[Cooper]

Hi,

My textbook states that at constant volume, the molar heat capacity is...

[tex]C_{V,m}=\frac{dU_m}{dT}[/tex]

And I get that, but then it goes on to substitute for a monatomic gas...

[tex]U_m=U_{m(0)}+\frac{3}{2}RT\rightarrow C_{V,m}=\frac{d(U_{m(0)}+\frac{3}{2}RT)}{dT}=\frac{3}{2}R.[/tex]

Now I have two questions...

First of all, how do you know the derivative of the first term (U_m(0)) WRT T is equal to 0? My book said this term was the energy of the atom at 0 K, therefore it represents the potential energy of the atom. Second of all, how can the heat capacity be a constant number (3R/2) when heat capacity is a function of temperature?

(EDIT: I think I get my first question now. The first term would just be a certain number of joules, correct? It doesn't depend on T and is a constant.)

I am probably misunderstanding something...

Thanks

[Corribus]

The heat capacity is kind of a measure of how many ways energy can be stored by a particle (or molecules). In an ideal monatomic gas, the only place that energy can go is into the translational kinetic energy of the atoms - i.e., making them move faster in random directions. In molecules, energy can also be funneled into internal vibrations and rotations - this is why the heat capacity of an ideal diatomic gas is larger than the heat capacity of an ideal monatomic gas. There is a single vibration that can store some energy.

An ideal gas is one in which there are no significant interactions between neighboring particles. If a monatomic gas does not behave ideally, there are interactions between neighboring atoms. These interactions can be treated, in a way, as intermolecular bonds, which also have characteristic "vibrations" and "rotations" of a sort that can also store energy. Which means that the heat capacity of a nonideal monatomic gas is different from that of an ideal monatomic gas. Note that gasses become less ideal as the pressure goes up or the temperature goes down. This means that the number of "vibrations" and "rotations" into which energy can be stored changes as a function of temperature... for a real gas! A hypothetical monatomic gas that remains ideal AT ALL TEMPERATURES need only ever consider translational kinetic energy, and the number of translational modes does not change as a function of temperature, so the heat capacity of an ideal monatomic gas is not a function of temperature, which is supported by your equation. Of course, no gas behaves truly ideally over all temperature, so it's a theoretical model only.

[Cooper]

Makes sense, thank you.