Source: High school physics (Chinese)
Problem
In a hydrogen atom, the distance between the electron and the proton in the ground state is $r = 5.29 \times 10^{-11}$ m. The energy required to move the electron from its ground state to an infinite distance is the ionization energy.
- What is this ionization energy in electron-volts?
- What is this ionization energy in joules?
[Q1] $E_{ion} = 13.6$ eV [Q2] $E_{ion} = 2.18 \times 10^{-18}$ J
The ionization energy $E_{ion}$ is the energy required to overcome the electrostatic attraction, equal to the negative of the total energy of the electron in its orbit, $E_{ion} = -E_{total}$. For an orbit under a $1/r^2$ force (like the Coulomb force), the Virial theorem states that the total energy is half the potential energy: $E_{total} = \frac{1}{2}U$. The electrostatic potential energy is $U = -k\frac{e^2}{r}$.
$$E_{ion} = -E_{total} = -\frac{1}{2}U = -\frac{1}{2}\left(-k\frac{e^2}{r}\right) = \frac{ke^2}{2r}$$First, calculate the energy in Joules, then convert to electron-volts by dividing by the elementary charge $e$.