# Degree of Freedom

1. Nov 4, 2009

### User Name

The homework assignment that I've been given in Physics deals with degrees of freedom (which my assignment refers to as "modes").

We're given the equation E$$_{thermal}$$= (# of modes) $$\times$$ $$\frac{1}{2}$$ $$\times$$ $$k$$ $$\times$$ T.

(k being the Boltzmann's Constant).

We are also told that gases have 3 modes of energy while solids/liquids have 6 modes of energy.

Then we're given a a list of two things to compare and to determine which has the higher thermal energy; most of them I have been fine with, but there has been one that is confusing me to no end.

I'm comparing 1 mole of a monatomic gas at 290 K versus 1 mole of a monatomic solid at 290 K.

If I use the equation given, then the solid would have a higher thermal energy (since both have 1 mole of the substance, which is $$N_{A}\ =\ 6.02214199(47)\ \times\ 10^{23} mol^{-1}$$ and the number of modes are dependent on how many particles there are.. so a gas would have 3 $$\times$$ Avagadro's number while a solid would have 6 $$\times$$ Avagadro's number, which results in solids having more thermal energy).

However, this doesn't really make sense to me. Shouldn't a gas have a higher thermal energy than a solid?

The subsequent question also raised this issue with me, because I had to explain how to use that particular equation I had to determine whether gases or liquids (specifically H$$\_{2}\$$O liquid and O$$\_{2}$$ gas) move faster at thermal equilibrium. If I similarly use the equation like I did in the previous problem, then I would again have that the liquid would have a higher thermal energy than the gas since it has 6 modes of energy versus 3 modes for gases.

However, I believe that thermal energy is directly related to kinetic energy at the microscopic level, so a higher thermal energy would then mean higher kinetic energy, which would then mean the liquid is moving faster than the gas.

And yet again, this doesn't make sense to me, because gases most definitely move much quicker than liquids.

Any help would be much appreciated!