Rationalize the following lattice energy values: $$ \begin{array}{|lc|} \hline \text { Compound } & \text { Lattice Energy (kJ/mol) } \\ \hline \text { CaSe } & -2862 \\ \mathrm{Na}_{2} \mathrm{Se} & -2130 \\ \mathrm{CaTe} & -2721 \\ \mathrm{Na}_{2} \mathrm{Te} & -2095 \\ \hline \end{array} $$

Understanding ionic radii is crucial when examining the stability and strength of ionic bonds. This term specifically refers to the measured size of an ion. An ionic radius is influenced by the ion's charge, with cations (positively charged ions) generally having smaller radii than anions (negatively charged ions) due to the loss of electrons and subsequent contraction of the electron cloud.

For example, when we consider the ionic lattice of calcium selenide (CaSe) versus sodium selenide (Na₂Se), the size difference between the cations, calcium (Ca²⁺) and sodium (Na⁺), is instrumental in explaining the disparity in their lattice energies. The smaller calcium ion leads to a stronger attraction between ions in the crystal lattice structure. This stronger attraction results in a higher lattice energy compared to the larger sodium ion.

It is essential to recognize that as the ionic radii decrease, the distance between the ions in the crystal lattice also decreases, leading to increased electrostatic forces of attraction and thus higher lattice energies.

The Born-Lande equation offers a theoretical means to calculate the lattice energy of an ionic solid. This equation is influential in predicting and explaining the factors that affect lattice energy:

\[\begin{equation}E_l = -\frac{N_AMz^+z^-e^2}{4\pi\epsilon_0r}\end{equation}\]
This formula encapsulates several variables.

• N_A is Avogadro's number, representing the number of pairs of ions in one mole of the solid.
• M is the Madelung constant that accounts for the geometric arrangement of the ions.
• z⁺ and z^- are the charges on the cations and anions, respectively.
• e is the elementary charge, equivalent to the charge of a proton.
• r is the internuclear distance between the ions.
• \(\epsilon_0\) is the permittivity of free space, a constant value in the equation.

Impact on Lattice Energy

With a fixed ionic arrangement, higher ionic charges and smaller ionic radii result in a more negative lattice energy according to the Born-Lande equation, which implies a stronger ionic bond. Hence, it's instrumental in explaining the differences in lattice energies for compounds like CaSe and Na₂Se, where different ionic charges play a significant role.

The role of ionic charges is a fundamental aspect when evaluating lattice energies. Ions are atoms or molecules that have gained or lost electrons, resulting in a net positive or negative charge. A higher charge magnitude on an ion leads to stronger electrostatic forces of attraction within a crystal lattice, and therefore, a higher lattice energy.

For example, comparing calcium ions (Ca²⁺) and sodium ions (Na⁺) illustrates that calcium ions, with their +2 charge, create stronger ionic bonds in the lattice than the +1 charge of sodium ions. Consequently, calcium selenide (CaSe) has a more substantial lattice energy than sodium selenide (Na₂Se).

The influence of ionic charges on lattice energy can be directly observed in the Born-Lande equation, where the product of the cationic and anionic charges significantly determines the magnitude of the lattice energy. Compounds with ions of higher charges should, in theory, possess larger lattice energies, which is typically observed in practice and reflected in exercises involving lattice energy comparisons, as seen in the example above.