For an \(\mathrm{Na}^{+}-\mathrm{Cl}^{-}\)ion pair, attractive and repulsive energies \(E_{A}\) and \(E_{R}\), respectively, depend on the distance between the ions \(r\), according to $$ \begin{aligned} &E_{A}=-\frac{1.436}{r} \\ &E_{R}=\frac{7.32 \times 10^{-6}}{r^{8}} \end{aligned} $$ For these expressions, energies are expressed in electron volts per \(\mathrm{Na}^{+}-\mathrm{Cl}^{-}\)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 \(\mathrm{Na}^{+}\)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 2.18, and compare these with the graphical results from part (b).

Bonding energy, often referred to as bond energy, is crucial to understanding the stability and behavior of molecules and compounds. It represents the amount of energy required to break a bond between two atoms. In the context of ionic compounds, like the Na+ and Cl- ion pair from our exercise, the bonding energy is a measure of the ionic bond's strength.

To calculate the bonding energy, we need an equation that describes the interaction between the ions. For the pair described in the exercise, the net energy (EN) is given by a sum of attractive (EA) and repulsive (ER) energy components, which vary with the ion separation distance, r. To find the bonding energy (E0), we mathematically determine the equilibrium spacing (r0), where the net energy is at its minimum. This r0 is the optimal distance where the ions have the least potential energy, signifying a stable ionic bond.

By setting the derivative of the net energy function with respect to r to zero, we can solve for the equilibrium spacing and subsequently determine the bonding energy. Precision in calculations is key, as minor errors can lead to a significant difference in the final bonding energy value, which affects the accuracy in predicting the stability of the Na+-Cl- pair.

Equilibrium ion spacing is a fundamental concept in the study of ionic bonds, defined as the distance between the centers of two ions in an ionic solid where the net energy is minimal. In the case of Na+ and Cl-, this spacing corresponds to the most stable configuration of the ions.

To find this equilibrium spacing, labeled r0, we look for the minimum point in the energy versus distance plot. It's essential to understand that at this point, the attractive and repulsive forces between the ions balance each other out. Physically, it means the ions are as close as possible without repelling each other due to electron cloud overlap, and this specific distance is a unique characteristic of each ion pair.

The graphical approach to finding r0 involves analyzing the point where the net energy plot (EN) reaches its lowest value. However, for more accuracy, a mathematical method where we differentiate the energy equation with respect to distance r and set it to zero gives a precise value for r0. Students should note that correct application of calculus is essential in these calculations to obtain accurate and reliable results.

The energy versus distance plot is a visual tool that illustrates how the potential energy of an ionic pair like Na+ and Cl- changes as the distance between the ions varies. This plot helps us understand the behavior of the ionic bond and identify critical points, such as the equilibrium spacing and the bonding energy.

When constructing this plot with the given equations, one axis represents the distance between the ions, while the other represents energy. The plot typically shows a curve that decreases to a minimum point, which indicates the equilibrium ion spacing (r0), and then rises again. The minimum point represents the most energetically favorable condition for the ion pair, corresponding to the optimal distance where the ionic bond is strongest.

Students can use graphing software to create the plot and visually determine r0 and the bonding energy (E0). Producing accurate plots is key to understanding not just the concept of equilibrium ion spacing and bonding energy but also general principles of interatomic interactions and the potential energy landscapes of molecular systems. Visual learners will find energy versus distance plots particularly enlightening for grasping how energy changes with interatomic distances.