Determine the nearest neighbour equilibrium distance

[didy]

Can teach me homework to do?

2. Energy per atom for argon crystal is given by
U(R) = −(A /R6 )+ (B/R12 )
where A = 1.03 ×10−77 Jm6 and B = 1.62 ×10−134 Jm12 are constants and R is the nearest
neighbour distance.
(a) Explain both expressions on the right hand side in the above equation.
(b) Determine the nearest neighbour equilibrium distance.
(c) Compare the van der Waals and repulsion between cores contributions to the energy at
the equilibrium distance.
(d) Estimate the magnitude of the van der Waals force between 2 argon atoms at
equilibrium distance, assuming that the van der Waals energy is only due to the nearest
neighbour interaction.

Thanks a lot...

 

[Jhero]

Sure, I would be happy to help you with your homework on argon crystal energy per atom.

(a) The first expression, -(A/R6), represents the van der Waals attraction between two argon atoms. This attraction is due to the temporary dipoles that form between atoms, creating a weak attractive force. The second expression, (B/R12), represents the repulsion between the two atoms due to their electron clouds overlapping. This repulsion is much stronger than the van der Waals attraction and is due to the Pauli exclusion principle, which states that no two electrons can occupy the same space.

(b) To determine the nearest neighbour equilibrium distance, we need to find the distance at which the energy is at its minimum. This can be done by taking the derivative of the energy equation with respect to R and setting it equal to 0. Solving for R gives us the equilibrium distance of R = (2B/A)^(1/6) = 3.71 x 10^-10 m.

(c) At the equilibrium distance, the van der Waals energy is equal to -(A/R6) = -2.05 x 10^-4 J. The repulsion between cores energy is equal to (B/R12) = 1.48 x 10^-4 J. This means that the repulsion between cores contributes more to the total energy than the van der Waals attraction.

(d) To estimate the magnitude of the van der Waals force between two argon atoms at equilibrium distance, we can use the force equation F = -dU/dR = 6A/R7 = 3.61 x 10^-8 N. This is a very weak force, but it is still strong enough to keep the atoms bonded together in a crystal structure.

I hope this helps you understand the energy per atom for argon crystal and how to determine the nearest neighbour equilibrium distance. Let me know if you have any other questions or need further clarification. Good luck with your homework!