Question

The potential of an atom is given by $V=V_{0}lo{g}_e\left(r/r_0\right)$ where $r_0$ is a constant and r is the radius of the orbit. Assuming Bohr's model to be applicable, which variation of $r_n$ with n is possible (n being principal quantum number)?

• A. $r_n\propto n$
• B. $r_n\propto 1/n$
• C. $r_n\propto n^2$
• D. $r_n\propto 1/n^2$

Answer 1

The correct option is A $r_n\propto n$

Given that, $V=V_{0}lo{g}_e\left(r/r_0\right)$
Field $E=-\frac{dV}{dr}=-V_0\left(\frac{r}{r_0}\right)$
$eE=\frac{m{V}^2}{r}$ or $\frac{e{V}_0{r}_0}{r}=\frac{m{V}^2}{r}$
$V={\left(\frac{e{V}_0{r}_0}{m}\right)}^{1/2}$
$mV=(m_e{V}_0{r}_0{)}^{1/2}=$ constant
$\therefore -mvr=\frac{nh}{2\pi }$, Bohr's quantum condition
or $r\propto n$.