Question

The potential energy as a function of the force between two atoms in a diatomic molecules is given by $U\left(x\right)=\frac{A}{x^{12}}-\frac{B}{x^6}$, where $A$ and $B$ are positive constants and x-refers to the distance between atoms. The position of stable equilibrium for the system of the two atoms is given as:

• A. $x=\frac{A}{B}$
• B. $x=\sqrt{\frac{A}{B}}$
• C. $x=\sqrt{\frac{3A}{B}}$
• D. $x={\left(\frac{2A}{B}\right)}^{\frac{1}{6}}$

Answer 1

The correct option is D $x={\left(\frac{2A}{B}\right)}^{\frac{1}{6}}$

Force on an atom can be given as the negative of gradient of potential energy.

$F=-\frac{dU}{dx}$

$=-\frac{d}{dx}(\frac{A}{x^{12}}-\frac{B}{x^6})$

$=\frac{12A}{x^{13}}-\frac{6B}{x^7}$

For equilibrium, $F=0$

$⟹x^6=\frac{2A}{B}$ ............(i)

Real solutions of above equation are:

$x=\pm {\left(\frac{2A}{B}\right)}^{1/6}$

For stable equilibrium, potential energy must be minimum,

$\frac{d^{2}U}{d{x}^2}>0$ at points of equilibrium

$\frac{d^{2}U}{d{x}^2}=\frac{13\times 12\times A}{x^{14}}-\frac{6\times 7\times B}{x^8}$

$=\frac{6B}{x^8}(\frac{26A}{B{x}^6}-7)$

Substituting (i),

$=\frac{36B}{x^8}>0$

Hence, both the points correspond to stable equilibrium.