Question

If potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U\left(x\right)=\frac{a}{x^8}-\frac{b}{x^4}$ , where $a$ and $b$ are constants in standard SI units and $x$ is in metres. Find the dissociation energy of the molecule (in $J$ ). [Take $a=4J{m}^8$ and $b=20J{m}^4$ ]

Answer 1

Answer: 25

$U\left(x\right)=\frac{4}{x^8}-\frac{20}{x^4}$
$\therefore \frac{dU}{dx}=0\Rightarrow \frac{-8\times 4}{x^9}+\frac{20\times 4}{x^5}=0$
$\Rightarrow \frac{4}{x^5}\left(20-\frac{8}{x^4}\right)=0$
$x=\infty$ and $x^4=\frac{8}{20}$
Also $\frac{d^{2}U}{d{x}^2}=+ve\text{at}{x}^4=\frac{8}{20}$
$\therefore U_{\min }=\frac{4}{x^4}\left(\frac{1}{x^4}-5\right)=\frac{4\times 20}{8}\left(\frac{20}{8}-5\right)$
$\Rightarrow U_{\min }=-\frac{4\times 20}{8}\times \frac{20}{8}=-25J$
$\therefore$ Dissociation energy $=\left(U_{\text{infinite}}-U_{\min }\right)=0-\left(-25\right)=25J$