Question

The potential energy between two atoms in a particular molecule has the form U(x) = 2.1 x 8 − 5.2 x4 where the units of x are length and the numbers 2.1 and 5.2 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

`x\approx 0.948`

Step-by-step explanation:

The correct formula for the potential energy between two atoms in a particular molecule is:

`U(x) = (2.1)/(x^(8))-(5.2)/(x^(4))`

Where
`x` is the distance.

According to the definitions of potential energy and work, as well as the Work-Energy Theorem and the Principle of Energy Conservation. The relation between that and related force is:

`F = -(dU)/(dx)`

The function is derived in terms of distance:

`F (x) = (84)/(5\cdot x^(9)) -(104)/(5\cdot x^(5))`

Then, it is needed to find at least of x so that F(x) equals to 0.

`(84)/(5\cdot x^(9))-(104)/(5\cdot x^(5))=0`

`(84)/(x^(4))-104 = 0`

`84-104\cdot x^(4) = 0`

`x=\sqrt[4]{(84)/(104) }`

`x\approx 0.948`