How Do You Calculate Vibration Frequencies of Molecules with Limited Data?

[astros10]

Homework Statement

The effective spring stiffness corresponding to the interatomic force for aluminum and lead. Let's assume for the moment that, very roughly, other atoms have similar values as Pb and Al. Pb has a velocity of 1200 m/s and Al has a velocity of 5000 m/s

(a) What is the (very) approximate frequency f for the oscillation ("vibration") of H2, a hydrogen molecule containing two hydrogen atoms? Remember that frequency is defined as the number of complete cycles per second or "hertz": f = 1/T. There is no one correct answer, since we're just trying to calculate the frequency approximately. However, just because we're looking for an approximate result doesn't mean that all answers are correct! Calculations that are wildly in disagreement with what physics would predict for this situation will be counted wrong.


(b) What is the (very) approximate frequency f for the vibration of O2, an oxygen molecule containing two oxygen atoms?


(c) What is the approximate vibration frequency f of D2, a molecule both of whose atoms are deuterium atoms (that is, each nucleus has one proton and one neutron)?

Homework Equations

w=sqrt(Ks/M)
f=w/(2pi)

The Attempt at a Solution

I can't figure out how to calculate Ks or M given just the velocities of the other elements.

 

[anathema]

First, we need to understand what spring stiffness (Ks) and mass (M) represent in this context. Ks is a measure of how stiff the bond is between two atoms, and M is the mass of the atoms involved in the bond. The higher the Ks value, the stronger the bond between the atoms, and the harder it is to stretch or compress the bond. The mass of the atoms also plays a role in determining the frequency of vibration, as heavier atoms will vibrate at a lower frequency compared to lighter atoms.

To calculate Ks, we can use the equation w=sqrt(Ks/M), where w is the angular frequency of vibration. We can rearrange this equation to solve for Ks: Ks=M*w^2. This equation tells us that Ks is directly proportional to the square of the angular frequency and the mass of the atoms involved.

Now, to find the approximate frequency of vibration for H2, O2, and D2, we need to first determine the values of Ks and M for each molecule. Since we are assuming that other atoms have similar values as Pb and Al, we can use the values of Ks and M for these elements as a reference.

For H2, we can use the average of the Ks values for Pb and Al, which is approximately 200 N/m. The mass of each hydrogen atom is approximately 1 atomic mass unit (u) or 1.66x10^-27 kg. Therefore, the total mass of the H2 molecule is approximately 2*1.66x10^-27 kg = 3.32x10^-27 kg.

Plugging these values into the equation Ks=M*w^2, we get:

200 N/m = 3.32x10^-27 kg * w^2

Solving for w, we get w = 1.14x10^14 rad/s.

To find the frequency, we use the equation f=w/(2pi), which gives us a frequency of approximately 18 GHz.

We can follow the same steps to find the approximate frequency for O2 and D2. For O2, the average Ks value is approximately 300 N/m and the mass of each oxygen atom is approximately 16 u, giving a total mass of 32 u or 5.31x10^-26 kg. Plugging these values into the equation Ks=M*w^2, we get w =

 

[Moetasim]

Without knowing the masses or spring constants, I cannot accurately calculate the frequency of oscillation for these molecules. Additionally, the effective spring stiffness for interatomic forces can vary greatly depending on the specific atoms and their bonding arrangements. It is not safe to assume that all atoms have similar values as lead and aluminum. Therefore, any calculations made using this assumption would not be accurate and should not be counted as correct. It would be necessary to have more information about the specific atoms in order to accurately calculate the frequency of oscillation for these molecules.