Question:
The potential energy (U) of a diatomic molecule is a function dependent on $r$ (interatomic distance) as
$\mathrm{U}=\frac{\alpha}{\mathrm{r}^{10}}-\frac{\beta}{\mathrm{r}^{5}}-3$
where, $\alpha$ and $\beta$ are positive constants. The equilibrium distance between two atoms will be
$\left(\frac{2 \alpha}{\beta}\right)^{\frac{a}{b}}$, where $a=$_________.
Solution:
For equilibrium
$\frac{\mathrm{dU}}{\mathrm{dr}}=0$
$\frac{-10 \alpha}{r^{11}}+\frac{5 \beta}{r^{6}}=0$
$\frac{5 \beta}{r^{6}}=\frac{10 \alpha}{r^{11}}$
$r^{5}=\frac{2 \alpha}{\beta}$
$r=\left(\frac{2 \alpha}{\beta}\right)^{\frac{1}{5}}$
$a=1$
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