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2.11 THE MOLECULAR NATURE OF INTERNAL ENERGY Section 2.11
The Molecular Nature
Internal energy is energy at the molecular level. The molecular description of internal of Internal Energy
energy is outside the scope of thermodynamics, but a qualitative understanding of
molecular energies is helpful.
Consider first a gas. The molecules are moving through space. A molecule has a
1
2
translational kinetic energy mv , where m and v are the mass and speed of the mole-
2
cule. A translation is a motion in which every point of the body moves the same dis-
tance in the same direction. We shall later use statistical mechanics to show that the
total molecular translational kinetic energy U of one mole of a gas is directly pro-
tr,m
3
portional to the absolute temperature and is given by [Eq. (14.14)] U RT, where
tr,m 2
R is the gas constant.
If each gas molecule has more than one atom, then the molecules undergo rota-
tional and vibrational motions in addition to translation. A rotation is a motion in
which the spatial orientation of the body changes, but the distances between all points
in the body remain fixed and the center of mass of the body does not move (so that
there is no translational motion). In Chapter 21, we shall use statistical mechanics to
show that except at very low temperatures the energy of molecular rotation U rot,m
3
in one mole of gas is RT for linear molecules and RT for nonlinear molecules
2
3
[Eq. (21.112)]: U RT; U RT.
rot,lin,m rot,nonlin,m 2
Besides translational and rotational energies, the atoms in a molecule have vibra-
tional energy. In a molecular vibration, the atoms oscillate about their equilibrium po-
sitions in the molecule. A molecule has various characteristic ways of vibrating, each
way being called a vibrational normal mode (see, for example, Figs. 20.26 and 20.27).
Quantum mechanics shows that the lowest possible vibrational energy is not zero but
is equal to a certain quantity called the molecular zero-point vibrational energy
(so-called because it is present even at absolute zero temperature). The vibrational
energy contribution U to the internal energy of a gas is a complicated function of
vib
temperature [Eq. (21.113)]. For most light diatomic (two-atom) molecules (for ex-
ample, H ,N ,HF, CO) at low and moderate temperatures (up to several hundred
2 2
kelvins), the average molecular vibrational energy remains nearly fixed at the zero-
point energy as the temperature increases. For polyatomic molecules (especially
those with five or more atoms) and for heavy diatomic molecules (for example, I )
2
at room temperature, the molecules usually have significant amounts of vibrational
energy above the zero-point energy.
Figure 2.14 shows translational, rotational, and vibrational motions in CO .
2
In classical mechanics, energy has a continuous range of possible values. Quantum
mechanics (Chapter 17) shows that the possible energies of a molecule are restricted to
O C O
certain values called the energy levels. Forexample, the possible rotational-energy val-
ues of a diatomic molecule are J(J 1)b [Eq. (17.81)], where b is a constant for a given A translation
molecule and J can have the values 0, 1, 2, etc. One finds (Sec. 21.5) that there is a dis-
tribution of molecules over the possible energy levels. For example, for CO gas at 298 K,
0.93% of the molecules are in the J 0level, 2.7% are in the J 1level, 4.4% are in O C O
the J 2level,..., 3.1% are in the J 15 level,....Asthetemperature increases,
more molecules are found in higher energy levels, the average molecular energy A rotation
increases, and the thermodynamic internal energy and enthalpy increase (Fig. 5.11).
Besides translational, rotational, and vibrational energies, a molecule possesses
electronic energy e (epsilon el). We define this energy as e e e , where O C O
el el eq q
e is the energy of the molecule with the nuclei at rest (no translation, rotation, or
eq A vibration
vibration) at positions corresponding to the equilibrium molecular geometry, and e
q
is the energy when all the nuclei and electrons are at rest at positions infinitely far Figure 2.14
apart from one another, so as to make the electrical interactions between all the
Kinds of motions in the CO 2
charged particles vanish. (The quantity e is given by the special theory of relativity molecule.
q