Page 49 - Physical chemistry eng

26 CHAPTER 2 Heat, Work, Internal Energy, Enthalpy, and the First Law of Thermodynamics

translation, rotation, vibration, and electronic excitation. Each of the degrees of free-
TABLE 2.2 Energy Level
dom has its own set of energy levels and the probability of an individual molecule
Spacings for Different Degrees
occupying a higher energy level increases as it gains energy. Except for translation, the
of Freedom
energy levels for atoms and molecules are independent of the container size.
Degree of Energy Level The amount of energy needed to move up the ladder of energy levels is very differ-
Freedom Spacing ent for the different degrees of freedom: ¢E electronic W¢E vibration 77¢E rotation 77
¢E translation . Values for these ¢E are molecule dependent, but order of magnitude num-
Electronic 5 * 10 -19 J
bers are shown in Table 2.2.
Vibration 2 * 10 -20 J Energy is gained or lost by a molecule through collisions with other molecules. An
Rotation 2 * 10 -23 J order of magnitude estimate of the energy that can be gained or lost by a molecule in a
B
Translation 2 * 10 -41 J collision is k T , where k = R>N A is the Boltzmann constant, and T is the absolute
temperature. A given degree of freedom in a molecule can only take up energy through
molecular collisions if the spacing between adjacent energy levels and the temperature
satisfies the relationship ¢E L k T , which has the value 4.1 * 10 -21 J at 300 K. At
B
300 K, ¢E L k T is always satisfied for translation and rotation, but not for vibration
B
and electronic excitation. We formulate the following general rule relating the heat
capacity C V,m and the degrees of freedom in a molecule, which will be discussed in
more detail in Chapter 32.

The heat capacity C V,m for a gas at temperature T not much lower than 300 K is
R>2 for each translation and rotational degree of freedom, where R is the ideal
gas constant. Each vibrational degree of freedom for which the relation
¢E>kT 6 0.1 is obeyed contributes R to C V,m . If ¢E>k T 7 10 , the degree of
B
freedom does not contribute to C V,m . For 10 7¢E>k T 7 0.1 , the degree of
B
freedom contributes partially to C V,m .

Figure 2.8 shows the variation of C V,m for a monatomic gas and several molecu-
lar gases. Atoms only have three translational degrees of freedom. Linear molecules
have an additional 2 rotational degrees of freedom and 3n-5 vibrational degrees of
freedom where n is the number of atoms in the molecule. Nonlinear molecules have
80
C H 3 translational degrees of freedom, 3 rotational degrees of freedom, and 3n-6 vibra-
2 4
tional degrees of freedom. A He atom has only 3 translational degrees of freedom,
70
and all electronic transitions are of high energy compared to kT. Therefore,
C = 3R>2 over the entire temperature range as shown in the figure. CO is a lin-
60 ear diatomic molecule that has two rotational degrees of freedom for which
V,m
C V, m /J K 1 mol 1 50 CO 2 degree of freedom begins to contribute to C B V,m below 1000. K. CO 2 has 4 vibrational
¢E>k T 6 0.1
at 200. K. The single vibrational
at 200. K. Therefore, C
= 5R>2
V,m
B
above 200. K, but does not contribute
fully for T 6 1000. K
40
because 10 7¢E>k T
30 degrees of freedom, some of which contribute to C V,m near 200. K. However, C V,m
does not reach its maximum value of 13R>2
below 1000. K. Similarly, C
for
V,m
C H , which has 12 vibrational degrees of freedom, does not reach its maximum
2 4
20 CO
value of 15 R below 1000. K, because 10 7¢E>k T for some vibrational degrees
B
of freedom. Electronic energy levels are too far apart for any of the molecular gases
10 He
to give a contribution to C V,m .
To this point, we have only discussed C V,m for gases. It is easier to measure C P,m
than C for liquids and solids because liquids and solids generally expand with
0 200 400 600 800 1000 V,m
increasing temperature and exert enormous pressure on a container at constant vol-
Temperature/K
ume (see Example Problem 3.2.) An example of how C P,m depends on T for solids
FIGURE 2.8 and liquids is illustrated in Figure 2.9 for Cl 2 . To make the functional form of C P,m (T)
Molar heat capacities C V,m are shown for
understandable, we briefly discuss the relative magnitudes of C P,m in the solid, liquid,
a number of gases. Atoms have only
and gaseous phases using a molecular level model. A solid can be thought of as a set
translational degrees of freedom and,
of interconnected harmonic oscillators, and heat uptake leads to the excitations of the
therefore, have comparatively low values
collective vibrations of the solid. At very low temperatures these vibrations cannot be
for C V,m that are independent of tempera-
activated, because the spacing of the vibrational energy levels is large compared to
ture. Molecules with vibrational degrees
of freedom have higher values of C V,m at k T . As a consequence, energy cannot be taken up by the solid. Hence, C P,m
B
temperatures sufficiently high to activate approaches zero as T approaches zero. For the solid, C P,m rises rapidly with T because
the vibrations. the thermal energy available as T increases is sufficient to activate the vibrations of