Q. 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 $D = \lbrack U(x = \infty) - U_{at\ equilibrium}\rbrack$, $D$ is
A) $\frac{b^{2}}{6a}$
B) $\frac{b^{2}}{2a}$
C) $\frac{b^{2}}{12a}$
D) $\frac{b^{2}}{4a}$
Explanation
(d) $U = \frac{a}{x^{12}} - \frac{b}{x^{6}}$
$F = - \frac{dU}{dx} = + 12\frac{a}{x^{13}} - \frac{6b}{x^{7}} = 0 \Rightarrow x = \left( \frac{2a}{b} \right)^{1/6}$
$U(x = \infty) = 0$
$U_{equilibrium} = \frac{a}{\left( \frac{2a}{b} \right)^{2}} - \frac{b}{\left( \frac{2a}{b} \right)} = \frac{b^{2}}{4a}$
$\therefore U(x = \infty) - U_{equilibrium} = 0 - \left( - \frac{b^{2}}{4a} \right) = \frac{b^{2}}{4a}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.