Problem 13 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Construct a Born-Haber cycle for the formation of MgO (s) from \(\mathrm{Mg}(\mathrm{s})\) and \(\mathrm{O}_{2}(\mathrm{g})\) and use the information below to calculate the lattice enthalpy of MgO. (Section 6.5) $$\begin{array}{ll} \Delta_{f} H^{\ominus}(\mathrm{MgO}) & -602 \mathrm{kJ} \mathrm{mol}^{-1} \\ \Delta_{\mathrm{i}} H(1)^{\mathrm{e}}(\mathrm{Mg}) & +738 \mathrm{kJ} \mathrm{mol}^{-1} \\ \Delta_{\mathrm{i}} H(2)^{\circ}(\mathrm{Mg}) & +1451 \mathrm{kJmol}^{-1} \\ \Delta_{\mathrm{a}} H^{\circ}(\mathrm{Mg}) & +147 \mathrm{kJ} \mathrm{mol}^{-1} \\ \Delta_{\mathrm{a}} H^{\circ}(\mathrm{O}) & +249 \mathrm{kJ} \mathrm{mol}^{-1} \\\ \Delta_{\mathrm{eg}} H(1)^{\circ}(\mathrm{O}) & -141 \mathrm{kJ} \mathrm{mol}^{-1} \\ \Delta_{\mathrm{eg}} H(2)^{\ominus}(\mathrm{O}) & +798 \mathrm{kJ} \mathrm{mol}^{-1} \end{array}$$

Step-by-Step Solution

Answer

The lattice enthalpy of MgO is -3844 kJ/mol.

Step 1: Understanding the Born-Haber Cycle

The Born-Haber cycle is a thermochemical cycle that relates the various enthalpy changes involved in the formation of an ionic compound from its elements with the lattice enthalpy of the compound. For MgO, this cycle involves the sublimation of Mg, ionization of Mg, breaking of O_2 bond, electron affinity of O, and the formation of MgO from ions.

Step 2: Sublimation of Magnesium

The first step is sublimating solid Magnesium to gaseous Magnesium. This step is represented by the sublimation enthalpy \(\Delta_{\text{a}} H^{\circ}(\text{Mg}) = 147 \text{ kJ mol}^{-1}\).

Step 3: Ionization of Magnesium

The next step involves the ionization of gaseous Magnesium to form \(\text{Mg}^{2+}\). This consists of two ionization energies: \(\Delta_{\text{i}} H(1)^{\circ}(\text{Mg}) = 738 \text{ kJ mol}^{-1}\) and \(\Delta_{\text{i}} H(2)^{\circ}(\text{Mg}) = 1451 \text{ kJ mol}^{-1}\).

Step 4: Dissociation of Oxygen Molecule

The transition from oxygen gas \(\text{O}_2\) to atomic oxygen involves bond dissociation. The bond dissociation energy here is the enthalpy of formation for 1 mole of atomic oxygen: \(\frac{1}{2} \times \Delta_{\text{a}} H^{\circ}(\text{O}_2) = 249 \text{ kJ mol}^{-1}\).

Step 5: Formation of Oxide Ions

Oxygen atoms gain electrons to form \(\text{O}^{2-}\) ions. The first electron affinity is \(\Delta_{\text{eg}} H(1)^{\circ}(\text{O}) = -141 \text{ kJ mol}^{-1}\). The second electron affinity is more endothermic \(\Delta_{\text{eg}} H(2)^{\ominus}(\text{O}) = 798 \text{ kJ mol}^{-1}\).

Step 6: Enthalpy Change for Formation of MgO

The enthalpy of formation \(\Delta_{f} H^{\ominus}(\text{MgO}) = -602 \text{ kJ mol}^{-1}\) gives the change when MgO is directly formed from its elements.

Step 7: Calculate the Lattice Enthalpy

Using Hess's Law, we sum the enthalpies: \[ \text{Lattice enthalpy} = \Delta_{f} H^{\ominus}(\text{MgO}) - \sum(\text{Steps 2 to 5}) \].Calculate the total enthalpy from steps 2 through 5, and rearrange the expression to solve for the lattice enthalpy.

Step 8: Summing and Solving

Sum up all the changes from steps 2 to 5:\[ 147 + 738 + 1451 + 249 - 141 + 798 = 3242 \text{ kJ mol}^{-1} \]Then calculate the lattice enthalpy:\[ \text{Lattice enthalpy of MgO} = -602 - 3242 = -3844 \text{ kJ mol}^{-1} \]

Key Concepts

Lattice Enthalpy

Lattice enthalpy is a fundamental concept when discussing ionic compounds like magnesium oxide (MgO). It reflects the energy required to separate one mole of a solid ionic compound into its gaseous ions.
In this context, for MgO, lattice enthalpy provides insight into the strength of the ionic bonds within the compound. Strongly bonded ionic compounds, such as MgO, have high lattice enthalpy values because breaking those strong bonds requires significant energy. Knowing the lattice enthalpy is crucial for understanding the stability and solubility of ionic compounds.
When calculating lattice enthalpy using the Born-Haber cycle, you'd typically bring into account several enthalpy changes, including sublimation and ionization of magnesium, dissociation of oxygen, and electron affinities. These individual energy changes build up to ultimately reveal the hidden strength of the ionic bonds.

Thermochemical Cycle

The Born-Haber cycle is a thermochemical cycle that is crucial for analyzing the formation of an ionic compound from its elements.
Through various steps, this method helps to visually and mathematically relate the energy changes that accompany the formation of a compound like MgO from its metallic magnesium and diatomic oxygen.
The cycle accounts for several key energy transformations, which include:

• The sublimation of solid magnesium into gaseous atoms.
• The ionization of gaseous magnesium to form cations.
• The dissociation of molecular oxygen into individual atoms.
• The capture of electrons by oxygen atoms to form oxide ions.

Each step in the cycle represents a change in energy or enthalpy, giving a full picture of the energy dynamics involved. This thermochemical cycle is a powerful tool for chemists to understand complex ionic systems in a structured way.

Ionic Compound Formation

The formation of an ionic compound, like magnesium oxide, stems from a series of energy transformations that occur between neutral elements.
In the case of MgO, magnesium atoms lose electrons, becoming Mg²⁺ ions, while oxygen atoms gain those electrons to become O²⁻ ions.
This electron exchange is crucial in forming strong ionic bonds, where the resulting compound adheres due to electrostatic interactions between positively charged cations and negatively charged anions.
The formation of ionic compounds involves key processes such as sublimation, ionization, electron gain, and the lattice arrangement of ions. Understanding these steps provides insight into why ionic compounds generally have high melting points, high boiling points, and the ability to conduct electricity when melted or dissolved - all due to their strong internal lattice structures that result from these initial formation steps.

Enthalpy Changes

Enthalpy changes accompany many chemical processes, and understanding these changes is vital in the study of reaction energetics.
In the formation of magnesium oxide, several enthalpy changes occur, each representing distinct processes. For instance:

• Sublimation enthalpy is the energy required to convert magnesium from solid to gas.
• Ionization energies involve removing electrons from gaseous atoms to form cations.
• Bond dissociation refers to breaking the oxygen-oxygen bond in diatomic oxygen to yield atomic oxygen.
• Electron affinities relate to the energy changes when oxygen atoms gain electrons to become oxide ions.

Each enthalpy change, whether endothermic (absorbing heat) or exothermic (releasing heat), contributes to the overall energetic picture leading to the compound's stability. By analyzing these changes, chemists can predict the likelihood and conditions under which ionic compounds will form, crucial for both theoretical studies and practical applications.