Coulomb’s Law of 20 Electrostatic Forces Charge Interactions Are Described by Coulomb’s Law Electrical interactions between charges govern much of physics, chemistry, and biology. They are the basis for chemical bonding, weak and strong. Salts dis- solve in water to form the solutions of charged ions that cover three-quarters of the Earth’s surface. Salt water forms the working fluid of living cells. pH and salts regulate the associations of proteins, DNA, cells, and colloids, and the conformations of biopolymers. Nervous systems would not function without ion fluxes. Electrostatic interactions are also important in batteries, corrosion, and electroplating. Charge interactions obey Coulomb’s law. When more than two charged par- ticles interact, the energies are sums of Coulombic interactions. To calculate such sums, we introduce the concepts of the electric field, Gauss’s law, and the electrostatic potential. With these tools, you can determine the electrostatic force exerted by one charged object on another, as when an ion interacts with a protein, DNA molecule, or membrane, or when a charged polymer changes conformation. Coulomb’s law was discovered in careful experiments by H Cavendish (1731–1810), J Priestley (1733–1804), and CA Coulomb (1736–1806) on macro- scopic objects such as magnets, glass rods, charged spheres, and silk cloths. Coulomb’s law applies to a wide range of size scales, including atoms, molecules, and biological cells. It states that the interaction energy u(r ) 385 between two charges in a vacuum is Cq q u(r ) = 1 2 , (20.1) r where q1 and q2 are the magnitudes of the two charges, r is the distance sep- arating them, and C = 1/4πε0 is a proportionality constant (see box below). Units The proportionality constant C in Equation (20.1) depends on the units used to measure the charge and the distance. In the older c.g.s. system, the units were defined so that C = 1. We use the SI system, in which the unit of charge is the coulomb C, the unit of energy is the joule J, and the unit of distance is the meter −1 m. The corresponding constant C equals (4πε0) . The factor ε0 is called the −12 −1 permittivity of the vacuum. In SI units, ε0 = 8.85 × 10 Fm . The farad F is the SI unit of capacitance, equal to 1 C V−1. The volt V is the SI unit of electri- −1 −12 2 −1 −1 cal potential, which is equal to 1 J C .Soε0 = 8.85 × 10 C J m . In c.g.s. units, the unit of charge is 1 statcoulomb or 1 esu = 3.00 × 109 C, the unit of potential is 1 statvolt = (1/300) V, and the unit of capacitance is 1 statfarad = 9 × 1011 F. In SI units, the charge on a proton is e = 1.60 × 10−19 C (with N e = 9.647 × − 104 C mol 1, where N is Avogadro’s number), and the charge on an elec- tron is −e =−1.60 × 10−19 C. A useful quantity for calculations is Ce2N= 2 −4 −1 e N /4πε0 = 1.386 × 10 J m mol . EXAMPLE 20.1 The Coulombic attraction between Na+ and Cl−. Compute the pair interaction u(r ) between a sodium ion and a chloride ion at r = 2.8Å, the bond length of NaCl, in a vacuum. Equation (20.1) gives −C 2N − . × −4 −1 u(r ) = e = 1 39 10 J m mol r (2.8 × 10−10 m) − − =−496 kJ mol 1 ≈−119 kcal mol 1. EXAMPLE 20.2 Coulombic interactions hold materials together. Coulomb’s law explains the attraction of positive for negative charges. Coulomb’s law also explains how even when charges have the same sign, they can cluster together into stable structures. Positive charges will cluster together if there is enough negative charge ‘glue’ holding them together. There are many examples: salt crystals have alternating positive and negative charges, molecules are constel- lations of positively charged atomic nuclei held together by negatively charged electron density in covalent bonds, and charge clustering is common among col- loids and biological complexes. Electrostatic bonding can be understood using the following simple model. 