# Energy of an Electron in a Bohr Orbit

## Calculating the Energy of an Electron in a Bohr Orbit

Early researchers were very excited when they were able to predict the energy of an electron at a particular distance from the nucleus in a hydrogen atom. If a spark promotes the electron in a hydrogen atom into an orbit with n = 3, what is the calculated energy, in joules, of the electron?

### Solution

The energy of the electron is given by this equation:

$$E=\phantom{\rule{0.2em}{0ex}}\cfrac{-k{Z}^{2}}{{n}^{2}}$$

The atomic number, Z, of hydrogen is 1; k = 2.179 $$\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}$$ 10–18 J; and the electron is characterized by an n value of 3. Thus,

$$E=\phantom{\rule{0.2em}{0ex}}\cfrac{-\left(2.179\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-18}\phantom{\rule{0.2em}{0ex}}\text{J}\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{\left(1\right)}^{2}}{{\left(3\right)}^{2}}\phantom{\rule{0.2em}{0ex}}=-2.421\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-19}\phantom{\rule{0.2em}{0ex}}\text{J}$$

The horizontal lines show the relative energy of orbits in the Bohr model of the hydrogen atom, and the vertical arrows depict the energy of photons absorbed (left) or emitted (right) as electrons move between these orbits.