If the outermost electron in an atom is excited to a very high energy state, its orbit is far beyond that of the other electrons. To a good approximation, we can think of the electron as orbiting a compact core with a charge equal to the charge of a single proton. The outer electron in such a Rydberg atom thus has energy levels corresponding to those of hydrogen.

a. What is the radius of the $n=100$ state of the Bohr hydrogen atom?

b. Sodium is a common element for such studies. How does the radius you calculated in part a compare to the approximately $0.2 \mathrm{~nm}$ radius of a typical sodium atom?

Solution

Verifieda)

The radius of the electron orbit in $n=100$ state is:

$\begin{align*} r&=r_1n^2\\ &=0.53\cdot 10^{-10}\,\text{m}100^2\\ &=\boxed{0.53\,\mu\text{m}} \end{align*}$

b)

The radius from part a) is more then 1000 times bigger then the radius of the sodium atom.