Harmonic vibrational frequency

[Ajs51]

1. A molecule of 39^K 35^Cl consists of two ions,39^K+ and 35^Cl−,which are attracted to each other by electrostatic forces. The short-range repulsive force results in an interatomic distance of 2.7 ˚A at equilibrium. Considering only the electrostatic attraction, derive an expression for the force constant using Hooke’s law.Hence, determine the harmonic vibrational frequency of the 39^K 35^Cl molecule in wavenumbers.



3. I just can't seem to do this any help would be appreciated

 

[badphysicist]

There is one positive charge on the potassium and one negative charge on the chlorine. The attractive force is from these two opposite charges. It sounds to me like you are supposed to treat the repulsive force as being spring-like. You also know the distance between the atoms when the forces cancel each other out exactly, so that should give you all of the information that you need. Hopefully that points you in the right direction.

 

[Ajs51]

Oh that makes sense..would it make sense to set the Coulomb force, equal to the hooke law...solve to find k...then use k to find w (angular frequency) ?

 

[Ajs51]

Also thanks for the help!

 

[paranormal]

I would be happy to provide assistance with this problem.

First, let's define some terms. The force constant, denoted as k, is a measure of the strength of the chemical bond between two atoms in a molecule. It is related to the bond length, which is the distance between the two atoms at equilibrium, and the vibrational frequency, which is the frequency at which the atoms vibrate around their equilibrium position.

To derive an expression for the force constant using Hooke's law, we can start by considering the electrostatic attraction between the two ions in the 39^K 35^Cl molecule. This attraction can be described by Coulomb's law, which states that the force between two charged particles is proportional to the product of their charges and inversely proportional to the square of the distance between them. In this case, the charges are the charges of the two ions and the distance is the interatomic distance of 2.7 ˚A.

Using this information, we can write the following equation for the electrostatic force:

F = k_e * (q1 * q2) / r^2

Where F is the force, k_e is the Coulomb constant, q1 and q2 are the charges of the two ions, and r is the interatomic distance.

Now, we need to relate this force to the force constant, which is defined as the force required to stretch or compress a bond by a certain amount. This can be achieved through Hooke's law, which states that the force required to deform an object is proportional to the amount of deformation. In this case, the deformation is the change in bond length, which we can denote as Δr.

Using Hooke's law, we can write the following equation:

F = k * Δr

Combining this with the previous equation, we get:

k * Δr = k_e * (q1 * q2) / r^2

Solving for the force constant, we get:

k = (k_e * (q1 * q2)) / (r^2 * Δr)

Now, we can plug in the values for k_e, q1, q2, and r to get an expression for the force constant in terms of the equilibrium bond length and the change in bond length.

To determine the harmonic vibrational frequency, we can use the equation:

v = (1 / 2π