The potential energy (U) of a diatomic molecule

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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