scale is selected.
Principal quantum numbern = 2
Angular quantum numberl = 1
Radial nodes / n − l − 10
Angular nodes / l1
What are the model's mathematical and numerical limits?
The spatial wavefunction is ψ(r,θ,φ) = Rnl(r)Y(θ,φ). Rnl uses the normalized hydrogenic associated-Laguerre expression with Z = 1. The pz, dz² and dxy shapes use normalized real spherical harmonics. In particular, dxy is a real combination of ml = +2 and −2 states; it is not a separate definite-ml eigenstate.
Each field is sampled on a 72 × 72 × 72 Cartesian grid. A common |ψ| threshold for the two signs is selected by summing grid probabilities to approximately 90%, and the boundaries are extracted as triangle meshes. Mesh discretization and finite grid extent introduce small numerical errors. Tiny inner features may be less accurate than the large-scale shape.
In sample mode, the 5,000 positions are independent draws from the discrete |ψ|² distribution with sub-voxel jitter. They represent repeated position measurements on identically prepared systems, not 5,000 electrons in one orbital and not a trajectory. Their colors are theoretical phase annotations, not an extra position measurement.
For a stationary state, the probability density is time-independent. Rotation here changes the camera orientation; accumulating dots reveals a distribution. Neither operation simulates a classical electron moving around the nucleus.
MIT OpenCourseWare: hydrogen atom wavefunctions · Hydrogen radial functions.
03 / Density is not the same as radial probability
Where is an electron likely to be found?
The 3D surface shows a spatial probability density. The graph below integrates over directions and asks a different question: how likely is a measurement within a thin spherical layer at distance r?
A spherical layer has volume
P(r) = r² |Rnl(r)|²
With normalized angular functions, P(r)dr is the probability of measuring a radius between r and r + dr. Distances are shown in Bohr radii, a0 ≈ 0.0529 nm.
For hydrogen 1s, the spatial density is largest at the nucleus, but the radial distribution peaks at r = a0. There is very little volume at extremely small r, so these statements do not conflict.
A radial node is a spherical surface where the radial wavefunction vanishes. A pz angular node is the xy plane. Nodes are places where the ideal wavefunction is zero, not solid barriers.
Select a different orbital above to compare the number of radial peaks and nodes.
The plotted radial functions are the same ones used for the 3D meshes. MIT: wavefunctions and probability.
04 / From orbitals to atoms
Build an electron configuration
Choose a neutral atom among the first 20 elements, or a listed common ion. The shell-count diagram and orbital boxes are bookkeeping aids, not electron paths. This panel uses the standard introductory ground-state filling sequence for these examples.
Main-group positions are shown; transition-metal columns are omitted. This is a limited element palette, not the entire periodic table.
Atomic number / protons
Electrons
Charge in units of e
Oxygen / ground state
Orbital occupancy diagram
One box is one spatial orbital. Two arrows in a box mean two electrons with opposite spin projections. The left-to-right order within a degenerate p subshell is conventional.
Three filling rules
Aufbau: occupy lower-energy available orbitals first in this ground-state model.
Pauli exclusion: a spatial orbital holds no more than two electrons with opposite spin projections.
Hund's rule: within a set of degenerate orbitals, place one electron in each with parallel spin before pairing.
Scope: the simple filling engine is restricted to the first 20 neutral atoms and the explicitly listed ions. It is not a general configuration solver for transition metals, excited states or arbitrary ions. Listing a common chemical ion does not predict its stability as an isolated gas-phase ion. The hydrogen 3D shapes above do not change into exact many-electron atomic orbitals when an element is selected here.
Why potassium is 2, 8, 8, 1: neutral potassium begins filling 4s after 3p; it does not first put 18 electrons into the n = 3 shell. The third shell's capacity remains 18. Shell capacities alone do not determine the filling order.
05 / Energy changes become light
Hydrogen energy-transition laboratory
Choose an initial and final principal energy level. A transition to a lower energy releases a photon; a transition to a higher energy requires energy input. The moving marker on the diagram represents a change of state, not motion along a spatial orbit.
Process
Photon energy
Wavelength
En ≈ −13.6 / n² eV
λ = hc / |ΔE|
Visible Balmer lines / n → 2
400 nm500 nm600 nm700 nm
Displayed colors are approximate screen colors. Ultraviolet and infrared photons do not have a human-visible color; any animation of those photons is a symbolic marker.
This is a principal-level energy calculation for hydrogen, using a rounded 13.6 eV constant. It omits fine structure, isotope corrections, lifetimes and transition strengths. It does not assert that every arbitrary pair of orbital states is connected by a permitted single-photon electric-dipole transition; angular-momentum selection rules also matter.
OpenStax: hydrogen energy levels and spectra · NIST: measured hydrogen energy levels.
06 / Guided investigations
Make the distinctions yourself
Surface versus probability
- Choose 2s and turn on the cutaway.
- Compare it with 2pz.
- Accumulate samples and identify where points are scarce.
Expected observation
2s has one radial node; 2p has no radial node but one angular nodal plane. The dots sample probability, and low-density areas can appear sparse without being exact nodes. The color change marks wavefunction sign.
Capacity is not occupancy
- Select argon, then potassium, then calcium.
- Read both the configuration and shell counts.
- Explain why n = 3 does not contain 18 electrons.
Expected observation
Argon is [Ne]3s²3p&sup6. Potassium and calcium add electrons to 4s, producing shell counts 2,8,8,1 and 2,8,8,2. The 3d subshell exists but is unoccupied in these ground-state examples.
Energy and wavelength
- Compare 3 → 2 with 6 → 2.
- Predict which photon has the shorter wavelength.
- Reverse one transition and identify absorption.
Expected observation
6 → 2 has a larger energy difference and a shorter wavelength than 3 → 2. Reversing the same pair preserves the required photon-energy magnitude but changes emission to absorption.
07 / Knowledge check
Check your understanding
Grade the answers to reveal explanations. Nothing is transmitted or stored outside this page.
Read further
Atomic structure and quantum resources
Quantum numbers and orbitals
Develop the distinction between shells, subshells, orbitals and quantum numbers.
Open resource →Electronic structure of atoms
Study Aufbau, Pauli exclusion, Hund's rule and exceptions beyond this tutorial's element range.
Open resource →Hydrogen levels and the Bohr model
Understand the historical model, its successful energy predictions and its limitations.
Open resource →Hydrogen atom wavefunctions
A university lecture linking orbital mathematics, shapes and probability distributions.
Open resource →Hydrogen energy levels
Inspect measured energy levels and fine structure rather than rounded teaching values.
Open resource →Build an Atom
Practice the relationships among proton number, electron number, isotope and charge.
Open resource →Models of the Hydrogen Atom
Compare historical and quantum descriptions of hydrogen with an independent simulation.
Open resource →Hydrogen radial wavefunctions
Work through radial solutions and their relation to probability distributions.
Open resource →