Force between polar molecule and an ion

[Westin]

Homework Statement

Under certain conditions the interaction between a "polar" molecule such as HCl located at the origin and an ion located along the x-axis can be described by a potential energy U=−b[PLAIN]https://s3.lite.msu.edu/adm/jsMath/fonts/cmmi10/alpha/144/char3D.pngx^2, where b is a constant.

What is Fx, the x component of the force on the ion?

Fx=

What is Fy, the y component of the force on the ion?

Fy= 0[/B]

Homework Equations

U=-b/x^2[/B]

The Attempt at a Solution

The y component of the force on the ion was fairly obvious but I am getting stuck on the x component.

I know that the force must me positive but it was incorrect when I tried +b[PLAIN]https://s3.lite.msu.edu/adm/jsMath/fonts/cmmi10/alpha/144/char3D.pngx^3[/B][/B]

 

[TESL@]

I think you need to know the partial charges and their separation/distance.

 

[Westin]

these are the choices i have
A). +2b/x^3
B). 0
C). +b/x^3
D). -2b/x^3
E). -b/(2x)
F). -b/x^3
G). -b/x

 

[DrClaude]

What is the relation between force and potential?

 

[DrClaude]

I know that the force must me positive

Why?

 

[DrClaude]

I think you need to know the partial charges and their separation/distance.

Why do you say that?

 

[TESL@]

The molecule will rotate in such way that H will be at the farthest and Cl at the closest point to the ion. Then the force will depend on partial charges and the distance between both H and Cl, and ion and molecule.

 

[DrClaude]

The molecule will rotate in such way that H will be at the farthest and Cl at the closest point to the ion. Then the force will depend on partial charges and the distance between both H and Cl, and ion and molecule.

On can infer from the equation U=−b/x² that the molecule is taken as a point dipole with its orientation assumed to be always favorable.

 

[Westin]

Wait, wouldn't the answer be negative because the force on an object is the negative of the derivative of the potential function? The potential energy U is equal to the work you must do against that force to move an object from the U=0 reference point to the position r. The force you must exert to move it must be equal but oppositely directed, and that is the source of the negative sign.

 

[Westin]

I'm almost positive the answer is -b/(2x) after what I just said. Any feedback?

 

[DrClaude]

I'm almost positive the answer is -b/(2x) after what I just said. Any feedback?

Again, what is the equation that relates force and potential energy?

 

[Westin]

Again, what is the equation that relates force and potential energy?

F = -dU/dx

 

[DrClaude]

F = -dU/dx

Great, now apply that equation to the problem given.

 

[Westin]

Im confused with on how to.. F = −b/x^2 take derive then get -b/2x

 

[DrClaude]

Im confused with on how to.. F = −b/x^2 take derive then get -b/2x

I guess you mean U. What is the derivative of ##x^{-2}##?

 

[Westin]

Great, now apply that equation to the problem given.

F = -b/x^2 ---> -bx^-2 ---> 2b/x^3 this is it

 

[Westin]

F = -b/x^2 ---> -bx^-2 ---> 2b/x^3 this is it

It was incorrect... I have one try left , I don't get what else it could be

 

[DrClaude]

F = -b/x^2 ---> -bx^-2 ---> 2b/x^3 this is it

Doesn't this contradict what you wrote earlier?

Wait, wouldn't the answer be negative because the force on an object is the negative of the derivative of the potential function? The potential energy U is equal to the work you must do against that force to move an object from the U=0 reference point to the position r. The force you must exert to move it must be equal but oppositely directed, and that is the source of the negative sign.

It was incorrect... I have one try left , I don't get what else it could be

Before submitting answers, let's check if they make sense...

 

[Westin]

Ahh yes it does,-2b/x^3 should be correct? I am sorry for my personal confusion. I'm in a poor experimental physics class at Michigan state where you have to teach yourself online everything..

 

[DrClaude]

It's ok to be confused. I'm just hoping that you can gain some good habits. From the formula for energy, you can see that the energy increases (decreases in magnitude but with a negative sign) when the ion moves away from the origin, so the force must be directed towards the molecule. That takes care of the sign. Then it is just a question of going back to fundamentals, ie, F = -dU/dx, to get the force. Then you can check that the answer makes sense, considering the physics (force must be directed towards the origin). It's easy to lose track of minus signs, especially in a case like this where you get 3 in a row.