Conflicting definition of degree of freedom in Kinetic Theory of Gases

[vcsharp2003]

TL;DR Summary

Is degree of freedom just an independent term/variable/coordinate or the number of independent terms/variables/coordinates?

I am seeing conflicting definitions of degree of freedom in my textbook. If I look at the definition given as per screenshot below then it is the number of independent terms/variables/coordinates used to define the energy of a molecule. But, if I look at the statement of Equipartition of energy that is given below the definition, then it seems that degree of freedom is any one of independent terms/variables/coordinates used to get energy of a molecule.

I think degree of freedom should just be the independent term/variable/coordinate.

 

[Doc Al]

Why do you think the two paragraphs conflict?

 

[vcsharp2003]

Why do you think the two paragraphs conflict?

The first paragraph says that degree of freedom is the  number of independent variables.

The second paragraph is is meaning degree of freedom to be the independent variable.

 

[Doc Al]

Ah, I think I see what you're saying. I can appreciate the confusion.

I would have said something like: A monatomic gas molecule has three degrees of freedom, a diatomic molecule has five, etc. And the equipartition of energy gives the average energy associated with each degree of freedom.

 

[vanhees71]

For me the excerpt from the book is formulated so inaccurately that it's even errorneous and misleading. I trcommend to use another book. I'd rdcommend the Berkeley physics course volume on Statistical Physics (the "little Reif").

 

[vcsharp2003]

Ah, I think I see what you're saying. I can appreciate the confusion.

I would have said something like: A monatomic gas molecule has three degrees of freedom, a diatomic molecule has five, etc. And the equipartition of energy gives the average energy associated with each degree of freedom.

Yes, that makes perfect sense; I mean the last paragraph.

It appears that degree of freedom means the independent coordinate/variable/term.

 

[vcsharp2003]

For me the excerpt from the book is formulated so inaccurately that it's even errorneous and misleading. I trcommend to use another book. I'd rdcommend the Berkeley physics course volume on Statistical Physics (the "little Reif").

I agree. Its clearly confusing since a student reading these paragraphs from that book would be left wondering "What exactly is a degree of freedom".

 

[vanhees71]

The correct statement for the equipartition theorem is that any phase-space-degree of freedom, which enters the Hamiltonian quadratically (!!!) contributes ##k_{\text{B}} T/2## per particle to the (mean) energy. E.g., for an ideal mon-atomic gas, the single-particle Hamiltonian is ##H=\vec{p}^2/(2m)##, and thus the mean energy per particle is ##3 k_{\text{B}} T/2##. For two-atomic (and more-atomic) molecules you have two (three) more rotational degrees of freedom. This is when the vibrational modes are still frozen, i.e., at not too high temperatures. At higher temperatures you get additional vibrational degrees of freedom and the corresponding contributions to the mean energy per particle (in the harmoni-oscillator approximation).