Physical Chemistry I
Kinetic Molecular Theory of Gases and the Principle of Equipartition
Kinetic Molecular Theory of Gases
We have seen that the volume of a gas increases with temperature. But what happens on the atomic or molecular level as the temperature of a gas increases? Most of us have a sense that the velocity (or kinetic energy) of gas molecules increases at the temperatures increases. We can more carefully determine the connection between temperature and the velocity of gases by looking at the molecular dynamics of a gas which produce a pressure inside a container.
We take as our container the rectangular box shown below. The volume of the box is a x A where a is the size of the box along the x-axis and A is the area of the wall of the box that is perpendicular to the x-axis.
Pressure is defined as the force per unit area.
To calculate the pressure we need to determine the force exerted by gas molecules colliding with wall A. The force exerted by a molecule of mass m colliding with wall A can be calculated from
.
The last quantity in the above equation can be determined if we know the change in velocity per collision with wall A and the time between collisions with wall A. A collision with wall A will reverse only the x-component of the velocity . If we assign the average initial x-component of the velocity before collision as -vx and the final x-component of the velocity after collision to vx then the change in velocity with collision with wall A is
The time between collisions with wall A will again depend upon the x-component of the velocity and the distance travelled by the gas molecule (along x) between collisions. In our box a gas molecule, after colliding with wall A, would have to travel along x to the opposite wall, a distance of a, and back again to wall A, for a total distance travelled along x of 2a. Thus the time between collisions with wall A would be
and the force exerted by one gas molecule of mass m colliding with wall A becomes
Rearranging we find
pV = mvx2 per gas molecule.
Recognizing that the velocity is related to its components by the Pythagorean Theorem and that, on average, each of the components are equal we find:
v2 = vx2 + vy2 + vz2 = 3 vx2
and
pV = 1/3 mv2 per gas molecule,
or
pVm = 1/3 NA mv2 per mole of gas molecules,
where Vm is the volume occupied by a mole of gas molecules and NA is Avogadro's number (6.022 x 1023 molecules per mole).
Drawing on our results from the ideal gas temperature scale, i.e., pVm = RT, we can finally connect the velocity and kinetic energy of the gas molecules to the temperature.
pVm = 1/3 NA mv2 = RT and
The velocity, v, is the root mean square velocity. At room temperature (300 K) the velocity of nitrogen molecules ( m = 4.65 x 10-26 kg) is 519 meters/second. (*Note: k is called Boltzmann's constant and is related to the gas constant R such that k = R/NA = 1.38 x 10-23 joule per Kelvin per molecule.)
The last two equations for the energy of the gas molecules amount to what is called the equipartition principle. The gas that we have used in this description are monoatomic (single atom) and therefore have no internal motions such as rotation or vibration. The only motion that these molecules experience is translation as depicted in the box above. Each gas atom has three degrees of translational freedom, motion along x, y, or z. The average energy then per degree of freedom for the translating atomic gas is 1/2 kT per degree of freedom per gas atom or 1/2 RT per degree of freedom per mole of gas atoms. By the equipartition principle the total energy is equally distributed among the degrees of freedom.
For polyatomic molecules thermal energy will also be distributed among the rotations and vibrations of the molecule. In the same way that translating molecules could move along x, y, or z, so too can each of the atoms in a molecule. Thus molecules have a total of 3N degrees of freedom, where N is the number of atoms in the molecule. Of the total 3N degrees of freedom only 3 will be translations of the whole molecule through space. The remainder are internal degrees of freedom: vibrations and rotations. Non-linear polyatomic molecules have three degrees of rotational freedom while linear polyatomic molecules have only two rotational degrees of freedom. Rotation of a linear molecule along its molecular axis does not consume thermal energy (It's easy to roll a pencil). Each rotation is allotted 1/2 kT per rotation (or 1/2 RT per mole of rotations) according to the equipartition principle. A mole of water molecules (water is a non-linear molecule), for example, has 3 rotations and 3/2 RT of rotational energy according to the equipartition principle. The water rotations are shown below.
Equipartition of energy among vibrations is similar to that for translations and rotations except that thermal energy may go into potential energy, i.e., into the stiffness of the hypothetical spring connecting vibrating atoms, or into kinetic energy, the frequency of the vibration. Each of these vibrational degrees of freedom obtains 1/2 kT according to the equipartition principle or a full kT per vibration per molecule (a full RT per vibration per mole of molecules). Non-linear molecules have 3N - 6 vibrations, while linear molecules have 3N - 5 vibrations. Water, for example, has 3(3) - 6 = 3 vibrations. The vibrations of water are shown below.
According to the equipartition principle the total energy of a mole of water vapor is
Utotal = Utrans + Urot + Uvib = 3/2 RT + 3/2 RT + 3RT = 6RT.
It is important to recognize that the equipartition principle is a classical idea that fails to correctly account for the true quantum energies of molecules, with particularly poor applicability to vibrations.
