Question

The potential energy (U) of a diatomic molecule is $U = \frac{\alpha}{r^{10}} - \frac{\beta}{r^5} - 3$. The equilibrium distance between two atoms will be $\left( \frac{2\alpha}{\beta} \right)^{\frac{a}{b}}$ , where a = _________.

Hint:

Equilibrium always occurs at the minimum of the Potential Energy curve where the derivative (slope) is zero and the force is zero.

Answer 1

Correct Answer: 1

Step 1: At equilibrium, the force $F = -\frac{dU}{dr} = 0$.
Step 2: $\frac{dU}{dr} = \alpha(-10r^{-11}) - \beta(-5r^{-6}) = 0$.
Step 3: $-\frac{10\alpha}{r^{11}} + \frac{5\beta}{r^6} = 0 \Rightarrow \frac{5\beta}{r^6} = \frac{10\alpha}{r^{11}}$.
Step 4: $r^5 = \frac{10\alpha}{5\beta} = \frac{2\alpha}{\beta}$.
Step 5: $r = \left( \frac{2\alpha}{\beta} \right)^{1/5}$.
Step 6: Comparing with $\left( \frac{2\alpha}{\beta} \right)^{a/b}$, we get $a=1$ (and $b=5$).