Calculating Molar Specific Heat w/ Equipartition Theorem for Hydrogen Gas

[jaejoon89]

How do you use the equipartition theorem to calculate molar specific heat for hydrogen gas?

Can somebody please explain how it works?

 

[Bacat]

Thanks for posting to the forum.

Can you list any equations you have?

What have you tried to do so far?

 

[Count Iblis]

The average energy per molecule in every degree of freedom is

1/2 k T

It is k T for vibrational degrees of freedom, but these are "frozen" at room temperature. Also, the rotational degree of freedom that has the axis of rotation along the line between the two H-atoms does not contribute either due to its high excitation energy (and this degree of freedom actually corresponds to the angular momentum of the two atoms).

So, what you need to do is figure out how many degrees of freedom of each sort there are. You can do that as follows. On the one hand, since H_2 is just two hydrogen atoms bound together, the total number of degrees of freedom must be twice that of a single atom. A single atom (ignoring electronic excitations) has 3 degrees of freedom, corresponding to the motion in three independent directions.

So, the total number of degrees of freedom is 6.

If you now look at the H_2 molecule, you can account for these 6 degrees of freedom as follows:

We have 3 degrees of freedom for the center of mass translational motion.

We have one vibrational degree of freedom.

We have 2 degrees of freedom for rotation as there are two independent chpoices for the rotational axis. The choice of the rotational axis along the line between the two atoms would, in the separate two atoms picture correspond to giving the two atoms an angular momentum which is not included in the total of 6 degrees of freedom.

Since 3 + 1 + 2 = 6, we can be sure to have accounted for all the degrees of freedom.

Then what you do is you assign an energy of 1/2 k T to the three translational and the 2 rotational degrees of freedom. The 1 vibrational degree of freedom is frozen at room temperature, so this doesn't count.

 

[kingkool]

There are 3 degrees of translational freedom, 2 of rotational freedom, 1 degree of vibrational freedom. The vibrational freedom is not accessible at room temperature, at higher temperatures it is but let's stick with room temperature.

So that's 5 degrees of freedom per molecule. So if you have a lot of molecules (N number of molecules), there are 5N degrees of freedom. each contributes 1/2kT to internal energy (U), for a total of 5/2kt*N..

So U = 5/2kT*N
dU/dT = heat capacity = 5/2K*N = 5/2R (for 1 mole, k*N_avagadro = R)

 

[Dadface]

There are 3 degrees of translational freedom, 2 of rotational freedom, 1 degree of vibrational freedom. The vibrational freedom is not accessible at room temperature, at higher temperatures it is but let's stick with room temperature.

So that's 5 degrees of freedom per molecule. So if you have a lot of molecules (N number of molecules), there are 5N degrees of freedom. each contributes 1/2kT to internal energy (U), for a total of 5/2kt*N..

So U = 5/2kT*N
dU/dT = heat capacity = 5/2K*N = 5/2R (for 1 mole, k*N_avagadro = R)

And if the gas is allowed to expand the specific heat capacity is greater because more heat has to flow into do external work, 5R/2 is the minimum specific heat capacity when the volume is kept constant and 7R/2 is the maximum when the pressure is kept constant.