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 equlibrium must be

Options

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

Answer

B

Explanation

Here, `U=(M)/(r^(6))-(N)/(r^(12)),`
`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))` or ` r^(6)=(2N)/(r^(13))`
Hence, `U=(M)/((2N//M))-(N)/((2N//M)^(2))=(M^(2))/(4N)`