(a) Use Bohr's postulate to prove that the radius of n^th orbit in a hydrogen atom is proportional to n^2. (b) How will the energy of a hydrogen atom…

centripetal force = electrostatic force of attraction

$$\frac{m v^{$\displaystyle 2$}}{r}=\frac{k e^{$\displaystyle 2$}}{r^{$\displaystyle 2$}}

$$$$ \begin{equation*} \mathrm{r}=\frac{k e^{$\displaystyle 2$}}{m v^{$\displaystyle 2$}} \tag{i} \end{equation*}

$$From Bohr's II postulate of quantization

$$\begin{equation*} \mathrm{L}=\mathrm{mvr}=\frac{n h}{2 \pi} \tag{ii} \end{equation*}

$$Substitute v from eq (ii) into eq (i)

$$\mathrm{r}=\frac{\mathrm{ke}^{$\displaystyle 2$}}{\mathrm{~m} \cdot \mathrm{n}^{$\displaystyle 2$} \mathrm{~h}^{$\displaystyle 2$}} \times $\displaystyle 4$ \pi^{$\displaystyle 2$} \mathrm{~m}^{$\displaystyle 2$} \mathrm{r}^{$\displaystyle 2$}

$$$\mathrm{r}=\frac{n^{2} h^{2}}{4 \pi^{2} m K e^{2}}$

$$\mathrm{r} \propto \mathrm{n}^{$\displaystyle 2$}

$$For $\displaystyle \mathrm{n}=1, \mathrm{E}_{1}=-13.6 \mathrm{eV}$, for $\displaystyle \mathrm{n}=\infty, \mathrm{E}_{\infty}=0$

As n increases from n = $\displaystyle 1$ to $\displaystyle \infty$,energy increases