Kinetic Theory of Gases

The Equipartition of Energy For monatomic gases, there is

excellent agreementbetween experimental measurements and theoretical predictions of C_{P}, C_{V}, and C_{P}/C_{V}.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

equallyby 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 isfully specifiedby 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

diatomicmolecule. Three coordinates specify its center of mass.Three additionalcoordinates specify its orientation in space or its rotation about the center of mass. However, forpoint masses, the moment of inertia about the symmetry axis iszero. This means there is no kinetic energy for rotation about this axis. That effectively reduces our degrees of freedom from six tofive.By the equipartition theorem, we expect each "degree of freedom" to contribute

( ^{1}/_{2}) N k Tor

( ^{1}/_{2}) n R Tto the total internal energy. Therefore, we expect

U = ( ^{5}/_{2}) N k Tor

U = ( ^{5}/_{2}) n R Tfor a diatomic molecule. With U = (

^{5}/_{2}) n R T, we havedU = n C _{V}dTThat means

C _{P}= C_{V}+ R = (^{7}/_{2}) R= C

_{P}/C_{V}= [(^{7}/_{2})R]/[(^{5}/_{2})R] =^{7}/_{5}= 1.40And this, too, is in very

good agreementwith experimental values.And

two additionalcoordinates specify such a diatomic molecule's contraction or elongation -- its state ofvibration. That would give usC _{V}= (^{7}/_{2}) Rand

= C _{P}/C_{V}= [(^{9}/_{2})R]/[(^{7}/_{2})R] =^{9}/_{7}= 1.28Why is the experimental agreement better with

outconsidering the vibration?In particular, look at the molar specific heat, C

_{V}, for molecular hydrogen (H_{2}) 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 H_{2}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.Brieflythese 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

solidsvaries with temperature and approaches zero at the temperature becomes very low.

Return to Ch21 ToC(c) Doug Davis, 2002; all rights reserved