# The Equipartition of Energy

For monatomic gases, there is excellent agreement between experimental measurements and theoretical predictions of CP, CV, and CP/CV.

What is different about more complex gases that would lead to different results?

Any atom can have translational kinetic energy -- even a monatomic atom. A more complex atom can also have kinetic energy of rotation and of vibration.

It can be shown that the total energy is shared equally by each independent "degree of freedom". A "degree of freedom" is, essentially, an independent variable that is necessary to specify the location or speed or movement of a system. For a monatomic gas, the atoms are considered tiny, "point particles" and each is fully specified by the variables x, y, and z. A monatomic gas has three degrees of freedom. Or three coordinates fully specify the motion of a point.

Now, move on to a diatomic molecule. Three coordinates specify its center of mass. Three additional coordinates specify its orientation in space or its rotation about the center of mass. However, for point masses, the moment of inertia about the symmetry axis is zero. This means there is no kinetic energy for rotation about this axis. That effectively reduces our degrees of freedom from six to five.

By the equipartition theorem, we expect each "degree of freedom" to contribute

(1/2) N k T

or

(1/2) n R T

to the total internal energy. Therefore, we expect

U = (5/2) N k T

or

U = (5/2) n R T

for a diatomic molecule. With U = (5/2) n R T, we have

dU = n CV dT

That means

CP = CV + R = (7/2) R

= CP/CV = [(7/2)R]/[(5/2)R] = 7/5 = 1.40

And this, too, is in very good agreement with experimental values.

And two additional coordinates specify such a diatomic molecule's contraction or elongation -- its state of vibration. That would give us

CV = (7/2) R

and

= CP/CV = [(9/2)R]/[(7/2)R] = 9/7 = 1.28

Why is the experimental agreement better without considering the vibration?

In particular, look at the molar specific heat, CV, for molecular hydrogen (H2) gas. At very low temperatures the molar specific heat is very near to (3/2) R -- there is only energy due to the translational kinetic energy of the center of mass of the molecules. At somewhat higher temperatures, the rotational nature becomes availabe and the molar specific heat is near to (5/2) R -- but there is still no effect of vibration. Only at higher temperatures do we see the vibrational effects of the H2 molecule as the molar specific heat approaches (7/2) R.

In PHY 1370 we will look in detail at energy quantization and some very unexpected results and characteristics due to quantization effects. Briefly these effects mean that there are only certain, particular, discrete values that energy may have. There are only certain, particular, discrete values that the rotational energy can have and the energies available at low temperatures are not great enough to move the molecule from one rotational energy to the next higher rotational energy level or energy state. There are only certain, particular, discrete values that the vibrational energy can have and the energies available at low temperatures are not great enough to move the molecule from one vibrational energy to the next higher rotational energy level or energy state.

This temperature dependence of the value of the molar specific heat is a result of the ideas of "modern Physics" or energy quantization. Consider this a preview of interesting and unexpected results that you will encounter is PHY 1370.

For the same reasons -- the quantization of available energy states -- the molar specific heat of solids varies with temperature and approaches zero at the temperature becomes very low.

 Adiabatic Processes Molecular Speeds Return to Ch21 ToC

(c) Doug Davis, 2002; all rights reserved