Question

The potential energy $U$ of a diatomic molecule is a function dependent on $r$ $($interatomic distance$)$ as $U=\frac{\alpha }{r^{10}}-\frac{\beta }{r^5}-3$. Where, $a$ and $b$ are positive constants. The equilibrium distance between two atoms will be ${\left(\frac{2\alpha }{\beta }\right)}^{\frac{a}{b}}$, then $a=$________

Answer 1

$F=-\frac{dU}{dr}$

$\Rightarrow F=-[-\frac{10\alpha }{r^{11}}+\frac{5\beta }{r^6}]$

For equilibrium, $F=0$

$\frac{10\alpha }{r^{11}}=\frac{5\beta }{r^6}$

$\Rightarrow \frac{2\alpha }{\beta }=r^5$

$\Rightarrow r={\left(\frac{2\alpha }{\beta }\right)}^{\frac{1}{5}}={\left(\frac{2\alpha }{\beta }\right)}^{\frac{a}{b}}$

$\Rightarrow a=1$