Question

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

Short answer

The standard 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.

Step-by-step solution

Understanding the Born-Haber cycle

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.

Following the steps of the cycle

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.

Calculating the enthalpy of formation

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.

Key concepts

Enthalpy of Formation

Enthalpy of formation is a crucial concept in chemistry, especially when dealing with ionic compounds. It refers to the heat change that occurs when one mole of a compound is formed from its elements in their standard states. For example, in the formation of magnesium fluoride (MgF₂), the elements magnesium (Mg) and fluorine (F₂) react to form this compound.
Understanding the enthalpy of formation allows us to grasp how much energy is released or absorbed during the creation of a substance from its basic elements. This concept is generally expressed in kilojoules per mole (kJ/mol). When the enthalpy of formation is negative, it indicates that the reaction is exothermic, releasing energy into the surroundings.
Estimating the enthalpy of formation for compounds like MgF₂ often involves a series of calculated steps, which is where the Born-Haber cycle comes in handy. The cycle helps break down this complex process into understandable steps, utilizing various thermodynamic data to arrive at the final enthalpy of formation.

Lattice Energy

Lattice energy is an integral part of understanding ionic solids and their formation. It is defined as the energy released when ions in the gaseous state come together to form an ionic crystalline solid. This energy is crucial because it contributes significantly to the stability and strength of the ionic bond in the compound.
In the context of magnesium fluoride, the lattice energy corresponds to the process where gaseous Mg²⁺ ions and F⁻ ions combine to form the solid MgF₂ lattice. It is often a large exothermic value, reflecting the strong forces holding the ions together in the crystal lattice.
The lattice energy can be influenced by several factors such as the charge of the ions and the size of the ions involved. Generally, higher charges and smaller ionic sizes result in greater lattice energies due to stronger electrostatic forces. These energies are crucial in calculating the enthalpy of formation using the Born-Haber cycle.

Hess's Law

Hess's Law is a powerful principle in thermochemistry that states that the total enthalpy change for a reaction is the same, regardless of the number of steps it takes or the path it follows.
In the Born-Haber cycle, Hess's Law is pivotal because it allows us to calculate the enthalpy of formation of a compound through an indirect method. By adding the enthalpy changes of each step in the cycle, we can determine the total enthalpy change for the overall reaction, even if the direct measurement is difficult or impossible.
This law reflects the conservation of energy, meaning that energy cannot be created or destroyed. Therefore, the energy differences across different reaction pathways must balance out to give the same overall energy change for the system. This makes Hess's Law an essential tool when analyzing complex chemical reactions.

Ionization Energy

Ionization energy is an important concept describing the energy required to remove an electron from an atom or ion in the gas phase. It is typically expressed as the energy needed to ionize one mole of atoms.
For magnesium in the Born-Haber process, we consider two ionization energies because magnesium tends to form a 2⁺ ion by losing two electrons. The first ionization energy is the energy needed to remove the first electron, and the second ionization energy refers to the removal of the second electron. Both values contribute to the calculation of the total enthalpy change in forming the ionic compound MgF₂.
Ionization energy values are essential for calculating the overall energy changes within the Born-Haber cycle, as they represent the cost of converting neutral atoms into positive ions, a key step in the formation of ionic compounds.