Question

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by \[U\left( x \right)=\dfrac{a}{{{x}^{12}}}-\dfrac{b}{{{x}^{6}}}\], where a and b are constants and x is the distance between the atoms. If the dissociation energy of the molecule is \[D=[U(x=\infty )-{{U}_{equilibrium}}]\], then D is equal to
$A.\dfrac{{{b}^{2}}}{2a}$
$B.\dfrac{{{b}^{2}}}{12a}$
$C.\dfrac{{{b}^{2}}}{4a}$
$B.\dfrac{{{b}^{2}}}{6a}$

Answer 1

Hint: By using the potential energy function, we need to find \[U(x=\infty )\] and \[{{U}_{equilibrium}}\]. Obtain \[{{U}_{equilibrium}}\] by assuming F to be equal to 0. So D is the difference between \[U(x=\infty )\] and \[{{U}_{equilibrium}}\] which is the required answer.

Complete step by step answer:
Dissociation energy is the energy required to break a bond and form two atomic fragments or molecules. It is an endothermic reaction. For a stable bond to break it requires more energy so the dissociation energy is larger. This means if a bond is relatively unstable less energy is required to break it so the dissociation energy is less. When atoms come together to form new chemical bonds, the electrostatic forces bringing them together leave the bond with a large excess of energy.
We have the potential energy function which is
\[U\left( x \right)=\dfrac{a}{{{x}^{12}}}-\dfrac{b}{{{x}^{6}}}\]……….. (1)
We know that,
Dissociation energy \[D=[U(x=\infty )-{{U}_{equilibrium}}]\]
So we need to find \[U(x=\infty )\]and \[{{U}_{equilibrium}}\] from equation 1
Let us substitute $x=\infty $in equation 1
\[U(x=\infty )\]= 0
Now for U to be at equilibrium F=0
$F=-\dfrac{dU}{dx}=-[\dfrac{12a}{{{x}^{13}}}+\dfrac{6b}{{{x}^{7}}}]$
$\Rightarrow -[\dfrac{12a}{{{x}^{13}}}+\dfrac{6b}{{{x}^{7}}}]=0$
$\therefore {{x}^{6}}=\dfrac{2a}{b}$
Let us substitute $\therefore {{x}^{6}}=\dfrac{2a}{b}$ in equation 1
$\therefore {{U}_{equilibrium}}=\dfrac{a}{{{(\dfrac{2a}{b})}^{2}}}-\dfrac{b}{(\dfrac{2a}{b})}=\dfrac{-{{b}^{2}}}{4a}$
$\therefore D=[U(x=\infty )-{{U}_{equilibrium}}]=0-(-\dfrac{{{b}^{2}}}{4a})=\dfrac{{{b}^{2}}}{4a}$ .

Hence the correct option is (c).

Note:
The possibility of mistake can be option (a) due to miscalculation of \[{{U}_{equilibrium}}\]. The energy required for separating all the bonds in an element is known as the atomization enthalpy. The energy required for separating completely the atoms in one mole of covalent bonds is known as the bond dissociation enthalpy of that bond.