Section 12–3 Molecular forces

(Attractive force / Repulsive force / Proportional to the displacement)

In this section, Feynman discusses how molecular forces become attractive and repulsive as well as how molecular forces are proportional to the displacement.

1. Attractive force:

“For all nonpolar molecules, in which all the electrical forces are neutralized, it nevertheless turns out that the force at very large distances is an attraction and varies inversely as the seventh power of the distance… (Feynman et al., 1963, section 12–3 Molecular forces).”

Feynman states that molecular forces are forces between the atoms and are the origin of frictional forces. In a water molecule, the negative charges are located nearer to the oxygen, but the mean positions of the negative charges and of the positive charges are not at the same point; thus, a molecule nearby feels a dipole-dipole force. The molecular force at large distances is attractive and varies inversely as the seventh power of the distance, F = -k/r⁷, where k is a constant depending on the molecules. The molecular forces (or Van der Waals forces) can be classified as three types: (1) Keesom interaction (permanent dipole-permanent dipole interaction), (2) Debye interaction (permanent dipole-induced dipole interaction), and (3) London interaction (induced dipole-induced dipole interaction). Molecular forces can be demonstrated by a friction experiment using a sliding glass tumbler or Johansson blocks.

When Feynman was about twenty-one years old, he published his undergraduate thesis at MIT in The Physical Review. In his own words, “Van der Waals’ forces can also be interpreted as arising from charge distributions with higher concentration between the nuclei. The Schrodinger perturbation theory for two interacting atoms at a separation R, large compared to the radii of the atoms, leads to the result that the charge distribution of each is distorted from central symmetry, a dipole moment of order 1/R⁷ being induced in each atom. The negative charge distribution of each atom has its center of gravity moved slightly toward the other. It is not the interaction of these dipoles which leads to van der Waals' force, but rather the attraction of each nucleus for the distorted charge distribution of its own electrons that gives the attractive 1/R⁷ force… (Feynman, 1939, p. 343).” In 1936, Hans Hellman derived a molecular force theorem and it is now known as Hellman-Feynman theorem.

2. Repulsive force:

“When atoms or molecules get too close they repel with a very large repulsion; that is what keeps us from falling through the floor! (Feynman et al., 1963, section 12–3 Molecular forces).”

By using a graph as shown in Fig. 12–2, it illustrates that the molecular forces attract at long distances and repel at short distances. This indicates that all the atoms are held together by their attractions to form solids, but held apart by their repulsions that set in when they are too close together. More important, when atoms or molecules are closer together, they repel with a very large repulsion. This explains what keeps us from falling through the floor. Feynman elaborates that at a certain distance d, the forces are zero where the graph crosses the axis. It means that the molecular forces are all balanced such that the molecules stay at that distance apart from one another. However, Feynman did not include the r⁻¹² term, which represents the repulsive molecular force.

Some teachers explain that the r⁻¹² term, which is the repulsive term, is due to Pauli repulsion at short ranges as a result of overlapping electron orbitals. In Statistical Mechanics, Huang (1987) writes that “[t]he attractive part of the potential energy originates from the mutual electric polarization of the two molecules and the repulsive part from the Coulomb repulsion of the overlapping electronic clouds of the molecules (p. 38).” Feynman did not use Pauli exclusion principle to elaborate the repulsive forces possibly because he was unable to find a simple way of explaining Pauli’s principle at an elementary level. In his words, “[t]his probably means that we do not have a complete understanding of the fundamental principle involved (Feynman et al., 1966, Section 4–1 Bose particles and Fermi particles).”

3. Proportional to the displacement:

“Therefore, in many circumstances, if the displacement is not too great the force is proportional to the displacement (Feynman et al., 1963, section 12–3 Molecular forces).”

If molecules are separated by a very short distance closer or farther than their equilibrium positions, then the force is proportional to the displacement. This principle is known as Hooke’s law of elasticity, which specifies that the restoring force of a body is proportional to the extension of the body. The validity of Hooke’s law is dependent on the materials; for instance, the force on dough or putty is quite small, but the force on a piece of steel is relatively large. Feynman explains that Hooke’s law can be experimentally demonstrated with a vertically suspended coil spring that is long and made of steel. An advantage of using a long coil spring allows us to measure the extension of the spring that is relatively short. Furthermore, the length of the spring that is made of steel may increase the total force to extend the spring due to its weight.

An idealization of a vertically suspended spring is that it has no mass. If the ideal spring is suspended horizontally, we may assume that it slides on a frictionless horizontal surface. Essentially, Hooke's law is a first-order linear approximation to Hookean materials that are not stretched beyond its elastic limit. On the contrary, rubber is a non-Hookean material because its elasticity is stress dependent and sensitive to temperature. In Volume II of Feynman Lectures, he briefly clarifies that: “the general theory of elasticity, the atomic machinery that determine the elastic properties, and finally the limitations of elastic laws when the forces become so great that plastic flow and fracture occur (Feynman et al., 1964, section 38-1 Hooke’s law).”

Questions for discussion:

1. How would you explain that molecular forces are attractive?

2. How would you explain that molecular forces are repulsive?

3. How would you explain that molecular forces are proportional to the displacement?

The moral of the lesson: the molecular force at large distances is attractive and varies inversely as the seventh power of the distance, F = -k/r⁷, but it is repulsive at short distances; it is proportional to the displacement near the equilibrium position.

References:

1. Feynman, R. P. (1939). Forces in molecules. Physical Review, 56(4), 340-343.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley. 

4. Feynman, R. P., Leighton, R. B., & Sands, M. (1966). The Feynman lectures on physics Vol III: Quantum Mechanics. Reading, MA: Addison-Wesley.

5. Huang, K. (1987). Statistical Mechanics (Second Edition). New York: John Wiley & Sons.