Atomic Models

Models explaining atomic spectra through discrete energy levels. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Atomic models grew from a stubborn clue: atoms do not emit any color of light, but separated spectral lines. That means the electron cannot have just any internal state. In a classical picture, an orbiting electron should radiate continuously and fall into the nucleus, yet real atoms are stable. The quantum idea changes the stage: only certain states are allowed, labeled by $n$, and each one has a definite $E_n$. An atom is therefore not a tiny free planetary system, but a microscopic energy staircase.

🟢 Essential level

A quantized atomic model explains spectra by restricting the possible electron states. The electron is not assigned any arbitrary size or energy: it may occupy only allowed levels labeled by $n$.

Each level has a definite $E_n$, and when the electron changes level, the atom exchanges a specific $\Delta E$. That energy appears as a photon whose $\lambda$ fixes one spectral line. This is why hydrogen does not emit a continuous rainbow, but selected colors. Discreteness is not a measurement defect; it is a property of the quantum system. The Bohr model represents this with allowed orbits and sizes such as $r_n$, although its deeper value is showing that atomic stability and spectra come from quantum restrictions, not classical freedom. It is like the rungs of a ladder: a foot can rest on one rung or the next, never on a floating point in between, and each change of rung costs or releases exactly the same amount of effort every time.

🔵 Formal level

Basic formulas

In the Bohr model for hydrogen-like atoms, the state is labeled by $n$, which takes positive integer values. The $E_n$ is set by the scale $E_R$ and by the nuclear charge $Z$. The central relation is:

Bohr energy level

The model also assigns an allowed spatial scale, the $r_n$, related to $a_0$, $n$, and $Z$:

Bohr orbital radius

Real student questions

Why can the electron not sit between two levels? Because those intermediate possibilities do not satisfy the model's quantum conditions; they are not allowed stationary states.

Does the electron jump like a tiny ball? No. “Jump” means a change of quantum state, not a filmable trajectory.

Why is $E_n$ usually negative? Because it is measured relative to a free electron; energy must be supplied to separate it.

Is $r_n$ real? It is a useful Bohr-model scale, not an exact classical orbit.

Why do we see lines? Because each $\Delta E$ produces a specific $\lambda$.

Final synthesis

Atomic models explain stability and spectra by replacing continuous energies with discrete levels. $n$, $E_n$, $r_n$, and $\Delta E$ connect internal structure with observed light.