Use the following data to estimate \(\Delta H_{\mathrm{f}}^{\circ}\) for magnesium fluoride. $$\mathrm{Mg}(s)+\mathrm{F}_{2}(g) \longrightarrow \mathrm{MgF}_{2}(s)$$ $\begin{array}{l}{\text { Lattice energy }} & {-22913 . \mathrm{kJ} / \mathrm{mol}} \\ {\text { First ionization energy of } \mathrm{Mg}} & \quad{735 \mathrm{kJ} / \mathrm{mol}} \\ {\text {Second ionization energy of } \mathrm{Mg}} & \quad {1445 \mathrm{kJ} / \mathrm{mol}}\\\\{\text { Electron affinity of } \mathrm{F}} & {-328 \mathrm{kJ} / \mathrm{mol}} \\ {\text { Bond energy of } \mathrm{F}_{2}} & \quad {154 \mathrm{kJ} / \mathrm{mol}} \\\ {\text { Enthalpy of sublimation for } \mathrm{Mg}} & \quad {150 . \mathrm{kJ} / \mathrm{mol}} \end{array}$

The Born-Haber cycle is a thermochemical cycle that can help us determine the enthalpy of formation of an ionic compound using the related values like lattice energy, ionization energy, electron affinity, etc. The cycle as a whole is based on Hess's law, which states that the total energy change in a reaction is the same regardless of the path taken. For this problem, we use the Born-Haber cycle to calculate the enthalpy of formation of magnesium fluoride, MgF₂, from the given data.

Here we will carry out the cycle in a step-by-step manner: 1. Sublimation of Magnesium: This step describes the transformation of solid magnesium to gaseous magnesium i.e., Mg(s) -> Mg(g). The enthalpy change for this step is the enthalpy of sublimation, which is given as 150 kJ/mol. 2. Ionization of Magnesium: In this step, gaseous magnesium loses two electrons to form Mg²⁺(g). This process occurs in two steps, as there are two ionization energies given for magnesium. The total ionization energy required is the sum of the first and second ionization energies. Total Ionization Energy = 735 kJ/mol + 1445 kJ/mol = 2180 kJ/mol 3. Bond dissociation of F₂: The bond energy for F₂ (g) is given as 154 kJ/mol. In this step, we break the bond in the F₂ molecule. 4. Electron affinity of Fluorine: This step describes the process of gaseous fluorine gaining an electron to form F⁻(g). The electron affinity for fluorine is given as -328 kJ/mol. Since we have two fluorine atoms, the total electron affinity change is 2*(-328 kJ/mol) = -656 kJ/mol. 5. Formation of MgF₂: Mg²⁺ and 2F⁻ combine to form MgF₂(http://latex.codecogs.com/gif.latex?%28s%29). The enthalpy change in this step is the lattice energy, which is given as -22913 kJ/mol.

Hess's law states that the total energy change in a reaction is the same regardless of the path taken. Therefore, the enthalpy of formation for magnesium fluoride, \(\Delta H_{\mathrm{f}}^{\circ}\), is the sum of the enthalpy changes in each step of the Born-Haber cycle. \(\Delta H_{\mathrm{f}}^{\circ}\) = Enthalpy of sublimation + Total ionization energy + Bond dissociation energy + Total electron affinity + Lattice energy \(\Delta H_{\mathrm{f}}^{\circ}\) = 150 kJ/mol + 2180 kJ/mol + 154 kJ/mol - 656 kJ/mol - 22913 kJ/mol \(\Delta H_{\mathrm{f}}^{\circ}\) = -21185 kJ/mol Thus, the estimated standard enthalpy of formation for magnesium fluoride is -21185 kJ/mol.