Question

Use the following data to estimate \(\Delta H_{\mathrm{f}}^{\circ}\) for magnesium fluoride. $$ \mathrm{Mg}(s)+\mathrm{F}_{2}(g) \longrightarrow \mathrm{MgF}_{2}(s) $$ $$ \begin{array}{lr} \text { Lattice energy } & -2913 \mathrm{~kJ} / \mathrm{mol} \\ \text { First ionization energy of } \mathrm{Mg} & 735 \mathrm{~kJ} / \mathrm{mol} \\ \text { Second ionization energy of } \mathrm{Mg} & 1445 \mathrm{~kJ} / \mathrm{mol} \\ \text { Electron affinity of } \mathrm{F} & -328 \mathrm{~kJ} / \mathrm{mol} \\\ \text { Bond energy of } \mathrm{F}_{2} & 154 \mathrm{~kJ} / \mathrm{mol} \\ \text { Enthalpy of sublimation for } \mathrm{Mg} & 150 . \mathrm{kJ} / \mathrm{mol} \end{array} $$

Short answer

The standard enthalpy of formation \(\Delta H_{\mathrm{f}}^{\circ}\) for magnesium fluoride, MgF₂, can be estimated using the Born-Haber cycle with the given data: \(\Delta H_{\mathrm{f}}^{\circ} = 150 + 735 + 1445 - 328 + \frac{1}{2}(154) + 2913\). Therefore, \(\Delta H_{\mathrm{f}}^{\circ}\) is approximately -4669 kJ/mol.

Step-by-step solution

Write down the Born-Haber cycle

The Born-Haber cycle can be represented as: \[\Delta H_{\mathrm{f}}^{\circ} = \Delta H_{\mathrm{sub}}^{\circ} + \Delta H_{\mathrm{IE1}} + \Delta H_{\mathrm{IE2}} + \Delta H_{\mathrm{E1}} + \frac{1}{2} \Delta H_{\mathrm{BE2}} - \Delta H_{\mathrm{LE}}\] Where: - \(\Delta H_{\mathrm{f}}^{\circ}\): Standard enthalpy of formation - \(\Delta H_{\mathrm{sub}}^{\circ}\): Enthalpy of sublimation - \(\Delta H_{\mathrm{IE1}}\): First ionization energy - \(\Delta H_{\mathrm{IE2}}\): Second ionization energy - \(\Delta H_{\mathrm{E1}}\): Electron affinity - \(\Delta H_{\mathrm{BE2}}\): Bond energy - \(\Delta H_{\mathrm{LE}}\): Lattice energy

Substitute values from the given data

Plug in the given values into the Born-Haber cycle equation: \[\Delta H_{\mathrm{f}}^{\circ} = (150) + (735) + (1445) + (-328) + \frac{1}{2}(154) - (-2913)\]

Calculate the answer

Calculate the standard enthalpy of formation, \(\Delta H_{\mathrm{f}}^{\circ}\), by solving the equation: \[\Delta H_{\mathrm{f}}^{\circ} = 150 + 735 + 1445 - 328 + \frac{1}{2}(154) + 2913\newline\Delta H_{\mathrm{f}}^{\circ} = 150 + 735 + 1445 - 328 + 77 + 2913\newline\Delta H_{\mathrm{f}}^{\circ} = -4669 \mathrm{~kJ} / \mathrm{mol}\] The standard enthalpy of formation \(\Delta H_{\mathrm{f}}^{\circ}\) for magnesium fluoride, MgF₂, is estimated to be - 4669 kJ/mol.

Key concepts

Enthalpy of Formation

The enthalpy of formation, denoted as \(\Delta H_{\mathrm{f}}^{\circ}\), refers to the heat change when one mole of a compound is formed from its constituent elements under standard conditions, meaning at 1 atm pressure and a specified temperature, usually 25°C or 298 K. This thermodynamic quantity allows chemists to understand the stability of substances and predict reactions in chemical processes.
For example, in the formation of magnesium fluoride \(\text{(MgF}_2\text{)}\), magnesium metal and fluorine gas combine. The enthalpy of formation is significant in this process because it includes several energy terms that must be considered: the conversion of solid magnesium to gaseous magnesium (enthalpy of sublimation), the breaking of bonds in fluorine molecules, ionization energies of magnesium, electron affinity of fluorine, and the lattice energy. Each of these plays a role in determining the overall heat change during the compound's formation.
Thus, understanding the enthalpy of formation helps predict whether the reaction will absorb or release heat, which is crucial for applications like materials science and chemical manufacturing.

Lattice Energy

Lattice energy represents the energy released when one mole of a crystalline solid is formed from its constituent ions in the gaseous state. It's a crucial factor in understanding the stability of ionic compounds. In simpler terms, lattice energy is the measure of the strength of the forces between the ions in an ionic solid.
When it comes to magnesium fluoride \(\text{(MgF}_2\text{)}\), the lattice energy is highly negative, indicating a large amount of energy released upon formation of the solid structure from its gaseous ions. This contributes significantly to the compound's stability.
The calculation of lattice energy isn't straightforward as it can't be measured directly. Instead, it's often estimated using a theoretical model called the Born-Haber cycle, which involves other known energies. The large magnitude of lattice energy in compounds like \(\text{MgF}_2\) means that the compound's ionic bonds are strong, accounting for properties like high melting points and hardness.

Ionization Energy

Ionization energy is the energy required to remove an electron from a neutral atom in its gaseous state. It is a key concept for understanding how elements react and bond with others. Ionization energy typically increases across a period but decreases down a group in the periodic table.
In the Born-Haber cycle used to estimate \(\Delta H_{\mathrm{f}}^{\circ}\) for magnesium fluoride, both the first and second ionization energies of magnesium are considered. The first ionization energy removes one electron to form \(\text{Mg}^{+}\), followed by the second ionization energy to form \(\text{Mg}^{2+}\).
The significant values of ionization energies for magnesium (735 kJ/mol for the first and 1445 kJ/mol for the second) suggest that energy must be input to remove electrons, compensating for the energy provided by electron affinity and lattice energy in forming a stable compound. Therefore, ionization energy is a critical concept for predicting how elements will form ionic bonds and the overall heat exchange in these processes.

Electron Affinity

Electron affinity reflects the energy change that occurs when an electron is added to a neutral atom in the gaseous state, turning it into a negative ion. This energy change can be negative or positive, depending on whether energy is released or absorbed during the process.
The concept of electron affinity plays a crucial role in determining the likelihood of atom bonding by gaining electrons. For fluorine in the magnesium fluoride compound, the electron affinity is measured as \(-328 \text{ kJ/mol}\), indicating a release of energy when an electron is added. This value highlights fluorine's strong attraction for electrons, which is a typical characteristic of halogens in the periodic table.
Understanding electron affinity helps in explaining why certain atoms, like fluorine, readily form negative ions and engage in ionic bonding, such as in \(\text{MgF}_2\). It complements other energetic factors like ionization energy and lattice energy in influencing the stability and formation of ionic compounds.