Question

The potential energy between two atoms in a molecule is gives by, U (x) = $\frac{a}{x^{12}}$ - $\frac{b}{x^6}$ , where a and b are postive constant and x is the distant between the atome. The system is in stable equilibrium when

• A. x = 0
• B. x =
• C. x =
• D. x = ()

Answer 1

The correct option is C

x =

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

At equilibrium Force = 0

F = - $\frac{dv}{dx}$ = 0

$-\frac{dv}{dx}=-(-12\frac{a}{x^{13}}+6\frac{b}{x^7})$=0

$\frac{6}{x^7}(\frac{2a}{x^6}-b)=0$
$x^6=\frac{2a}{b}\left(1\right)$

Equilibrium exits at x = ${\left(\frac{2a}{b}\right)}^{1/6}$
The system has a stable equilibrium at minimum potential energy state, so to check whether $U_x$ is minimum at x = ${\left(\frac{2a}{b}\right)}^{1/6}$ or not

$\frac{d^{2}u}{d{x}^2}$ should be $>$ 0 at

x=${\left(\frac{2a}{b}\right)}^{1/6}$Let's check

$\frac{d}{dx}\left(\frac{du}{dx}\right)=12x^{13}\frac{a}{x^{14}}-6x^7\frac{b}{x^8}Atx={\left(\frac{2a}{b}\right)}^{\frac{1}{6}}\frac{d^{2}u}{d{x}^2}>0$