Question

The potential energy between two atoms in a particular molecule has the form U(x) = 2.6 x8 − 5.1 x4 where the units of x are length and the numbers 2.6 and 5.1 have appropriate units so that U(x) has units of energy. What is the equilibrium separation of the atoms (that is the distance at which the force between the atoms is zero)?

Answer 1

Explanation:

The given equation is as follows.

         U = $\frac{2.6}{x^{8}} - \frac{5.1}{x^{4}}$

or,  $\frac{dU}{dx} = \frac{2.6 \times (-8)}{x^{9}} - \frac{5.1 \times (-4)}{x^{5}}$

                    = $\frac{-20.8}{x^{9}} + \frac{20.4}{x^{5}}$

We know that for equilibrium, F = $-\frac{dU}{dx}$ = 0

               $\frac{dU}{dx}$ = 0

  $\frac{-20.8}{x^{9}} + \frac{20.4}{x^{5}}$ = 0

     $\frac{20.4}{x^{5}}$ = $\frac{20.8}{x^{9}}$

                     x = 1.004 m

                        = 1 m (approx)

Thus, we can conclude that the equilibrium separation of the given atoms is 1 m.