Question

List the individual steps used in constructing a Born-Haber cycle for the formation of \(\mathrm{Bal}_{2}\) from the elements. Which of the steps would you expect to be exothermic?

Short answer

The individual steps for constructing a Born-Haber cycle for the formation of \(\mathrm{BaI}_{2}\) from the elements are: 1. Sublimation of barium and iodine (endothermic) 2. Ionization energy of barium (endothermic) 3. Electron affinity of iodine (exothermic) 4. Lattice energy (exothermic) Among these steps, the exothermic ones are the electron affinity of iodine and lattice energy.

Step-by-step solution

Sublimation of barium and iodine

Sublimate both barium and iodine to convert them into gaseous atoms. Write the balanced equation for the sublimation of each element: \[\mathrm{Ba}(s) \rightarrow \mathrm{Ba}(g)\] \[\mathrm{I}_{2}(s) \rightarrow \mathrm{2I}(g)\] This step is endothermic since energy is required to break the intermolecular forces in both barium and iodine.

Ionization energy of barium

Gaseous barium loses two electrons to form a gaseous Ba2+ ion. Write the balanced equation for the ionization of barium to form Ba2+: \[\mathrm{Ba}(g) \rightarrow \mathrm{Ba}^{2+}(g) + 2e^{-}\] This step is endothermic since energy is required to remove electrons from barium.

Electron affinity of iodine

Gaseous iodine atoms each gain one electron to form gaseous I- ions. Write the balanced equation for the electron affinity of iodine to form I-: \[\mathrm{I}(g) + e^{-} \rightarrow \mathrm{I}^{-}(g)\] This step is exothermic because energy is released when an electron is added to an iodine atom.

Lattice energy

The gaseous Ba2+ and I- ions come together to form a solid BaI2 compound. Write the balanced equation for the formation of solid BaI2 compound from gaseous ions: \[\mathrm{Ba}^{2+}(g) + 2\mathrm{I}^{-}(g) \rightarrow \mathrm{BaI}_{2}(s)\] This step is exothermic because energy is released when the ionic bond is formed between Ba2+ and I- ions. To summarize, the individual steps for constructing a Born-Haber cycle for the formation of \(\mathrm{BaI}_{2}\) from the elements are: sublimation of barium and iodine (endothermic), ionization energy of barium (endothermic), electron affinity of iodine (exothermic), and lattice energy (exothermic).

Key concepts

Sublimation of Barium and Iodine

Sublimation is the process where a substance transitions from its solid to gaseous state without passing through the liquid phase. This step is essential in the Born-Haber cycle to prepare atoms for the formation of ionic compounds. Regarding barium (\textsc{Ba}) and iodine (\textsc{I}\(_2\)), the sublimation process requires energy input, since you must overcome the forces holding the solid lattice together.

\textsc{Ba}(s) \(\rightarrow\) \textsc{Ba}(g)
\textsc{I}\(_2\)(s) \(\rightarrow\) \textsc{2I}(g)

For barium, the strong metallic bonds must be broken, which consumes a significant amount of energy. In iodine, the van der Waals forces between the diatomic molecules need to be overcome. The energy changes associated with sublimation are always endothermic, as they need heat from the surroundings to proceed.

Ionization Energy

Ionization energy is the amount of energy required to remove an electron from an atom or ion in its gaseous state. In the Born-Haber cycle for forming \textsc{BaI}\(_2\), gaseous barium (\textsc{Ba}(g)) must lose two electrons to form a \textsc{Ba}\(^{2+}\) ion:

\textsc{Ba}(g) \(\rightarrow\) \textsc{Ba}\(^{2+}\)(g) + 2e\(^-\)

This process is endothermic and is a crucial step because it contributes to the overall energy needed to form an ionic compound. The difficulty of removing these electrons—and hence the amount of ionization energy required—depends on the atomic structure of the element. For barium, with relatively low nuclear charge and larger radius, the first ionization energy is not exceedingly high, but the second ionization energy is significantly higher since it's harder to remove an electron from a positively charged ion.

Electron Affinity

Electron affinity refers to the energy change when an electron is added to a neutral atom in the gas phase to form a negative ion. For iodine atoms (\textsc{I}(g)) becoming iodide ions (\textsc{I}\(^-\)), the process can be represented as follows:

\textsc{I}(g) + e\(^-\) \(\rightarrow\) \textsc{I}\(^-\)(g)

This step in the Born-Haber cycle releases energy, making it exothermic. Iodine has a high electron affinity because it's a halogen and requires only one additional electron to complete its valence shell, leading to a more stable electronic arrangement. As electrons are added to an atom, energy is given off due to the attraction between the negatively charged electrons and the positively charged nucleus.

Lattice Energy

Lattice energy is the energy released when ions in the gas phase come together to form an ionic solid. In the Born-Haber cycle for \textsc{BaI}\(_2\), the \textsc{Ba}\(^{2+}\) and \textsc{I}\(^-\) gaseous ions create the ionic lattice of barium iodide:

\textsc{Ba}\(^{2+}\)(g) + \textsc{2I}\(^-\)(g) \(\rightarrow\) \textsc{BaI}\(_2\)(s)

This step is highly exothermic. The more charge and closer the ions are within a crystal lattice, the greater the lattice energy. In \textsc{BaI}\(_2\), the strong electrostatic attraction between the \textsc{Ba}\(^{2+}\) cations and \textsc{I}\(^-\) anions results in a significant release of energy. The formation of the crystalline lattice is what stabilizes the ionic compound energetically, making it a favored process.