The potential energy function for the force between two atoms in a diatomic molecule is approximately given by U(x) = ax¹² - bx⁶, where a and b are constants and x is the distance between the atoms. If the dissociation energy of the molecule is D = [U(x = ∞) - U at equilibrium], then D is equal to?

The potential energy function is given by U(x) = ax¹² - bx⁶. To find the equilibrium position, we need to find the minimum of the potential energy function. We do this by taking the derivative with respect to x and setting it to zero:

dU(x)/dx = 12ax¹¹ - 6bx⁵ = 0

12ax¹¹ = 6bx⁵

2ax⁶ = b

x⁶ = b/(2a)

x = (b/(2a))^(1/6)

Now we substitute this equilibrium value of x back into the potential energy function to find the potential energy at equilibrium:

U(equilibrium) = a(b/(2a))² - b(b/(2a))

U(equilibrium) = a(b²/(4a²)) - b²/(2a)

U(equilibrium) = b²/(4a) - b²/(2a) = -b²/(4a)

At x = ∞, U(x) = 0. Therefore, the dissociation energy D is:

D = U(x = ∞) - U(equilibrium) = 0 - (-b²/(4a)) = b²/(4a)