## 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,

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