# Degrees of freedom and specific heat?

The specific heat of a given substance increases with an increase in temperature because more degrees of freedom become "unfrozen" at larger temperatures, yeah?

My doubt is this; why does an increase in the number of degrees of freedom directly relate to an increase in the specific heat of the substance? I realize that the energy itself is able to be distributed in more ways due to these additional degrees of freedom that become available but so what? Can this energy not be just as efficiently distributed in the existing degrees of freedom?

Additionally, I realize that there is a flaw in my logic in that more degrees of freedom would obviously make more paths available but why should this increase the "efficiency"?

If the explanation requires for me to have a basic understanding of quantum mechanics, please let me know because I have only just graduated high school.

MUCH thanks in advance :) Regards.

The key point you seem to be missing is that energies add but temperatures do not, since temperature is an average.

For instance, suppose you have two systems. System 1 has one degree of freedom. You give it an energy $E$ and it vibrates with a corresponding temperature $T$.

System 2, on the other hand, has two degrees of freedom. You give each degree of freedom an energy $E$ so each vibrates with a temperature $T$. The total energy of this system will therefore be $2E$ while the temperature will still be $T$.

So the more degrees of freedom you have, the more energy is stored at any given temperature (i.e. a higher heat capacity).

• OH! What I hadn't digested is the fact that temperature is the average of the K.E of the constituents of the sample! Thanks for clearing this up. – user106570 Sep 7 '16 at 10:41

Temperature is a measure for the mean energy that is stored in a degree of freedom.

Consider for instance a large number of pendula, that are all coupled (for instance with very soft springs). If one pendulum has a large amplitude in comparison with the others (i.e. it is 'hotter'), then this additional energy will quickly be disspiated and distributed to the other oscillators.

This is exactly what happens if you bring two bodies into contact: energy will flow from the hotter to the colder body, until the mean energy per DoF (i.e. the temperature) is equalized.

It does not have anything to do with the number of ways the energy can be distributed among a number of DoF's (or, as physicists call it, entropy). It's just that the more DoF's you have, the more of them you need to heat up, thus increasing the specific heat.