386 Chapter 20. Coulomb’s Law of Electrostatic Forces Figure 20.1 shows an in-line arrangement of two +1 charges, each separated by a distance r from a charge of −q in the middle. The total Coulombic energy is the sum of the three pairwise interactions: r r +1 −q +1 q q 1 u = Ce2 − − + . (20.2) r r 2r Figure 20.1 Two charges This arrangement of charges is stable when u<0, which happens when there of +1 are held together by is sufficient negative charge in the middle, −q<−0.25. This model illustrates electrostatic attraction if how charged objects can bond together. However, this model is not quantita- there is a sufficient negative charge −q in the middle. tively accurate for capturing covalent chemical bonding in molecules, where positive nuclei are held together by negative electron density. For that, you need quantum mechanics, which gives different mathematical forms for u(r ), including steric repulsions as r → 0. Charge Interactions Are Long-Ranged In previous chapters, we considered only the interactions between uncharged atoms. Uncharged atoms interact with each other through short-ranged inter- actions. That is, attractions between uncharged atoms are felt only when the two atoms are so close to each other that they nearly contact. Short-ranged interactions typically diminish with distance as r −6. We will explore this in more detail in Chapter 24. For now, it suffices to note that charge interactions are very different: they are long-ranged. Coulombic interactions diminish with distance as r −1. Counting lattice contacts is no longer sufficient. The difference between short-ranged and long-ranged interactions is much more profound than it may seem. The mathematics and physics of long-ranged and short-ranged interactions are very different. A particle that interacts through short-ranged interactions feels only its nearest neighbors, so system energies can be computed by counting the nearest-neighbor contacts, as we did in previous chapters. But when interactions involve charges, more distant neighbors contribute to the energies, so we need more sophisticated methods of summing energies. Example 20.3 shows how long-ranged electrostatic interactions hold a salt crystal together. EXAMPLE 20.3 Why are sodium chloride crystals stable? Let’s assume that a crystal of sodium chloride is composed of hard spherical ions that inter- act solely through electrostatics. To compute the total energy, we sum all the pairwise Coulombic interaction energies of one ion with every other ion in the crystal. Crystalline NaCl is arranged in a simple cubic lattice, with interionic spacings a = 2.81Åat25◦C. First sum the interactions of a sodium ion in row 1 shown in Figure 20.2 with all the other ions in the same row. Because there are two negative ions at distance a with unit charge −e, two positive ions at distance 2a with unit charge e, and two more negative ions at distance 3a, etc., the total Coulombic Charge Interactions Are Long-Ranged 387 Figure 20.2 Crystalline sodium chloride packs in a Adjacent Row cubic array of alternating sodium and chloride ions. Na Cl Row 1 Adjacent Row interaction along the row urow 1 is given by the series C 2 u = e −2 + 2 − 2 + 2 −··· row 1 a 1 2 3 4 2Ce2 1 1 1 = −1+ − + −··· . (20.3) a 2 3 4 This series is slow to converge, implying that two charges can have substantial interaction even when they are far apart. The series converges to − ln 2 per row (see Appendix J, Equation (J.4) with a = x = 1), so the energy of interaction of one sodium ion with all other ions in the same row is 2Ce2 1.386Ce2 u =− ln 2 =− . (20.4) row 1 a a Next, sum the interactions of the same sodium ion with all the ions in the four adjacent rows above, below, in front of, and behind row 1. In these four rows, the closest ions are the√ four negative charges at distance a, the eight positive√ charges at distance a 2, and the eight negative charges at dis- tance a 5. The pattern of pairwise distances gives a jth term involving eight charges at distance a(1+j2)1/2. These four adjacent rows contribute energies ufour adjacent rows: C 2 4 e √2 √2 √2 ufour adjacent rows = −1+ − + −··· . (20.5) a 2 5 10 When the pairwise energies are summed over all the rows of charges, the total Coulombic energy of the interaction between one ion and all the others in the crystal is found [1] to be Ce2N U =−1.747 a (− . )( . × −4 −1) = 1 747 1 386 10 J m mol 2.81 × 10−10 m − − =−862 kJ mol 1 =−206 kcal mol 1. (20.6) The energy holding each positive charge in the crystal is U/2, where the fac- tor of 2 corrects for the double-counting of interactions. To remove a NaCl + − − molecule, both a Na ion and a Cl ion, the energy is U = 206 kcal mol 1, according to this calculation. 388 Chapter 20. Coulomb’s Law of Electrostatic Forces The experimentally determined energy of vaporization (complete ion- − − ization) of crystalline NaCl is 183 kcal mol 1. The 23 kcal mol 1 discrepancy between the value we calculated and the experimental value is due mainly to the assumption that the ions are hard incompressible particles constrained to a separation of exactly a = 2.81 Å.