Page 87 - Physical Chemistry

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Chapter 2 as the sum of the rest-mass energies m c for the electrons and nuclei.) For a sta-
rest
The First Law of Thermodynamics ble molecule, e is less than e .
eq
q
The electronic energy e can be changed by exciting a molecule to a higher elec-
el
tronic energy level. Nearly all common molecules have a very large gap between the
lowest electronic energy level and higher electronic levels, so at temperatures below,
say, 5000 K, virtually all the molecules are in the lowest electronic level and the con-
tribution of electronic energy to the internal energy remains constant as the tempera-
ture increases (provided no chemical reactions occur).
In a chemical reaction, the electronic energies of the product molecules differ
from those of the reactant molecules, and a chemical reaction changes the thermody-
namic internal energy U primarily by changing the electronic energy. Although the
other kinds of molecular energy generally also change in a reaction, the electronic
energy undergoes the greatest change.
Besides translational, rotational, vibrational, and electronic energies, the gas
molecules possess energy due to attractions and repulsions between them (intermo-
lecular forces); intermolecular attractions cause gases to liquefy. The nature of inter-
molecular forces will be discussed in Sec. 21.10. Here, we shall just quote some key
results for forces between neutral molecules.
The force between two molecules depends on the orientation of one molecule rel-
ative to the other. For simplicity, one often ignores this orientation effect and uses a
force averaged over different orientations so that it is a function solely of the distance
r between the centers of the interacting molecules. Figure 21.21a shows the typical be-
havior of the potential energy v of interaction between two molecules as a function of
r; the quantity s (sigma) is the average diameter of the two molecules. Note that, when
1
the intermolecular distance r is greater than 2 or 3 times the molecular diameter s,
2
the intermolecular potential energy v is negligible. Intermolecular forces are gener-
ally short-range. When r decreases below 3s, the potential energy decreases at first,
indicating an attraction between the molecules, and then rapidly increases when r
becomes close to s, indicating a strong repulsion. Molecules initially attract each
other as they approach and then repel each other when they collide. The magnitude
of intermolecular attractions increases as the size of the molecules increases, and it
increases as the molecular dipole moments increase.
The average distance between centers of molecules in a gas at 1 atm and 25°C is
about 35 Å (Prob. 2.55), where the angstrom (Å) is
8
1 Å 10 cm 10 10 m 0.1 nm (2.87)*
Typical diameters of reasonably small molecules are 3 to 6 Å [see (15.26)]. The aver-
age distance between gas molecules at 1 atm and 25°C is 6 to 12 times the molecular
diameter. Since intermolecular forces are negligible for separations beyond 3 times the
molecular diameter, the intermolecular forces in a gas at 1 atm and 25°C are quite
small and make very little contribution to the internal energy U. Of course, the spatial
distribution of gas molecules is not actually uniform, and even at 1 atm significant
numbers of molecules are quite close together, so intermolecular forces contribute
slightly to U. At 40 atm and 25°C, the average distance between gas molecules is only
10 Å, and intermolecular forces contribute substantially to U.
Let U be the contribution of intermolecular interactions to U . U
intermol,m m intermol,m
differs for different gases, depending on the strength of the intermolecular forces.
Problem 4.22 shows that, for a gas, U is typically 1 to 10 cal/mol at 1 atm
intermol,m
and 25°C, and 40 to 400 cal/mol at 40 atm and 25°C. (U is negative because
intermol
intermolecular attractions lower the internal energy.) These numbers may be com-
3
pared with the 25°C value U RT 900 cal/mol.
tr,m 2
The fact that it is very hard to compress liquids and solids tells us that in con-
densed phases the molecules are quite close to one another, with the average distance