Statistical mechanics - characteristics temperature of the HF molecule

In summary, the hydrogen fluoride molecule has a vibrational frequency of 7.8x10^14 rad/sec and a moment of inertia of I=1.35x10^-47 kg.m^2. Find the relevant characteristic temperatures of the HF molecule.
  • #1
Anabelle37
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Homework Statement



Spectroscopic data (rotational-vibrational lines) show that the hydrogen fluoride molecule has a vibrational frequency of 7.8x10^14 rad/sec and a moment of inertia of I=1.35x10^-47 kg.m^2. Find the relevant characteristic temperatures of the HF molecule.


Homework Equations





The Attempt at a Solution



I am completely lost on where to start and what formulas to use!
Please help!
 
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  • #2
Characteristic temperatures can be figured out easily like this:

the usual "energy unit" in statistical mechanics is k*T where k is boltzman's constant and T is the temperature.
Now different kind of motions have different "energy units" for example the energy for vibration is [tex]\hbar \omega [/tex].
So we shall obtain the characteristic temperature if we make this equal to the "statistical mechanics energy" : [tex]k\cdot T_v=\hbar \omega [/tex]

And from here you can express T_v.

Similarly for rotations, the energy of a quantum rotator is hbar^2/(2*I). and you can figure the characteristic temp. for rotations from this...
 
  • #3
thank you heaps!

the following part says:

Assuming that HF is a diatomic gas, use the equipartition theorem to predict the specific heat of this gas (quoted as C_v/molecule/k) at T=520K

Won't there be a contribution to the specific heat from each of the vibrational and rotational parts which don't depend on temperature?
 
  • #4
They do depend on temperature of course :)

Here what you have to do is figure out the different degrees of freedom. Since HF is a linear diatomic molecule it will have 2 rotational degrees of freedom (you can check this by looking on the molecule) then it will have 3 translational degrees of freedom (it can move in three directions in space) and finally it will have 1 vibrational degree of freedom (it can only vibrate with the atoms approaching each other, as if on a spring). But the vibrational part is tricky as one vibrational degree of freedom, means that there will be two square terms in the hamiltonian. These means that actually when using the equipartition theorem you will have to take this 2 times.

So totally there are 3+2+2*1=7 square terms in the hamiltonian. Hence the specific heat is ... and I will leave you to figure that out ;)
 
  • #5
Ok thanks,
So the internal energy will be (7/2)*k*T so the specific heat will be (7/2)*k which doesn't depend on temperature so why have they given us T=520K??
 

1. What is statistical mechanics?

Statistical mechanics is a branch of physics that uses statistical methods to explain the behavior of a large number of particles, such as molecules, in a system. It aims to predict the macroscopic properties of a system based on the microscopic properties of its individual components.

2. How is temperature related to the HF molecule in statistical mechanics?

In statistical mechanics, temperature is a measure of the average kinetic energy of the molecules in a system. For the HF molecule, temperature is directly related to the average speed and motion of the individual molecules, which in turn affects the overall behavior and properties of the system.

3. What are the characteristics of the HF molecule?

The HF molecule is a diatomic molecule composed of hydrogen and fluorine atoms. It has a polar covalent bond, with the fluorine atom having a higher electronegativity and therefore pulling the shared electrons closer to itself. This results in a dipole moment and makes the molecule polar, with a slightly positive charge on the hydrogen atom and a slightly negative charge on the fluorine atom.

4. How does the temperature of the HF molecule affect its properties?

The temperature of the HF molecule affects its properties in several ways. As the temperature increases, the molecules gain more kinetic energy and move faster, leading to a higher average speed and a wider distribution of velocities. This can affect properties such as diffusion, heat capacity, and viscosity. Additionally, at high temperatures, the hydrogen and fluorine atoms may dissociate, breaking the bond between them and changing the overall behavior of the system.

5. What is the Boltzmann distribution and how is it related to the temperature of the HF molecule?

The Boltzmann distribution is a statistical law that describes the distribution of energies among a system of particles. It states that the number of particles with a certain energy level is proportional to the exponential of the negative energy divided by the temperature. In the case of the HF molecule, this distribution can help explain the distribution of molecular speeds and energies at a given temperature.

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