Question

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U\left(x\right) = \frac{a}{x^{12}} - \frac{b}{x^{6}}$, where a and b are constants and x is the distance between the atoms. If the dissociation energy of the molecule is $d = \left[U\left(x = \infty\right) - U_{at \,equilibrium}\right], D$ is

• A. $\frac{b^{2}}{2a}$
• B. $\frac{b^{2}}{12a}$
• C. $\frac{b^{2}}{4a}$
• D. $\frac{b^{2}}{6a}$

Answer 1

Answer: (C)

$\frac{b^{2}}{4a}$

Answer 2

$U\left(x\right) = \frac{a}{x^{12}}-\frac{b}{x^{6}}$ $U\left(x = \infty\right) = 0$ as, $\quad F = -\frac{dU}{dx} = -\left[\frac{12a}{x^{13}}+\frac{6b}{x^{7}}\right]$ at equilibrium, $F = 0$ $\therefore\quad x^{6} = \frac{2a}{b}$ $\therefore\quad U_{at \,equilibrium} = \frac{a}{\left(\frac{2a}{b}\right)^{2}}-\frac{b}{\left(\frac{2a}{b}\right)} = \frac{-b^{2}}{4a}$ $\therefore\quad D = \left[U \left(x = \infty\right)-U_{at \,equilibrium}\right] = \frac{b^{2}}{4a}$