Bohr Model and Hydrogen Spectrum Simulator

What the Bohr model says

The Bohr model says the electron in a hydrogen atom can only circle the nucleus on certain orbits, each with a fixed energy, Eₙ = −13.6 Z²/n² eV, and that the atom gives out or takes in light only when the electron jumps from one orbit to another. The photon carries exactly the energy between the two levels, which is why hydrogen’s light is a set of sharp lines rather than a rainbow, and its wavelength follows from the Rydberg formula, 1/λ = RZ²(1/n₁² − 1/n₂²), with n₁ the lower level and n₂ the upper.

Niels Bohr reached it in 1913 from two conditions. The pull of the nucleus supplies the force that keeps the electron on its circle, and the orbit’s angular momentum is a whole number of ℏ, the reduced Planck constant. The first ties the speed to the radius; the second picks out the allowed orbits, numbered n = 1, 2, 3 and so on, with radii growing as n² and energies closing in on zero as 1/n². The same two conditions work for any atom with a single electron, such as He⁺ and Li²⁺, with Z the charge of the nucleus.

Using the simulator

Choose the atom, the level the electron starts in and the level it ends in. When the end is below the start the atom emits a photon, and when it is above, the atom absorbs one. The atom on the left shows the jump with the orbits to scale, the diagram on the right shows the same jump as an arrow between two energies, and the strip underneath shows where the line falls among the others in its series. Tap a level or an orbit to send the electron there from wherever it is now, or tap a line in the strip to pick it; the arrow keys do the same from the keyboard.

• Wavelength is in a vacuum, straight from the Rydberg formula, and In air is what a table of lines prints between 200 nm and 2 µm.
• Photon emitted or Photon absorbed is its energy in electronvolts, with joules, frequency and wavenumber beside it, the four forms exam questions ask for.
• Line names it: the series comes from the lower level and the Greek letter from how many levels the jump spans, so 3 to 2 is Balmer α, which for hydrogen is called Hα.
• Start and End give each level’s energy, with its orbit’s radius and the electron’s speed in the hint.

The plot turns the series into a straight line. Rydberg’s formula says the wavenumber falls linearly with 1/n₂², so every line of a series lies on one line of slope −RZ², and where it meets the axis, at 1/n₂² = 0, is the series limit. Plotted this way, lines measured with a spectrometer give R from the slope.

Worked example: the red line of hydrogen

Hydrogen falling from n = 3 to n = 2, which is where the simulator starts. With the proton’s recoil included, hydrogen’s Rydberg constant is R_H = 109,677.58 cm⁻¹, so

1/λ = 109,677.58 cm⁻¹ × (1/2² − 1/3²) = 109,677.58 × 5/36 = 15,233.00 cm⁻¹

and λ = 1/15,233.00 cm = 656.47 nm in a vacuum, the red Hα line. In air it is 656.29 nm, the 656.3 nm printed in most tables. The energy route gives the same photon: the two levels are at E₃ = −1.51092 eV and E₂ = −3.39957 eV, so the atom loses 1.88865 eV, which is 3.02595 × 10⁻¹⁹ J, and λ = hc/E = 1239.842 eV nm ÷ 1.88865 eV = 656.47 nm. Its frequency is c/λ = 4.5667 × 10¹⁴ Hz. The simulator shows each of these to five figures; the working carries six so that the last step still comes out at 656.47 nm.

Set the start level to 5 for Hγ, the jump NCERT works in Problem 2.10 of its chemistry chapter on the structure of the atom: the photon carries 2.8556 eV, which is 4.5752 × 10⁻¹⁹ J, at 434.17 nm. NCERT, using 2.18 × 10⁻¹⁸ J for the Rydberg energy, gets 4.58 × 10⁻¹⁹ J and 6.91 × 10¹⁴ Hz, where the simulator gives 6.9049 × 10¹⁴ Hz.

The energy levels of hydrogen

The energies close in on zero, the electron at rest far from the nucleus, so the gaps between levels shrink as n grows while the orbits spread out as n². With the nucleus held still the first radius is the Bohr radius, 52.918 pm, which textbooks round to 52.9 pm. The kinetic energy in any orbit is the size of its total energy and the potential energy is twice the total, so the ground state has 13.598 eV of kinetic energy and −27.197 eV of potential energy.

Hydrogen’s spectral series

Every line that ends on the same level belongs to one series, named after the scientist who first described it. Each series starts with its longest wavelength, the α line, and its lines crowd together towards a limit, the photon an electron gives out when it is captured from rest far away.

