Q.The Li$^{2+}$ ion is a hydrogen-like ion that can be described by the Bohr model. Calculate the Bohr radius of the third orbit and calculate the energy of an electron in the 4th orbit.

Niels Bohr rescued Rutherford's nuclear atom from classical electromagnetism's prediction of collapse by proposing that electrons occupy only certain fixed, non-radiating stationary orbits, in which the angular momentum is quantised as an integer multiple of $h/2\pi$: $mvr=nh/2\pi$. While an electron sits in a stationary orbit it does not radiate energy, no matter what classical theory says about accelerated charges - this single postulate is what makes atoms stable. An electron only emits or absorbs energy when it jumps between orbits, and the photon involved carries exactly the energy difference between the two orbits, $E_2-E_1=h\nu$.

For any one-electron ("hydrogen-like") species of atomic number $Z$ - hydrogen itself, or an ion such as $\text{He}^+$ or $\text{Li}^{2+}$ - solving the model gives closed-form results for the radius and energy of the $n$th orbit:

$r_n=\left(0.529\,\frac{n^2}{Z}\right)\text{Å}\qquad\qquad E_n=-\left(\frac{13.6\,Z^2}{n^2}\right)\text{eV atom}^{-1}=-\left(\frac{1312.8\,Z^2}{n^2}\right)\text{kJ mol}^{-1}$ …