Degrees of freedom in a molecule

[LCSphysicist]

Homework Statement

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Relevant Equations

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I am not understand the solution of the letter b and c given by the author.

Spoiler: solution

Since Cv = NfK/2, we have Cv/N = fK/2. Now, the degree of freedom of a N linear molecule is 3(trans) + 2(rot) + x, where x is the degree of freedom due the vibrational motion. I am having trouble to calc x, could you help me?

 

[ehild]

Homework Statement:: .
Relevant Equations:: .

View attachment 273247
I am not understand the solution of the letter b and c given by the author.

Spoiler: solution

View attachment 273248

Since Cv = NfK/2, we have Cv/N = fK/2. Now, the degree of freedom of a N linear molecule is 3(trans) + 2(rot) + x, where x is the degree of freedom due the vibrational motion. I am having trouble to calc x, could you help me?

What kind of energies belong to a single vibration?

 

[LCSphysicist]

What kind of energies belong to a single vibration?

The kinetic energy due to the vibration and the potential stored (as in kx²/2)

 

[ehild]

The kinetic energy due to the vibration and the potential stored (as in kx²/2)

Good! A two-atomic molecle has a single vibrational mode. When performig that motion, the molecule has both kinetic and potential energy. If you want to specify thevibrational state of the molecule, you have to give both the velocity and the phase of the vibrationThese are two degrees of freedom for each vibrational mode.
How many vibration modes has an N-atomic molecule?

 

[LCSphysicist]

Good! A two-atomic molecle has a single vibrational mode. When performig that motion, the molecule has both kinetic and potential energy. If you want to specify thevibrational state of the molecule, you have to give both the velocity and the phase of the vibrationThese are two degrees of freedom for each vibrational mode.
How many vibration modes has an N-atomic molecule?

"A two-atomic molecle has a single vibrational mode. " That's what i don't understand, two particles connected by a spring, as a model for a molecule, do not have two degree of freedom? So two vibrational mode which gives us 4 types of energy? The kinect and energy stored in one mode and in another mode.

 

[ehild]

"A two-atomic molecle has a single vibrational mode. " That's what i don't understand, two particles connected by a spring, as a model for a molecule, do not have two degree of freedom? So two vibrational mode which gives us 4 types of energy? The kinect and energy stored in one mode and in another mode.

Two particles conneted by a spring corresponds to a single oscillator, a single vibrational mode, but it has two degrees of vibrational freedom,.
A molecule of N atoms can perform 3N independent motions. From these , 3 are translations of the whole molecule: and 3 are rotations of the whole molecule. When the molecule is linear, it performs 2 indepemdent rotations. The other 3N-6 (3N-5) motions are vibrations. A translating or rotating molecule has only kinetic energy, 1/2 kT for each degree of freedom. Each of the 3N-6 (3N-5) vibrational modes have kT average energy.
Both atoms iof a two-wtomic molecule can move in the x, y, z directionsthat makes 6 different motions. From these, 3 motions are the same for both atoms,they are translations The moleculée can also rotate about two perpendicular axes . The translating or rotating molecule is rigid, it does not change shape. And there is a single motion, when the atoms move relative to each other: it is vibration.