It is important to understand why the energies of the stationary states are negative. Because the potential energy of two charged particles is $U=q_1{q}_2\text{/}4\pi {ϵ}_{0}r$, the zero of potential energy occurs at $r=\infty$ where the particles are infinitely far apart. The state of zero total energy corresponds to having the electron at rest $(K=0)$ and infinitely far from the proton $(U=0)$. This situation, which is the case of two “free particles,” occurs in the limit $n\to \infty$, for which $r_n\to \infty$ and $v_n\to 0$.

An electron and a proton bound into an atom have less energy than two free particles. We know this because we would have to do work (i.e., add energy) to pull the electron and proton apart. If the bound atom’s energy is lower than that of two free particles, and if the total energy of two free particles is zero, then it must be the case that the atom has a negative amount of energy.

Thus $\left|E_n\right|$ is the binding energy of the electron in stationary state $n$. In the ground state, where $E_1=-13.60\text{eV,}$ we would have to add 13.60 eV to the electron to free it from the proton and reach the zero energy state of two free particles. We can say that the electron in the ground state is “bound by 13.60 eV.” An electron in an $n=3$ orbit, where it is farther from the proton and moving more slowly, is bound by only 1.51 eV. That is the amount of energy you would have to supply to remove the electron from an $n=3$ orbit.

Removing the electron entirely leaves behind a positive ion, $\text{H}{}^{+}$ in the case of a hydrogen atom. (The fact that $\text{H}{}^{+}$ happens to be a proton does not alter the fact that it is also an atomic ion.) Because nearly all atoms are in their ground state, the binding energy $\left|E_1\right|$ of the ground state is called the ionization energy of an atom. Bohr’s analysis predicts that the ionization energy of hydrogen is 13.60 eV. FIGURE 38.20 illustrates the ideas of binding energy and ionization energy.

FIGURE 38.20 Binding energy and ionization energy.

We can test this prediction by shooting a beam of electrons at hydrogen atoms. A projectile electron can knock out an atomic electron if its kinetic energy K is greater than the atom’s ionization energy, leaving an ion behind. But a projectile electron will be unable to cause ionization if its kinetic energy is less than the atom’s ionization energy. This is a fairly straightforward experiment to carry out, and the evidence shows that the ionization energy of hydrogen is, indeed, 13.60 eV.