My textbook, Silberberg & Amateis, Chemistry: The Molecular Nature of Matter and Change (9th ed.), gives the following equation for calculating the energy level of an atom: $$E = \pu{-2.18E-18 J}\left(\frac{Z^2}{n^2}\right)$$ Where $Z$ is the charge of the nucleus.

They never explained what the constant term that is multiplying the (Z/n)^2 term means, or where it comes from. Where does this come from? It looks similar to the Rydberg equation, but I still can't see its relation to it exactly.

  • 1
    $\begingroup$ Well, if you figure out how to compute the energy associated with a particular wavelength of light, you can check for yourself if it is Rydbergs equation. $\endgroup$
    – Buck Thorn
    Mar 3, 2023 at 18:40
  • 3
    $\begingroup$ Rydberg formula is but an empirical observation, it doesn't explain anything either. The real explanation comes from quantum mechanics. $\endgroup$ Mar 3, 2023 at 19:02

1 Answer 1


Essentially, the equation conveys how the energy of a one-electron atom varies with factors such as the nuclear charge and electronic orbit number.

$-2.18\times10^{-18}\ \mathrm J$ is an empirically obtained factor, but you can also derive it from first principles using Bohr's atomic model.

$$2.18\times10^{-18}\ \mathrm J=\frac{(k_\mathrm ee^2)^2m_\mathrm e}{2\hbar^2}$$

Where $k_\mathrm e$ is Coulomb's constant, $e$ is the electronic charge, $m_\mathrm e$ is the mass of an electron, and $\hbar$ is the reduced Planck constant.

You can find the derivation on Wikipedia: https://en.wikipedia.org/wiki/Bohr_model

Intuitively, you can think of $-2.18\times10^{-18}\ \mathrm J$ as the energy of a hydrogen atom system with an electron in its first orbit. Here $Z=1$ and $n=1$.


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