Estimate the cohesive energy in aJ per \(\mathrm{NaCl}\) 'molecule' in a crystal, assuming that the cobesive forces are entirely Coulombic and that the interionic distance is \(0.28 \mathrm{~nm}\). (The cohesive energy is the difference between the energy of the crystal and that of the separated ions.)

Coulomb's law is a fundamental principle that describes the force between two point charges. According to this law, the force (\f\(F\f\)) between two charges (\f\(q_1\f\) and \f\(q_2\f\)) is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance (\f\(r\f\)) between them.

\f\(F = k \frac{q_1 \times q_2}{r^2}\f\)

Here, \f\(k\f\) is Coulomb's constant and has a value of approximately \f\(8.99 \times 10^9 \text{N}\text{m}^2\text{C}^{-2}\f\). When dealing with the cohesive energy of a crystal like sodium chloride (NaCl), we consider the electrostatic forces between the oppositely charged ions. The charges are usually multiples of the elementary charge, which is the charge of a single proton (or the magnitude of the charge of a single electron). For Na+ and Cl-, both have a charge equivalent to one elementary charge, \f\(1.60218 \times 10^{-19} \text{C}\f\).

Electrostatic potential energy is the energy that a charge possesses due to its position in an electric field created by another charge. This concept is vital when analyzing the stability of ionic crystals such as NaCl. The potential energy (\f\(E\f\)) associated with a pair of charges can be calculated using the formula derived from Coulomb's law:

\f\(E = k \frac{q_1 \times q_2}{r}\f\)

The negative sign usually found in this formula indicates that the potential energy is lower (more stable) when two opposite charges are close to each other, compared to when they are infinitely distant. This makes sense when you consider that work would need to be done to separate the charges against the attractive force they exert on one another. When calculating cohesive energy, this potential energy value will be viewed as negative, reflecting the attraction between the Na+ and Cl- ions.

Interionic distance is the space between the centers of adjacent ions in a crystal lattice. It's a significant factor in determining the strength of the electrostatic forces and, consequently, the cohesive energy of the crystal. A smaller distance means stronger attraction, and thus more energy is needed to pull the ions apart.

In the context of NaCl crystals, where ions arrange in a regular pattern, the interionic distance is the distance between a sodium ion and its neighboring chloride ion. The mentioned distance of \f\(0.28\text{ nm}\f\) (or \f\(0.28 \times 10^{-9}\text{ m}\f\)) is about ten times the diameter of a single atom and is key to calculating cohesive forces using Coulomb's law. This distance is particularly important because the cohesive energy in a crystal is fundamentally the electrostatic potential energy per 'molecule' or formula unit, represented by the pair of ions.