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, $$\alpha$$ and $$\beta$$ are positive constants. The equilibrium distance between two atoms will be $$\left(\frac{2\alpha}{\beta}\right)^{\frac{a}{b}}$$, where $$a$$ = ______.

The potential energy of the diatomic molecule is given by $$U = \frac{\alpha}{r^{10}} - \frac{\beta}{r^5} - 3$$.

At equilibrium, the net force on the molecule is zero, which means $$\frac{dU}{dr} = 0$$.

Differentiating with respect to $$r$$:

$$\frac{dU}{dr} = -\frac{10\alpha}{r^{11}} + \frac{5\beta}{r^6} = 0$$

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

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

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

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

Comparing with the given form $$\left(\frac{2\alpha}{\beta}\right)^{\frac{a}{b}}$$, where $$\frac{a}{b} = \frac{1}{5}$$, we get $$a = 1$$.

The answer is $$a = 1$$.