# Lennard-Jones Work

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1. Feb 27, 2017

### Nacho Verdugo

1. The problem statement, all variables and given/known data
I need to calculate the work donde by the Lennard-Jones Law, considering:

F(r)=F0 [2(σ/r)13-(σ/r)7]

when approximating two atoms from infinity to the equilibrium position between both atoms

2. Relevant equations

First thing I don't know how to calculate is the equilibrium position (x0) between two arbitrary atoms

3. The attempt at a solution

I just integrated this force, considering

WAB=∫F⋅dr ; r=A to r=B

As I don't know the x0, I just integrate from A to B, to later analyze the result. Assuming A=-∞ and B=x0

W= ∫ F0 [2(σ/r)13-(σ/r)7] dr ; r=A to r=B
= integrating...
= 2F0σ13 ( r-12 /-12) - F0σ7 (r-6/-6) ; still need to evaluate

Considering that A=-∞, I finally obtain:

W= F0σ7/6 (1/x06)-F0σ13/6(1/x012)

Is it right until here?

2. Feb 28, 2017

### kuruman

What is the force at the equilibrium position. What does "equilibrium" mean?
What are your limits of integration? Consider particle A already at the origin. Then calculate the work done by the force on particle B as it is brought from infinity to x0.

3. Feb 28, 2017

### Nacho Verdugo

The statement of this problems is:

In diatomic molecules, the constituent atoms exert attractive forces between themselves at great distances and repulsive forces at short distances. For many molecules the Lennard-Jones law is a good approximation for the modulus of these forces.

$$F(r)=F_0 \cdot [2(\frac{\sigma}{r})^{13}-(\frac{\sigma}{r})^7]$$

where r is the distance between the center of the nucleus of the atoms, $\sigma$ a length parameter and $F_0$ a constant.

Determine the work to be performed by this force or by an external agent to approximate two atoms from infinity to the position of balance between the two.

- So, what I mean to equilibrium position is the position of balance between the two atoms, but there is no more information about that.

- The limits of integration, as I don't know the position of balance, are $A=-\infty$ to $B=x_0$ (the so called position of balance).

- Didn't catch up the last advice

Last edited: Feb 28, 2017
4. Feb 28, 2017

### kuruman

How big is the force on molecule A when it is at equilibrium or, as you say, in balance with molecule B?

5. Feb 28, 2017

### Nacho Verdugo

I guess it is zero right? I that way it is in equilibrium.

6. Feb 28, 2017

### kuruman

Exactly. So at what inter-molecular distance r0 is the force zero?

7. Mar 2, 2017

### Nacho Verdugo

I don't know :C I took a while to answer cause of that. I think it should be at a huge distance, right?

8. Mar 2, 2017

### kuruman

It is simpler than you think. The force in general is
$$F(r)=F_0 \cdot [2(\frac{\sigma}{r})^{13}-(\frac{\sigma}{r})^7]$$
At r = r0 you have
$$0=F(r_0)=F_0 \cdot [2(\frac{\sigma}{r_0})^{13}-(\frac{\sigma}{r_0})^7]$$
Do you see what you have to do?

9. Mar 2, 2017

### Nacho Verdugo

yes! with $$F(r_0)=0$$ I can actually find $$r_0$$ and later evaluate that value in the integral.

10. Mar 2, 2017

### kuruman

Go for it!

11. Mar 2, 2017

### Nacho Verdugo

thank you very much!

12. Mar 27, 2017

### Nacho Verdugo

13. Mar 27, 2017

### Nacho Verdugo

14. Mar 27, 2017

### kuruman

Is there a question you wish to ask?

15. Mar 27, 2017

### Nacho Verdugo

No thanks! I just made a mistake editing this.