The potential energy for the force between two atoms in a diatomic molecule has

Question

The potential energy for the force between two atoms in a diatomic molecule has the approximate form:
U(x)= a/x6+ b/x12,
Where x is the distance between the atoms, and a, b are positive constants. Find the force.

Assuming that one of the atoms is relatively heavy, and remains at rest while the other, whose mass is m, moves in a straight line:
Find the equilibrium distance, and the period of small oscillations about the equilibrium position.

Other Comments: It has been about a year since my last physics or math course, therefore any solution to this problem should be thoroughly explained so that I may understand what you're doing (equations and variables) and why you are doing it. Thank you.

Explanation / Answer

U(x) = - (a/x6) + (b/x12)

force is given as

F = - dU(x)/dx

F = - (d/dx) (- (a/x6) + (b/x12))

F = (6a/x7) - (12b/x13)

at equilibrium

F = 0

(6a/x7) - (12b/x13) = 0

6a/x7 = 12b/x13

x6 = 2b/a

x = (2b/a)1/6

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