If the potential energy of a gas molecule is `U= M/(r^6) - N/(r^12) M` and N being positive constants, then the potential energy at equilibrium must be 

Options

A. zero
B. `(M^2)/(4N)`
C. `(N^2)/(4M)`
D. `(MN^2)/(4)`

Answer

B

Explanation

`F = (-(dU))/(dr) = -(d)/(dr) [(M)/(r^6) - N/(r^(12))] = - [(-6M)/(r^7) + (12N)/(r^(13))]`
In equilibrium position , `F = 0`
`:. (6M)/(r^7) - (12N)/(r^13) = 0 " or " r^6 = (2N)/(M)`
`:.` Potential energy at equilibrium position
`U = M/((2N//M)) - N/((2N//M)^2) = (M^2)/(2N) - (M^2)/(4N) = (M^2)/(4N)`