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 $a=4\,Jm^{8}$ and $b=20\,Jm^{4}$ ]

Answer 1

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