CBSE 2022 · Region 5 · Set 2 · Q9 · 3 marks
(a)Use Bohr's postulate to prove that the radius of $\displaystyle n^{\text {th }}$ orbit in a hydrogen atom is proportional to $\displaystyle n^{2}$.(b)How will the energy of a hydrogen atom change if $\displaystyle n$ increases from $\displaystyle 1$ to $\displaystyle \infty$ ?
(a)
Use Bohr's postulate to prove that the radius of $\displaystyle n^{\text {th }}$ orbit in a hydrogen atom is proportional to $\displaystyle n^{2}$.
(b)
How will the energy of a hydrogen atom change if $\displaystyle n$ increases from $\displaystyle 1$ to $\displaystyle \infty$ ?
Marking-scheme solution
(a)
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
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CBSE Class 12 Physics past-paper question from the 2022board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.