JEE Main · 2010 · 11 Apr 2010 · Offline · Work, Power & Energy

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by

U (x) = \frac{a}{x ^{12}} - \frac{b}{x ^{6}}

, 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 = \infty) - U_{at equilibrium}], D

Answer: (C)

Solution

Equilibrium is where the force vanishes, that is where

\frac{d U}{d x} = 0

. Differentiating

U = \frac{a}{x ^{12}} - \frac{b}{x ^{6}}

gives

\frac{d U}{d x} = - \frac{12 a}{x ^{13}} + \frac{6 b}{x ^{7}}

, and setting it to zero yields

x^{6} = \frac{2 a}{b}

. Substituting back,

U_{eq} = \frac{a}{( 2 a / b ) ^{2}} - \frac{b}{( 2 a / b )} = \frac{b ^{2}}{4 a} - \frac{b ^{2}}{2 a} = - \frac{b ^{2}}{4 a}

. Since

U \to 0

as the atoms separate to infinity, the dissociation energy is

D = 0 - (- \frac{b ^{2}}{4 a}) = \frac{b ^{2}}{4 a}

JEE Main · 2010 · 11 Apr 2010 · Offline · Single correct option