The potential energy of two atoms separated by a distance r is given by: (see Im
ID: 1525915 • Letter: T
Question
The potential energy of two atoms separated by a distance r is given by: (see Img)
where and are constant parameters. This is known as the Lennard-Jones potential, which describes the interaction of two inert gas atoms. The first term in the potential energy is due to a repulsive interaction from the overlapping charge distributions of the atoms, and the second term is due to an attractive dipole-dipole interaction, known as the van der Waals interaction.
(a) Find the value of r = r0 at which the force between the atoms vanishes (equilibrium point). (Hint: Recall that the force between atoms is obtained from the potential energy as F(r) = dV/dr.)
(b) Find the Taylor series expansion of V (r) about r = r0, neglecting terms O((r r0)3) and higher.
(c) Sketch the potential energy V (r). For what values of the total energy is the motion of the atoms bound, that is rmin < r < rmax? Find the maximum and minimum values of r. Describe qualitatively the motion of the two atoms in the vicinity of r0 (that is, for (rmax rmin)/r0 1).
The potential energy of two atoms separated by a distance r is given by: 12 where E and o are constant parameters. This is known as the Lennard-Jones potential, which describes the interaction of two inert gas atoms. The first term in the potential energy is due to a repulsive interaction from the overlapping charge distributions of the atoms, and the second term is due to an attractive dipole-dipole interaction, known as the van der Waals interaction. (a) Find the value of r ro at which the force between the atoms vanishes (equi- librium point). (Hint: Recall that the force between atoms is obtained from the potential energy as Fr dV/dr.) (b) Find the Taylor series expansion of V(r) about r To, neglecting terms O(r ro)s) and higher.Explanation / Answer
a) Here, - dV/dr = 0
=> - 12 * (sigma)12 * (1/r)13 + 6 * (sigma)6 * (1/r)7 = 0
=> - 2 * (1/r)6 + 1 = 0
=> r = (2)1/6
b) The Taylor series expansion of V (r) about r = r0
=> V(r) = 1 - r + r2/2! - r3/3! + r4/4! - ........
c) the maximum value of r = (2)1/2
the minimum value of r = (2)1/12
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