An ionic bond is formed between a cation \(\mathrm{A}^{+}\) and an anion \(\mathrm{B}^{-}\). How would the energy of the ionic bond [see Equation (9.2)\(]\) be affected by the following changes? (a) doubling the radius of \(\mathrm{A}^{+},\) (b) tripling the charge on \(A^{+},(\mathrm{c})\) doubling the charges on \(\mathrm{A}^{+}\) and \(\mathrm{B}^{-},\) (d) decreasing the radii of \(\mathrm{A}^{+}\) and \(\mathrm{B}^{-}\) to half their original values.

Coulomb's Law is a fundamental principle that helps us understand how charged particles interact with one another. It states that the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, it is expressed as:
\[ F = k \frac{{|q_1q_2|}}{{r^2}} \]
where \( F \) is the force between the charges, \( q_1 \) and \( q_2 \) represent the magnitudes of the charges, \( r \) is the distance between the centers of the two charges, and \( k \) is Coulomb's constant. In the context of ionic bonds, this law gives us the energy of the bond as the force times distance (work done to bring the charges together from infinity). When applying Coulomb's Law to ionic bonds, we consider the ions as point charges because their size is so much smaller compared to the distances between them in a crystalline structure. This makes Coulomb's Law an ideal way to predict how changes in charge and distance affect ionic bond energy.

Ionic bond energy can be thought of as the strength of the attraction between ions. According to Coulomb's Law, it's directly related to the force, which explains why changes in charge or distance have such significant effects on bond energy. As you work through problems involving ionic bonds, keeping Coulomb's Law in mind will help you understand the direction of changes in energy when the distance or charge varies.

The radius of an ion plays a crucial role in determining the bond energy of an ionic compound. As explained by Coulomb's Law, the energy of an ionic bond is inversely proportional to the distance between the ion centers which is effectively the sum of their radii for closely packed ions. When the radius of an ion doubles, the distance between the ions increases, leading to a weaker electrostatic attraction and thus a decrease in bond energy.

Imagine two magnets placed at a certain distance; when the distance is increased, the pull between them weakly feels weaker. Similarly, as the radius of an ion increases, the distance between ions in a bond increases, and they feel less attraction to each other therefore, the bond energy decreases. This concept is critical in understanding how the physical size of the ions can directly impact the strength and stability of ionic compounds. Conversely, when the ion radius decreases, the bond energy increases due to a stronger electrostatic attraction. Substances with high bond energies tend to have lower radii, contributing to their high melting and boiling points, making it a key factor in the physical properties of ionic compounds.

The charge of ions has a profound impact on the energy of an ionic bond. In accordance with Coulomb’s Law, the bond energy is directly proportional to the product of the charges of the cation and anion. For instance, when the charge on an ion is tripled, the interaction between the charged particles is stronger, which translates to a bond with higher energy.

To understand this in a real-world context, consider the analogy of magnets again; stronger magnets (akin to higher charges) have a stronger pull toward each other compared to weaker magnets. Therefore, in ionic compounds, an increase in the charge on the ions leads to a stronger bond. This direct relationship between charge and bond energy is crucial to predicting the strength and properties of different ionic compounds. If both ions involved in the bond have their charges doubled, the resulting bond energy increases by four times (since the charges are multiplied). This explains why compounds with ions of higher charges have significantly higher melting and boiling points; the stronger the charge, the stronger the bond, making it harder to break apart the ionic lattice.