Calculating the ionization energy for hydrogen


> 
> What energy would be needed to remove the electron from the $n = 4$ level of the hydrogen atom?
> 
> 
> * $\pu{−3.49 \* 10^{−17} J}$
> * $\pu{−1.36 \* 10^{−19} J}$
> * $\pu{+2.18 \* 10^{−18} J}$
> * $\pu{+1.36 \* 10^{−19} J}$
> 
> 
> 


I assume that the way to do this is to start from the [Rydberg formula](https://en.wikipedia.org/wiki/Rydberg_formula),


$$
E = \mathcal{R}Z^2 \left( \frac{1}{n\_i^2}-\frac{1}{n\_f^2} \right),
$$


set the initial level of the electron as $n\_i = 4$, and the final level corresponding to removing the electron (ionization) as $n\_f = \infty \implies \frac{1}{\infty^2} = 0$, leading to


$$
E = \pu{13.6\*\frac{1}{4^2} eV = 0.85 eV = 1.36\*10^{-19} J}.
$$




I assume you refer to [Bohr's atomic model](https://en.wikipedia.org/wiki/Bohr_model).


The energy of an electron in Bohr’s orbit of Hydrogen atom is given by the expression:


$$
\begin{align}
E\_{n} &= \frac{2\pi^{2}me^{4}Z^{2}}{n^{2}h^{2}\left(4\pi\epsilon\_{0}\right)^{2}} \\
&= \color{\navy}{-13.6\frac{Z^2}{n^2}\pu{ eV}}
\end{align}
$$


Since $Z = 1$ for hydrogen, the above equation can be further simplified. Now just plug in the required values.


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