Dissociation energy of two particles-variables only equation-mastering physics

[Same-same]

Homework Statement

The potential energy of two atoms in a diatomic molecule is approximated by U(r)= ar$^{-12}$ - βr$^{-6}$, where is the spacing between atoms and α and β are positive constants.
From earlier parts of the equation, it has been that determined that force between two atoms as a function of r is
F(r)= 12ar$^{-13}$ - 6βr$^{-7}$.
and that the equilibrium distance between them r$_{min}$ is
$\sqrt[6]{2α/β}$

Part a: Is the equilibrium stable? (My note: a stable equilibrium is a local minimum in a potential energy function. An unstable is a local maximum)

Part b:
Asssuming that the molecules are at said equlimbrium distance apart, find the energy required to bring the molecules an infiinite distance apart., and express this equation in terms of α and β

Homework Equations

W=∫ F dr= ΔU


3. The Attempt at a Solution
To part a:
Since potential energy is negatively related to displacement, and there are only two scenarios where Force could be 0 (at this distance or at infinity), I believe this equilibrium is unstable.

to part b:
∫$^{∞}_{equilibrium}$ (12ar$^{-13}$ - 6βr$^{-7}$) dx
which appears to be negative infinity, which I'm fairly certain is wrong.

Part b is what's really giving me problems. I'm not sure exactly what I'm looking for; I've honestly never dealt with dissociation before. I considered integrating the potential energy function from 0 to infinity, but that would give me the wrong units. Perhaps I'm looking for an energy equal to the maximum potential energy possible with the function?


Any help would be greatly appreciated
Thanks in advance!

 

[DukeLuke]

Part a: You might want to look up what a local maximum or minimum of a function is, and how to tell one from the other (hint: telling a maximum from a minimum has to do with the sign of the 2nd derivative at the max/min).

Part b: I don't think you are evaluating your limits correctly. Infinity to a negative power puts it in the denominator. If you divide a number by infinity what do you get?

 

[Same-same]

"Part b: I don't think you are evaluating your limits correctly. Infinity to a negative power puts it in the denominator. If you divide a number by infinity what do you get? "

My thought process (which may very well be wrong) was as follows "At infinity the function should be zero, but the area under the curve to get there appears to be infinite."

 

[DukeLuke]

You are right about the function approaching zero as it heads out to +- infinity, but that doesn't mean the area under the curve is infinite (unless you integrate over zero in the case of your function). Maybe if you plot it this will be easier to see.

 

[rwisz]

Your solution to part a is correct. The equilibrium in this scenario is unstable because it is a local maximum in the potential energy function.

For part b, you are on the right track. The energy required to bring the molecules an infinite distance apart is equal to the maximum potential energy of the system. This can be found by setting the potential energy function equal to 0 and solving for r. This will give you the distance at which the potential energy is at its maximum. Then, you can plug this value of r into the potential energy function to find the maximum potential energy in terms of α and β.

Another approach to solving part b would be to use the equation you provided in the homework statement, W = ∫ F dr = ΔU. This equation states that the work done by a force is equal to the change in potential energy. In this case, the force is given by F(r) = 12ar^-13 - 6βr^-7 and the potential energy is given by U(r) = ar^-12 - βr^-6. Integrating the force function from the equilibrium distance to infinity will give you the energy required to bring the molecules an infinite distance apart. You can then plug in the equilibrium distance and solve for the energy in terms of α and β.