Molecular Interaction

interaction between electrically neutral molecules or atoms; it determines the existence of liquids and molecular crystals and the difference between real and ideal gases and is manifested in various physical phenomena. Molecular interaction depends on the distance r between the molecules and usually is described by the potential energy of interaction U(r), the intermolecular potential, since it is precisely the average interaction potential that determines the state of matter and many of its properties.

Molecular interaction was first taken into consideration by J. D. van der Waals (1873) in explaining the properties of real gases and liquids. He proposed that at small intermolecular distances r there exist forces of repulsion, which with increasing distance are replaced by forces of attraction. On the basis of these concepts, without even considering the quantitative dependence of molecular interaction on distance, he derived the van der Waals equation for the state of a real gas.

Molecular interaction is electrical in nature and consists of forces of attractive forces (orientation, induction, and dispersion forces) and repulsive forces.

Orientation forces act between polar molecules (that is, molecules that have electric dipole moments). The attractive force between two polar molecules is a maximum when their dipole moments lie in the same line (Figure 1). This force arises because the distance between unlike charges is slightly smaller than the distance between like charges. Consequently, the attraction of the dipoles exceeds their repulsion. The interaction of dipoles depends on their relative orientation; therefore, the forces of dipole interaction are called orientation forces. Random thermal motion continually changes the orientation of polar molecules, but calculations show that the average value of the force for all possible orientations has a definite nonzero value. The potential energy of the orientation interaction is U_or(r) ˜ p₁p₂/r⁶, where p₁ and p₂ are the dipole moments of the interacting molecules. The corresponding force of interaction is F_or ˜ r⁷. The force F_or decreases with distance much more quickly than does the coulomb force of interaction of charged particles (F_coul ˜ r^-2).

Figure 1. The two electric dipoles ab and cd attract each other in this relative position, since the opposite charges at points b and c interact more strongly than do the like charges at points a and c (and also at b and d)

Induction (or polarization) forces act between polar and non-polar molecules. The polar molecule creates an electric field that polarizes the nonpolar molecule, whose electric charges are symmetrically distributed throughout the volume. Positive charges are displaced in the direction of the electric field, and negative charges are displaced in the opposite direction. As a result, a dipole moment is induced in the nonpolar molecule.

In this case the energy of molecular interaction is proportional to the dipole moment pi of the polar molecule and the polarizability a 2, which characterizes the ease with which the other molecule can be polarized: U_ind(r) ˜ P₁α₂/r⁶. This energy is called the induction energy, since it appears as a result of the molecular polarization induced by electrostatic induction. Induction forces (F_ind ˜ r^—7) also act between polar molecules.

Dispersion molecular interaction acts between nonpolar molecules. The nature of this interaction was fully elucidated only after the development of quantum mechanics. In atoms and molecules the electrons move around the nuclei in an intricate manner. The time average of the dipole moment of a nonpolar molecule is found to be zero, but at each instant the electrons occupy a certain position. Therefore, the instantaneous value of the dipole moment (for example, that of a hydrogen atom) is nonzero. The instantaneous dipole creates an electric field that polarizes the neighboring molecules. Interaction of the instantaneous dipoles occurs as a result. The energy of interaction of nonpolar molecules is the average result of the interaction of all possible instantaneous dipoles with the dipole moments that they induce in neighboring molecules. The potential energy of the dispersion molecular interaction is U_dis(r) ˜ α₁α₂, and F_dis ˜ r^-7 (here α₁ and α₂ are the polarizabilities of the interacting molecules). Interaction of this type is called dispersion interaction because it is determined by the same molecular properties as is the dispersion of light in matter. Dispersion forces act between all atoms and molecules, since the mechanism by which they appear is independent of whether the molecules (or atoms) have permanent dipole moments. These forces are usually greater in magnitude than orientation or induction forces. The quantity F_or is greater than F_dis only for molecules with large dipole moments—for example, water molecules, for which the difference is a factor of 3. However, for the interaction of such polar molecules as CO, HI, and HBr the dispersion forces exceed all. other forces by a factor of tens or hundreds. It is significant that all three types of molecular interaction decrease in the same way with distance:

U = U_or + U_ind + U_dis ˜ r^-6

Forces of repulsion act between molecules at very small distances, where the filled electron shells of the atoms making up the molecules come into contact. The Pauli exclusion principle in quantum mechanics prohibits the interpenetration of filled electron shells. The repulsive forces that arise here are more dependent on the molecular types than are the attractive forces. The assumption that the potential energy of the repulsive forces U_rep increases with decreasing distance according to the law U_rep (r) ˜ r^-12, and F_rep ˜ r^-13, leads to good agreement with experimental data.

If we assume that U(r) = 0 as r — ∞ and that the energy of attraction varies with distance as r^-6 and the energy of repulsion as r^-12, then the U(r) curve will have the shape depicted in Figure 2. The distance at which the forces of interaction of the molecules are equal to zero corresponds to the minimum of the potential energy curve.

Figure 2. Dependence of the Lennard-Jones potential U(r) of molecular interaction on the distance r between the molecules. The distance r = σ is the shortest possible distance between the molecules at rest, and ε is the depth of the potential well (the cohesion energy of the molecular pair).

It is impossible in practice to calculate U(r) with high accuracy on the basis of quantum mechanics for the great diversity of possible pairs of interacting molecules. Neither has it yet been possible to measure experimentally the force of interaction at intermolecular distances. Therefore, a formula for U(r) is usually selected in such a way that the calculations made with it are in good accord with experiment. The formula

U(r) = 4ε[(σ/r)¹² - (σ/r)⁶

called the Lennard-Jones potential, is most often used. The quantities cr and € in the formula are determined experimentally on the basis of the dependence on cr and € of the properties of the substance (for example, the diffusion coefficient, thermal conductivity, or viscosity).