Question

The electric potential between a proton and an electron is given by $V = V_{0} \text{ln} \frac{r}{r_{0}}$ where r₀ is a constant. Assuming Bohr’s model to be applicable, write variation of $r_{n}$ with $n$ , $n$ being the principal quantum number.

• A. $r_{n} \propto n$
• B. $r_{n} ∝ \frac{1}{n}$
• C. $r_{n} ∝ n^{2}$
• D. $r_{n} ∝ \frac{1}{n^{2}}$

Answer 1

Answer: (A)

U = eV = eV₀ ln $\left(\right. \frac{r}{r_{0}} \left.\right)$
$| F | = | - \frac{\text{d} U}{\text{d} r} | = \frac{e V_{0}}{r}$
This force provides necessary centripetal force.
Hence, $\frac{m v^{2}}{r} = \frac{e V_{0}}{r}$
Or $v = \sqrt{\frac{e V_{0}}{m}}$ ... (i)
Moreover, $m v r = \frac{n h}{2 \pi }$ ... (ii)
Dividing Eq. (ii) by (i), we have
$\textit{mr}=\left(\frac{\textit{nh}}{2 \pi }\right)\sqrt{\frac{\textit{m}}{\left(\text{eV}\right)_{0}}}$ or $\text{r}_{\text{n}} \propto \text{n}$