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The electric potential between a proton and an electron is given by $V=V_0 \ln \left(\frac{r}{r_0}\right)$ , where $r_0$ is a constant. Assuming Bohr's model to be applicable, write variation of $r_n$ with n, n being the principal quantum number:

Option: 1
$\mathrm{r}_{\mathrm{n}} \propto \mathrm{n}$

Option: 2
$\quad r_n \propto \frac{1}{n}$

Option: 3
$\quad \mathrm{r}_{\mathrm{n}} \propto \mathrm{n}^2$

Option: 4
$\quad \mathrm{r}_{\mathrm{n}} \propto \frac{1}{\mathrm{n}^2}$

Answers (1)

best_answer

Given:

$V=V_0 \ln \left(\frac{r}{r_0}\right)$
$\therefore \quad$ Potential energy, $\mathrm{U}=\mathrm{eV}$

or $\mathrm{U}=\mathrm{eV}_0 \ln \left(\frac{\mathrm{r}}{\mathrm{r}_0}\right) \therefore \frac{\mathrm{dU}}{\mathrm{dr}}=\mathrm{eV}_0\left(\frac{\mathrm{r}_0}{\mathrm{r}}\right) \frac{1}{\mathrm{I}_0}$

Force, $\mathrm{F}=-\frac{\mathrm{dU}}{\mathrm{dr}}=-\frac{\mathrm{e} \mathrm{V}_0}{\mathrm{r}} \ or\ |\mathrm{F}|=\frac{e \mathrm{~V}_0}{\mathrm{r}}$

This force provides the necessary centripetal force.

$\therefore \quad \frac{\mathrm{mv}^2}{\mathrm{r}}=\frac{\mathrm{eV}}{\mathrm{r}}$

or $\mathrm{v}=\sqrt{\frac{\mathrm{eV}_0}{\mathrm{~m}}}$
By Bohr's postulate, $\mathrm{mvr}=\frac{\mathrm{nh}}{2 \pi}$

or $\quad \mathrm{v}=\frac{\mathrm{nh}}{2 \pi \mathrm{mr}}$

From equations (i) and (ii), we get

$\frac{\mathrm{nh}}{2 \pi \mathrm{mr}}=\sqrt{\frac{\mathrm{eV}_0}{\mathrm{~m}}} \ or \ \mathrm{r}=\frac{\mathrm{nh}}{2 \pi \mathrm{m}} \times \sqrt{\frac{\mathrm{m}}{\mathrm{eV}_0}}$

or $\mathrm{r}=\left[\frac{\mathrm{h}}{2 \pi} \sqrt{\frac{1}{\mathrm{meV}_0}}\right] \times \mathbf{n}$

$\therefore\quad r_n \propto n$

Posted by

Devendra Khairwa

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