JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Work, Power & Energy

If the potential energy between two molecules in given by

U = \frac{A}{r ^{6}} + \frac{B}{r ^{12}}

, then at equilibrium, separation between molecules, and the potential energy are:

Answer: (C)

Solution

At equilibrium the net force on the molecule vanishes,

F = - \frac{d U}{d r} = 0

. With a short-range repulsive

r^{- 12}

term and a long-range attractive

r^{- 6}

term, differentiating gives

\frac{d U}{d r} = \frac{6 A}{r ^{7}} - \frac{12 B}{r ^{13}}

; setting this to zero yields

6 A r^{6} = 12 B

, so

r^{6} = \frac{2 B}{A}

and the equilibrium separation is

r_{0} = (\frac{2 B}{A})^{1/6}

. Substituting

r_{0}^{6} = \frac{2 B}{A}

back into the potential, the depth of the well comes out to

U (r_{0}) = - \frac{A ^{2}}{4 B}

JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Single correct option