Question

The force between two atoms in a diatomic molecule can be represented approximately by the potential energy function $U = U_{0} [{(\frac{a}{x})}^{12} - 2 {(\frac{a}{x})}^{6}],$ where $U_{0}$ and $a$ are constants.

(a) At what value of $x$ is the potential energy zero?
(b) Find the force $F_{x}$
(c) At what value of $x$ is the potential energy a minimum?

Accepted Answer

Given that

Potential energy, $U = U_{0} [{(\frac{a}{x})}^{12} - 2 {(\frac{a}{x})}^{6}]$

(a) If the potential energy is zero

$a$

We get,

$x = \frac{a}{2^{\frac{1}{6}}}$

(b) The force is negative derivative of potential energy function.

$F_{x} = - \frac{d U}{d x} \Rightarrow F_{x} = - \frac{d}{d x} [U_{0} \{{(\frac{a}{x})}^{12} - 2 {(\frac{a}{x})}^{6}\}] \Rightarrow F_{x} = 12 U_{0} [a^{12} {(\frac{1}{x})}^{13} - a^{6} {(\frac{1}{x})}^{7}] \dots (i)$

(c) The potential energy has its minimum value when

$\frac{d U}{d x} = 0 \Rightarrow 12 U_{0} [a^{12} {(\frac{1}{x})}^{13} - a^{6} {(\frac{1}{x})}^{7}] = 0 \Rightarrow a^{12} {(\frac{1}{x})}^{13} = a^{6} {(\frac{1}{x})}^{7}$

We get, $x = a .$ The minimum occurs at $x = a$ , which is the average spacing between atoms in such a molecule.

B M Sharma Solutions for Chapter: Work, Energy and Power, Exercise 45: ILLUSTRATION 8.43

B M Sharma Physics Solutions for Exercise - B M Sharma Solutions for Chapter: Work, Energy and Power, Exercise 45: ILLUSTRATION 8.43

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