Question

Suppose that the potential energy of an hypothetical atom consisting of a proton and an electron by $U=-k{e}^2/3r^3$. Then if Bohr's postulates are applied to this atom, then the radius of the be proportional to -

• A. $n^2$
• B. $1/n^2$
• C. $n^3$
• D. $1/n^3$

Answer 1

The correct option is B $1/n^2$

$\begin{array}{c}\text{Given,}\\ \text{Potential Energy}\left(U\right)=-\frac{K{e}^2}{3r^3}\end{array}$

$\text{we Know}$

$\begin{array}{cc}& F=-\frac{du}{dr}\\ F& =-\frac{d}{dr}(-\frac{K{e}^2}{3r^3})\\ F& =+\frac{K{e}^3}{3}(-3\frac{1}{r^4})\\ F& =-\frac{K{e}^2}{r^4}\end{array}$

$\begin{array}{c}\text{Now, it is given it follow Bohr's postalate}\\ F=\text{centripetal force}=\frac{m{v}^2}{r}\end{array}$

$\begin{array}{c}\frac{m{v}^2}{r}=\frac{k{e}^2}{r^4}\\ m{v}^2=\frac{k{e}^2}{r^3}-\left(i\right)\end{array}$

$\begin{array}{c}\text{By. conservation of angular momentum}\\ \begin{array}{c}mvr=\frac{nh}{2\pi }\\ v\alpha \frac{n}{r}-\left(ii\right)\end{array}\end{array}$

$\begin{array}{c}\text{from (i) and (ii)}\\ \begin{array}{cc}\frac{n^2}{r^2}\alpha \frac{1}{r^3}& \\ \text{or,}& r\alpha \frac{1}{n^2}\end{array}\end{array}$