Page 88 - Physical Chemistry

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between molecular centers being only slightly greater than the molecular diameter. Section 2.11
Here, intermolecular forces contribute very substantially to U. In a liquid, the molec- The Molecular Nature
of Internal Energy
ular translational, rotational, and vibrational energies are, to a good approximation
(Sec. 21.11), the same as in a gas at the same temperature. We can therefore find
U intermol in a liquid by measuring U when the liquid vaporizes to a low-pressure gas.
For common liquids, U for vaporization typically lies in the range 3 to 15 kcal/mol,
m
indicating U intermol,m values of 3000 to 15000 cal/mol, far greater in magnitude
than U intermol,m in gases and U tr,m in room-temperature liquids and gases.
Discussion of U in solids is complicated by the fact that there are several kinds of
solids (see Sec. 23.3). Here, we consider only molecular solids, those in which the
structural units are individual molecules, these molecules being held together by in-
termolecular forces. In solids, the molecules generally don’t undergo translation or
rotation, and the translational and rotational energies found in gases and liquids are
absent. Vibrations within the individual molecules contribute to the internal energy. In
addition, there is the contribution U intermol of intermolecular interactions to the internal
energy. Intermolecular interactions produce a potential-energy well (similar to that in
Fig. 21.21a) within which each entire molecule as a unit undergoes a vibrationlike
motion that involves both kinetic and potential energies. Estimates of U intermol,m from
heats of sublimation of solids to vapors indicate that for molecular crystals, U intermol,m
is in the same range as for liquids.
For a gas or liquid, the molar internal energy is
U U tr,m U rot,m U vib,m U el,m U intermol,m U rest,m
m
where U rest,m is the molar rest-mass energy of the electrons and nuclei, and is a con-
stant. Provided no chemical reactions occur and the temperature is not extremely high,
U el,m is a constant. U intermol,m is a function of T and P. U tr,m , U rot,m , and U vib,m are func-
tions of T.
3
3
For a perfect gas, U intermol,m 0. The use of U tr,m RT, U rot,nonlin,m RT, and
2
2
U rot,lin,m RT gives
3
3
U RT RT 1or RT2 U vib,m 1T2 const. perf. gas (2.88)
m
2
2
For monatomic gases (for example, He, Ne, Ar), U rot,m 0 U vib,m , so
3
U RT const. perf. monatomic gas (2.89)
2
m
The use of C V,m ( U / T) and C P,m C V,m R gives
V
m
5
3
C V,m R, C P,m R perf. monatomic gas (2.90)
2
2
provided T is not extremely high.
For polyatomic gases, the translational contribution to C V,m is C V,tr,m 3 2 R; the
rotational contribution is C R, C 3 R (provided T is not extremely
V,rot,lin,m V,rot,nonlin,m 2
low); C is a complicated function of T—for light diatomic molecules, C is
V,vib,m V,vib,m
negligible at room temperature.
Figure 2.15 plots C at 1 atm versus T for several substances. Note that C 5 R
P,m P,m 2
5 cal/(mol K) for He gas between 50 and 1000 K. For H O gas, C starts at 4R
2 P,m
8 cal/(mol K) at 373 K and increases as T increases. C 4R means C 3R. The
P,m V,m
value 3R for this nonlinear molecule comes from C C 3 R 3 R. The
V,tr,m V,rot,m 2 2
increase above 3R as T increases is due to the contribution from C as excited
V,vib,m
vibrational levels become populated.
The high value of C of liquid water compared with that for water vapor results
P,m
from the contribution of intermolecular interactions to U. Usually C for a liquid is
P
substantially greater than that for the corresponding vapor.
The theory of heat capacities of solids will be discussed in Sec. 23.12. For all
solids, C goes to zero as T goes to zero.
P,m