Finding the Equilibrium Separation in a Diatomic Molecule: A Scientific Approach

[Ginny Mac]

Oh boy...this problem has stumped me. I am not even sure I know where to start:

The potential energy of a diatomic molecule (a two-atom system like H2 or O2) is given by
U= (A/r^12) - (B/r^6)

where r is the separation of the two atoms of the molecule and A and B are positive constants. This potential energy is associated with the force that binds the two atoms together.
(a) Find the equilibrium separation - that is, the distance between the atoms at which the force on each atom is zero. Is the force repulsive (atoms are pushed apart) or attractive (atoms are pulled together) if their separation is (b) smaller and (c) larger than the equilibrium separation?

Okay. I started out thinking that because we are looking for the force, we would need to use F(x) = - (d/dx)(U(x)), and after differentiation

F(x) = (12A/r^13) + (6B/r^7)

So here is the big "now what?" Please help. Thank you - any help is greatly appreciated.
~Gin

 

[robphy]

(a) Find the equilibrium separation - that is, the distance between the atoms at which the force on each atom is zero.
[snip]

Okay. I started out thinking that because we are looking for the force, we would need to use F(x) = - (d/dx)(U(x)), and after differentiation

F(x) = (12A/r^13) + (6B/r^7)

So here is the big "now what?" Please help. Thank you - any help is greatly appreciated.
~Gin

You have the next step: you said "Find ... the distance between the atoms at which the force on each atom is zero." (that is, find the value of r that makes F=0)

 

[Ginny Mac]

okay...but there are three unknowns in my equation... Is there a step I am leaving out?? Even solving algebraically only reduces down to 2A-Br^6 = 0. Does something cancel out that I am not aware of? Or maybe we are supposed to be left with an equation?

 

[robphy]

You are told "A and B are positive constants". So, you can expect that your answer is in terms of those positive [but otherwise unspecified] constants.

 

[Ginny Mac]

Ahhh...so I got my equation down to (2A/B)^(1/6) = r. So that value of r is the equilibrium separation. Now for the next part, figuring out atomic behaviors based on larger/smaller values in relation to equilibrium sep. This should be interesting.

Equation: 2A-Br^6=0

If sep. is larger than equilibrium, we should have a negative force, and if it is smaller than eq., we should have a positive force. I think a negative force will draw the atoms together (?) Any thoughts are appreciated. Thank ya'll.

gin

 

[robphy]

Can you sketch the potential energy function U= (A/r^12) - (B/r^6) vs separation r, and locate your special value of r?