Bohr Model of the Atom: Energy Levels and the Hydrogen Spectrum

Here mₑ = 9.11·10⁻³¹ kg is the electron mass, vₙ and rₙ are its speed and orbit radius, and ħ = h/2π = 1.055·10⁻³⁴ J·s is the reduced Planck constant. The condition picks out only a discrete set from all the classical circular orbits.

The meaning of quantization was explained by de Broglie in 1924: the electron has a wave with wavelength λ = h/(mₑv). Substituting it into Bohr’s condition gives 2πrₙ = n·λ — a whole number of waves fits along the orbit, and the wave closes on itself as a standing wave. Turn on “de Broglie wave on the orbit” in the simulator: orbit n holds exactly n wavelengths.

Here h = 6.63·10⁻³⁴ J·s is Planck’s constant and ν is the photon frequency. The photon wavelength is λ = c/ν = hc/(Eₖ − Eₙ); it is handy to remember that hc ≈ 1240 eV·nm, so a 1.89 eV photon has λ ≈ 1240/1.89 ≈ 656 nm.

Two important things follow from the frequency rule. First, the emission spectrum of an atom consists of lines: the levels are discrete, so the energy differences take only certain values. Second, an atom absorbs only photons whose energy exactly matches the difference between two of its levels — that is why a gas’s absorption lines sit in the same places as its emission lines. The simulator shows this at once: a dark line in the absorption spectrum appears right under the bright emission line of the same transition.

The total energy is the sum of the electron’s kinetic energy and the potential energy of its attraction to the nucleus: E = mv²/2 − kZe²/r. For a circular orbit the kinetic energy is half the magnitude of the potential energy, so the total energy is negative and equals −kZe²/(2r). Substituting Bohr’s radii gives the level formula. Here e = 1.6·10⁻¹⁹ C is the elementary charge and ε₀ = 8.85·10⁻¹² F/m is the electric constant.

A negative energy means the electron is bound: to free it, the energy |Eₙ| must be supplied. For the hydrogen ground state this is the ionization energy of 13.6 eV. As n grows the levels crowd towards zero, and above zero lies the region of a free electron, where any energy is possible. For hydrogen-like ions (He⁺, Li²⁺) the energies are Z² times larger in magnitude: for He⁺ E₁ = −54.4 eV. An exact calculation also accounts for the motion of the nucleus: the electron mass is replaced by the reduced mass, and Ry for hydrogen becomes 13.598 eV.

The formula comes from two conditions. Newton’s second law: the Coulomb force gives the electron its centripetal acceleration, mv²/r = kZe²/r². Bohr’s first postulate: mvr = nħ. Eliminating the speed gives rₙ = a₀n²/Z, and the orbital speed is vₙ = v₁Z/n with v₁ = 2.19·10⁶ m/s — less than 1 % of the speed of light, so relativity can be ignored.

The quadratic growth of the radius is clearly visible in the simulator when you turn on “Real-scale orbits”: orbit n = 6 is 36 times larger than the first, and the inner orbits almost merge with the nucleus. In the normal mode the orbits are drawn to a schematic scale so that all six fit on the screen. A hydrogen atom excited to n = 100 would have a radius of about 0.5 µm — such giant “Rydberg atoms” are actually produced in laboratories.

The formula was found from experiment long before Bohr: in 1885 Balmer fitted it to the visible hydrogen lines, and Rydberg generalized it. Bohr was the first to derive it from theory: substituting Eₙ into the frequency rule hν = Eₖ − Eₙ with λ = c/ν gives R = Ry/(hc), and the calculated value agreed with the measured one — a triumph of the model.

Each lower level n₁ has its own series: n₁ = 1 is the Lyman series (ultraviolet, 91–122 nm), n₁ = 2 the Balmer series (visible and near ultraviolet, 365–656 nm), n₁ = 3 the Paschen series (infrared, 820–1875 nm), followed by the Brackett and Pfund series. As n₂ → ∞ the lines converge to the series limit λ = n₁²/(RZ²). Press a series button in the simulator and watch the lines crowd one after another towards this limit. Example: for Hβ (n₂ = 4, n₁ = 2) 1/λ = 1.097·10⁷·(1/4 − 1/16) ≈ 2.06·10⁶ m⁻¹, λ ≈ 486 nm.