Using Bohr's postulates, derive the expression for the radius of the $\displaystyle \mathrm{n}^{\text {th }}$ orbit of an electron in a hydrogen atom. Also find the numerical value of Bohr's radius $\displaystyle \mathrm{a}_{0}$.

From Bohr's second postulate: $\displaystyle m v r=\frac{n h}{2 \pi}$ ……($\displaystyle 1$) Also $\displaystyle \frac{m v^{2}}{r}=\frac{e^{2}}{4 \pi \varepsilon_{0} r^{2}}$ (z=$\displaystyle 1$) $\displaystyle v=\frac{e}{\sqrt{4 \pi \varepsilon_{0} m r}}$ Substituting in ($\displaystyle 1$) and simplifying: $\displaystyle r=\frac{n^{2} h^{2} \varepsilon_{0}}{\pi m e^{2}}$ For $\displaystyle n=1$, $\displaystyle r=a_{0}$ (Bohr's radius): $\displaystyle a_{0}=\frac{\left(6.63 \times 10^{-34}\right)^{2} \times 8.854 \times 10^{-12}}{3.14 \times 9.1 \times 10^{-31} \times\left(1.6 \times 10^{-19}\right)^{2}}$ $\displaystyle =5.29 \times 10^{-11} \mathrm{~m}$ $\displaystyle =0.53 \AA$