# What does the constant mean in this equation for the energy levels of an atom?

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.

• 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. Mar 3, 2023 at 18:40
• Rydberg formula is but an empirical observation, it doesn't explain anything either. The real explanation comes from quantum mechanics. Mar 3, 2023 at 19:02

$$-2.18\times10^{-18}\ \mathrm J$$ 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.
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 $$n=1$$.