Question

One description of the potential energy of a diatomic molecule is given by the Lennard-Jones potential. U= A/r^12 - B/r^6 where A and B are constants and r is the separation distance between the atoms. For the H2 molecule, take A=0.124 x 10^12eV m12 and B=1.488 x 10^-60eV m6. Find (a) the separation distance r0 at which the energy of the molecule is a minimum and (b) the energy E required to break up the H2 molecule.

Answer 1

Given expression for Potential is,

$U=\dfrac{A}{r^{12}}-\dfrac{B}{r^6}~~~~~~-(1)$

(a) When the energy is minimum ,it means the system is in equlibrium

Differentiate equation 1 with respect to r

$\to \dfrac{dU}{dr}=\dfrac{-12A}{r^{13}}+\dfrac{6B}{r^7}$

For energy minimum $\dfrac{du}{dr}=0$

$\therefore \dfrac{12A}{r^{13}}=\dfrac{6B}{r^7}$

$\to r^6=\dfrac{2A}{B}$

$\to r^6=\dfrac{2\times0.124\times 10^{12}}{1.488\times 10^{-60}}$

$\to r^6=\dfrac{0.248\times10^{72}}{1.488\times 10^{-60}}$

$\to r^6=0.16\times 10^{72}$

so r=$(0.16\times 10^{72})^{\frac{1}{6}}$ m

(b) Energy required to brake $H_2$ molecule is

$\to U=\frac{0.124\times 10^{12}}{0.16\times10^{72}}-\frac{1.488\times 10^{-60}}{0.16\times10^{72}}$

=0.775-9.3$\times 10^{-132}$ Joule