First slide

Bohr's atom model

Question

The electric potential between a proton and an electron is given by $V = V_{0} \ln \frac{r}{r_{0}}$ , where $r_{0}$ is a constant. Assuming Bohr's model to be applicable, write the variation of $r_{n}$ ,
n being the principal quantum number

Moderate

A

$r_{n} \propto n$
B

$r_{n} \propto 1 / n$
C

$r_{n} \propto n^{2}$
D

$r_{n} \propto 1 / n^{2}$

Solution

$Potential energy U = e V = e V_{0} \ln \frac{r}{r_{0}} Force F = - |\frac{d U}{d r}| = \frac{e V_{0}}{r r_{0}} .$
Therefore the force will provide the necessary centripetal force.
Hence,
$\frac{m v^{2}}{r} = \frac{e V_{0}}{r r_{0}} \Rightarrow v = \sqrt{\frac{e V_{0}}{m r_{0}}}$ ………(i)
and
$m v r = \frac{n h}{2 π}$ ………(ii)
$From equations (i) and (ii), m \sqrt{\frac{e V_{0}}{m r_{0}}} \cdot r = \frac{n h}{2 π} \Rightarrow r = \frac{n h}{2 π} \sqrt{\frac{r_{0}}{m e V_{0}}} \Rightarrow r_{n} \propto n$

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