Only the Balmer series reaches the visible, and only its first four lines are easy to see: Hα red, Hβ blue, Hγ and Hδ violet. To turn any of these wavelengths into a photon energy or a frequency on its own, use the photon energy calculator or the wavelength and frequency calculator.

Why hydrogen has more than one Rydberg constant

The Rydberg constant R∞ = 109,737 cm⁻¹, the CODATA 2022 value, belongs to a nucleus infinitely heavier than the electron. A real nucleus circles the centre of mass as well, so the electron behaves as if its mass were the reduced mass, μ = mₑM/(mₑ + M), and every energy and wavenumber is scaled by μ/mₑ, a little less than 1. For hydrogen that is 0.054 percent, which makes R_H = 109,678 cm⁻¹ and puts Hα 0.36 nm further to the red than R∞ does. Switch the correction off to see it move.

The heavier the nucleus, the smaller the shift, and two discoveries turned on it. Deuterium’s nucleus has about twice the proton’s mass, which moves its Hα line 0.179 nm short of hydrogen’s, and that shift is how Urey, Brickwedde and Murphy found deuterium, reporting it in 1932. And in 1896 Edward Pickering found lines in the star ζ Puppis that sat between hydrogen’s Balmer lines, which Bohr showed in 1913 came from He⁺: its lines ending on n = 4 fall alternately on and between the Balmer lines. Choose He⁺ and set 6 to 4: with the correction off, the line has exactly the wavelength of hydrogen’s Hα with the correction off, and with it on, the line is at 656.20 nm, 0.27 nm short of Hα, because the helium nucleus recoils less than the proton.

Vacuum and air wavelengths

The Rydberg formula gives wavelengths in a vacuum. Light is about 0.03 percent slower in air, so its wavelength there is shorter by the same fraction, and the NIST Atomic Spectra Database gives air wavelengths between 200 nm and 2 µm and vacuum wavelengths outside that range. The simulator does the same: its air value is for standard air, 15 °C and 101,325 Pa, from the formula of Peck and Reeder (1972) that NIST uses. That is why Hα is 656.47 nm from the formula and 656.28 nm in a table: 0.18 nm of the gap is the air, and the last hundredth comes mostly from fine structure, which the Bohr model leaves out.

Common mistakes

• Giving an emitted photon a negative energy. The change in the atom’s energy, E_f − E_i, is negative when it emits; the photon’s energy is the size of that change, and its wavelength is always positive.
• Swapping n₁ and n₂. n₁ is the lower level, the one the series is named after. The other way round, 1/λ comes out negative.
• Forgetting Z² for an ion. He⁺ levels are four times deeper than hydrogen’s and Li²⁺ levels nine times, so their lines sit at about a quarter and a ninth of hydrogen’s wavelengths.
• Using 13.6 eV where more figures matter. It is 13.606 eV for a nucleus held still and 13.598 eV for hydrogen. The 0.054 percent between them is larger than the 0.028 percent between Hα in air and in a vacuum.
• Comparing a vacuum wavelength with a table in air. Between 200 nm and 2 µm a table’s wavelength is shorter than the formula’s by about 0.03 percent. Say which one you mean.
• Reading the orbits as real paths. Quantum mechanics replaces them with orbitals, and it gives hydrogen’s ground state no orbital angular momentum at all, where Bohr’s gives it ℏ.

What this simulator leaves out

Everything the Bohr model does not have. Fine structure and the Lamb shift split each hydrogen line into components: NIST lists eight for Hα, spread over 0.020 nm. Hyperfine structure gives hydrogen its 21 cm radio line, a transition inside the ground state that no orbit picture can make. The model says nothing about how bright each line is, how long a level lasts or how wide a line is, and it cannot describe an atom with two or more electrons, which repel each other; how those electrons fill shells and subshells is what the electron configuration calculator works out. For heavy ions relativity starts to matter: C⁵⁺ is bound 0.043 percent more tightly than the model says.

The force holding the electron in place is Coulomb’s law, which the Coulomb’s law calculator works out for two charges, and the nucleus it pulls towards is the one the Rutherford scattering simulator finds. Light behaving as particles is the subject of the photoelectric effect simulator, and the electron behaving as a wave, which is one way to picture why only whole numbers of ℏ are allowed, is the subject of the de Broglie wavelength calculator.