Hydroxl Ion radius, binding energy and vibrational frequency

[spaderdabomb]

Homework Statement

The Hydroxl Ion (OH-) may be considered as a bound state of O^2- and a proton. Treating the O^2- ion as a nucleus of charge +8 with the electrons homogeneously distributed in a sphere of radius b, determine approximately the position of the proton, the hydroxl binding energy and its vibrational frequency.

Homework Equations

Vibrational energies E=(n + 1/2)hbarω n = 0, 1, 2...

E = hv=hc/λ where v is the vibrational frequency

The Attempt at a Solution

This was a test question and I got 1/20 points on it lol. So my attempt was pretty futile. There may be different methods you can use to solve this problem. One of my questions is what exactly the binding energy is. I thought it would just be the ground state vibrational energy but I could be wrong.

 

[mfb]

The binding energy is the ground-state energy of the proton in the potential well generated by the oxygen atom (neglecting motion of oxygen and the influence of the proton on the electron distribution). It is negative.
The (positive) vibrational energy has to be lower in magnitude than the binding energy, otherwise the molecule would break apart instead of oscillating.

 

[spaderdabomb]

Ok well I'm still not sure how to go about calculating the binding energy then. I understand how to get the vibrational energy but the binding energy and the distance between the two atoms is still confusing me.

I guess I know that the binding energy is at a minimum in potential energy which would correspond to the distance between the two molecules, but I just don't even know where to start. Help specifically on the math and calculations would be appreciated.

 

[mfb]

I have no idea how you want to determine the vibrational energy without knowledge of the binding energy.

I don't think you are supposed to solve the Schroedinger equation for that potential, so it is probably sufficient to calculate the minimum of the potential energy, using the given charge distribution.
Do you know how the potential of a point-charge looks like? The potential of a sphere?

 

[paranormal]

I am sorry to hear that you did not receive a high score on this test question. As a scientist, it is important to understand and analyze the information provided in the question to come up with a proper solution.

The binding energy of the Hydroxl Ion refers to the energy required to separate the O2- ion and the proton, essentially breaking the bond between them. This energy can be calculated using the Coulomb's law and the distance between the two particles.

To determine the position of the proton, we can use the concept of center of mass. Since the O2- ion is treated as a nucleus of charge +8, the center of mass will be closer to the O2- ion. Therefore, the proton will be located at a distance less than b from the center of the O2- ion.

To calculate the vibrational frequency, we can use the formula E = hv = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength. The vibrational energy can be calculated using the formula E = (n + 1/2)hbarω, where n is the quantum number and ω is the vibrational frequency.

Overall, solving this problem would require the use of various concepts and equations from physics and mathematics. It is important to carefully analyze the given information and use appropriate formulas to arrive at a solution.