The Bohr model was the first quantum picture of the atom: instead of a continuum of possible orbits, an electron can only occupy a fixed ladder of energy levels, and it moves between them by absorbing or emitting a photon of exactly the right energy. This calculator applies the model to hydrogen and hydrogen-like ions to compute energy, orbital size, speed, and transition energies.
Why energy is quantized and negative
In the Bohr model, an electron bound to a nucleus can only exist at specific, discrete energy levels indexed by the principal quantum number n = 1, 2, 3, … The energy of each level, Eₙ = −13.6·Z²/n² eV, is negative because the electron is bound — it takes positive energy input to pull it away from the nucleus. As n increases, Eₙ climbs toward zero, and at n = ∞ the electron is completely free (ionized). The magnitude of Eₙ at any level is therefore also that level's ionization energy: the energy needed to remove the electron entirely from that starting point.
How orbital radius and velocity scale
The orbital radius rₙ = 0.529·n²/Z Å grows with the square of n, meaning higher-energy orbits are dramatically larger — the n=2 orbit is four times the radius of n=1. A larger nuclear charge Z pulls the electron into a tighter orbit, shrinking rₙ proportionally. The orbital velocity vₙ = 2.188×10⁶·Z/n m/s comes from balancing the Coulomb attraction toward the nucleus against the centripetal force needed to keep the electron in a circular path — it scales up with Z (a stronger pull requires more speed to maintain orbit) and down with n (higher, more distant orbits move more slowly).
Reading a transition: absorption vs. emission
The Transition tab computes the energy exchanged when an electron moves from an initial level n1 to a final level n2. If n2 is higher than n1, the electron has gained energy — it absorbed a photon to make the jump. If n2 is lower than n1, the electron has released energy — it emitted a photon falling to a more tightly bound state. The calculator always reports the magnitude of that energy difference alongside a clear absorption/emission label, plus the corresponding photon wavelength using λ(nm) = 1240/ΔE(eV).
Where the simple Bohr model breaks down
The Bohr model is exact only for hydrogen-like systems — a single electron orbiting a nucleus of charge Z, such as hydrogen itself, He⁺, or Li²⁺. It ignores electron spin, relativistic corrections, and the electron-electron repulsion (shielding) present in any atom with more than one electron, so it cannot be applied directly to neutral helium or heavier elements. Modern quantum mechanics — the Schrödinger equation and its relativistic extension, the Dirac equation — is needed to model multi-electron atoms accurately, though the Bohr model's energy-level formula remains a remarkably good first approximation for one-electron systems.