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Characteristics of Bohr's Atomic Model

The electric ...

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 variation of $r_{n}$ with $n, n$ being the principal quantum number?

r_{n} \propto n

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r_{n} \propto 1 / n

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r_{n} \propto n^{2}

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

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Solution

The correct option is A $r_{n} \propto n$
Electric potential between proton and electron in nth orbit is given as: $V = V_{o} l n \frac{r_{n}}{r_{o}}$
Thus the coulomb force $| F_{c} | = e \frac{d V}{d r} = e \frac{V_{o}}{r_{n}}$
This coulombic force is balance by the centripetal force.
$∴ \frac{m v^{2}}{r_{n}} = e \frac{V_{o}}{r_{n}} ⟹ v = \sqrt{\frac{e V_{o}}{m}}$ means that speed is constant
Now from $m v r_{n} = \frac{n h}{2 π} ⟹ r_{n} \propto n$ (as $v = c o n s t a n t$ )

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