Lennard-Jones Work: Calculating Equilibrium Position

[Nacho Verdugo]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

I just integrated this force, considering

W_AB=∫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?

 

[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.

 

[Nacho Verdugo]

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.

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

 

[kuruman]

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

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

 

[Nacho Verdugo]

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

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

 

[kuruman]

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

 

[Nacho Verdugo]

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

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

 

[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?

 

[Nacho Verdugo]

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?

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

 

[kuruman]

Go for it!

 

[Nacho Verdugo]

Go for it!

thank you very much!

 

[Nacho Verdugo]

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

 

[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

 

[kuruman]

Is there a question you wish to ask?

 

[Nacho Verdugo]

Is there a question you wish to ask?

No thanks! I just made a mistake editing this.