For \(\mathrm{a} \mathrm{K}^{+}-\mathrm{Cl}^{-}\)ion pair, attractive and repulsive energies \(E_{A}\) and \(E_{R}\), respectively, depend on the distance between the ions \(r\), according to $$ \begin{aligned} E_{A} &=-\frac{1.436}{r} \\ E_{R} &=\frac{5.86 \times 10^{-6}}{r^{9}} \end{aligned} $$ For these expressions, energies are expressed in electron volts per \(\mathrm{K}^{+}-\mathrm{Cl}^{-}\)pair, and \(r\) is the distance in nanometers. The net energy \(E_{N}\) is just the sum of the preceding two expressions. (a) Superimpose on a single plot \(E_{N}, E_{R}\), and \(E_{A}\) versus \(r\) up to \(1.0 \mathrm{~nm}\). (b) On the basis of this plot, determine (i) the equilibrium spacing \(r_{0}\) between the \(\mathrm{K}^{+}\)and \(\mathrm{Cl}^{-}\)ions, and (ii) the magnitude of the bonding energy \(E_{0}\) between the two ions. (c) Mathematically determine the \(r_{0}\) and \(E_{0}\) values using the solutions to Problem \(2.14\) and compare these with the graphical results from part (b).

Equilibrium spacing, often represented as \( r_0 \), is a significant concept in the study of ionic bonds. In simple terms, it is the optimal distance between two oppositely charged ions at which the energies of attraction and repulsion balance each other out. At this point, the system achieves maximum stability and minimum potential energy.

When discussing the ionic bond between a cation, such as potassium (K+), and an anion, such as chloride (Cl-), the equilibrium spacing can be visualized on a graph where net energy is plotted against the distance between ions. The curve for net energy will typically have a distinct minimum point. This lowest point on the graph indicates the equilibrium spacing \( r_0 \), signifying the most stable arrangement of the ions within the compound.

In calculations and exercises like the given example, finding the equilibrium spacing involves determining where the derivative of the net energy with respect to the inter-ionic distance equals zero. This mathematical approach complements the graphical method and often serves to verify the accuracy of the plotted data.

When ions form a bond, there are two primary forces at play: attractive and repulsive energies. Attractive energies arise from the electrostatic force pulling ions with opposite charges together, such as the potassium cation and chloride anion. This energy is inversely proportional to the distance between the ions, meaning that as the ions draw closer, the attraction strengthens.

Conversely, repulsive energies work against this attraction. They emerge from the repulsion between electron clouds when ions are too close. Like attractive energies, repulsive energies are also dependent on the distance between ions, but they follow a considerably faster rate of change as ions approach each other, as illustrated in the given functions.

Understanding these concepts is critical for comprehending how the net energy of a system is calculated by summing the attractive and repulsive energies. By plotting these energies against various distances, students can visually grasp how these forces dictate the behavior of ionic systems and determine the point of equilibrium spacing where the energy is lowest and stability is the highest.

The electron volt (eV) is a unit of energy that's especially useful in the context of atomic and molecular scales. One electron volt is defined as the amount of kinetic energy gained or lost by a single electron as it moves through an electric potential difference of one volt. It's a practical unit because it relates to the energies typically involved in chemical bonds and changes in electron configurations.

In our exercise with \( \mathrm{K}^{+}-\mathrm{Cl}^{-} \) ion pairs, energies are expressed in electron volts per pair. This is convenient because it gives a direct sense of the scale of energies when ions interact. The electron volt is a much smaller unit than the Joule, the standard SI unit for energy, which makes it all the more suitable for describing the subtle shifts in energy as ions move closer or further apart. For reference, 1 eV equals approximately \( 1.602 \times 10^{-19} \) Joules.

The nanometer (nm) is a unit of length in the metric system, equivalent to one billionth of a meter (\( 10^{-9} \) meters). It's a preferred unit of measurement when dealing with the extremely small scales of atoms and molecules.

As seen in the provided exercise, distances between ions in ionic bonds are often expressed in nanometers. Utilizing nanometers allows scientists and students alike to easily communicate and calculate properties and behaviors of particles at the atomic level. Given that bonds in molecules and solids range on the scale of tenth to few nanometers, using such a precise unit ensures accuracy in descriptions and calculations of chemical structures, particularly when evaluating the effects of interatomic distances on bonding energy and force interactions.