Problem 106 - Chemistry: An Atoms-Focused Approach

Question

Calculate the lattice energy of sodium oxide \(\left(\mathrm{Na}_{2} \mathrm{O}\right)\) from the following data: Ionization energy of \(\mathrm{Na}(g)=495 \mathrm{kJ} / \mathrm{mol}\) Electron affinity of \(\mathrm{O}(g)\) for 2 electrons \(=603 \mathrm{kJ} / \mathrm{mol}\) Energy to sublime \(\mathrm{Na}(s)=109 \mathrm{kJ} / \mathrm{mol}\) Bond energy of \(\mathrm{O}_{2}(g)=498 \mathrm{kJ} / \mathrm{mol}\) \(\Delta H_{\mathrm{rxn}}\) for \(2 \mathrm{Na}(s)+\frac{1}{2} \mathrm{O}_{2}(g) \rightarrow \mathrm{Na}_{2} \mathrm{O}(s)=-416 \mathrm{kJ} / \mathrm{mol}\)

Step-by-Step Solution

Answer

Answer: The lattice energy of sodium oxide, Na₂O, is -1270 kJ/mol.

Step 1: Ionize sodium atoms

To ionize sodium atoms, we need to remove one electron from each of the two sodium atoms to form \(\mathrm{Na}^{+} (g)\). The energy needed for this process is called ionization energy, which is given as 495 kJ/mol for \(\mathrm{Na}(g)\). Since we have 2 sodium atoms, the total energy for this step is \(2\times495=990\,\mathrm{kJ/mol}\).

Step 2: Sublime sodium atoms

To convert the solid sodium atoms to gas state, we need to provide the energy to sublime \(\mathrm{Na}(s)\). This energy is given as 109 kJ/mol for \(\mathrm{Na}(s)\). Since we have 2 sodium atoms, the total energy for this step is \(2\times109=218\,\mathrm{kJ/mol}\).

Step 3: Break the oxygen bond

We need to break the oxygen bond to have oxygen atoms for the reaction. The bond energy of \(\mathrm{O}_{2}(g)\) is given as 498 kJ/mol. Since we have a half mole of oxygen in the reaction, the energy required for this step is \(\frac{1}{2}\times498=249\,\mathrm{kJ/mol}\)

Step 4: Add two electrons to the oxygen atom

To form the oxide ion, we need to add two electrons to the oxygen atom. The energy released in this process is the electron affinity of \(\mathrm{O}(g)\). For 2 electrons, it is given as 603 kJ/mol. The energy released due to this process is -\(603\,\mathrm{kJ/mol}\).

Step 5: Calculate the lattice energy

To calculate the lattice energy, \(\Delta H_{\mathrm{lattice}}\), we will use the Born-Haber cycle, which relates \(\Delta H_{\mathrm{rxn}}\) to ionization energies, electron affinities, and bond energies: $$ \Delta H_{\mathrm{lattice}} = \Delta H_{\mathrm{rxn}} - (\Delta H_{\mathrm{ionization}} + \Delta H_{\mathrm{sublimation}} + \Delta H_{\mathrm{bond-breaking}} + \Delta H_{\mathrm{electron-affinity}}) $$ Substituting the values: $$ \Delta H_{\mathrm{lattice}} = -416 - (990 + 218 + 249 - 603) $$ $$ \Delta H_{\mathrm{lattice}} = -416 - 854 $$ $$ \Delta H_{\mathrm{lattice}} = -1270\,\mathrm{kJ/mol} $$ So, the lattice energy of sodium oxide, \(\mathrm{Na}_{2}\mathrm{O}\), is \(-1270\,\mathrm{kJ/mol}\).

Key Concepts

Born-Haber Cycle

The Born-Haber Cycle is a crucial tool for understanding the formation and stability of ionic compounds. It allows us to calculate the lattice energy, which is the energy released when ions bond to form a crystalline lattice. To imagine it, think of it as a flowchart of energy changes that occur during the formation of an ionic compound. Each step in the cycle represents an energy change needed to transform reactants into products.

Here's how it works:

• Begin with the conversion of elements in their standard states to gaseous atoms through processes like sublimation and dissociation.
• Proceed to ionize the gaseous atoms by removing or adding electrons, using ionization energy and electron affinity.
• Finally, calculate the energy changes from forming ions and arranging them into a lattice.

By balancing these energy changes, the Born-Haber Cycle helps determine the lattice energy by comparing them to the overall reaction enthalpy ( delta H_{rxn}) of forming the compound.

Ionization Energy

Ionization energy is the energy required to remove an electron from an atom in the gas phase, forming a cation. It's a vital step in processes like the Born-Haber Cycle. To put it simply, it measures how strongly an atom holds onto its electrons. Higher ionization energies mean electrons are held tightly and are harder to remove.

For Sodium ( a), we need to ionize two sodium atoms in our exercise. This involves two sets of ionization energies. Each sodium atom requires 495 kJ/mol to lose one electron. Therefore, removing electrons from two sodium atoms costs a total of 990 kJ/mol. Understanding ionization energy helps us see why certain elements form cations more easily than others.

Electron Affinity

Electron affinity is the energy change that occurs when an electron is added to an atom in the gas phase, forming an anion. This process is usually exothermic, meaning it releases energy. Think of it as the opposite of ionization energy.

In our example, oxygen needs to gain two electrons to become an oxide ion. The energy released during this process is given as 603 kJ/mol for two electrons. This release of energy is crucial for balancing the energy costs in forming ionic compounds. Generally, elements with a high electron affinity readily form anions.

Sublimation Energy

Sublimation energy is the energy required to change a substance from a solid to a gaseous state without passing through the liquid state. It's part of the pathway in forming gaseous ions from solid elements.

For sodium ( a), the sublimation energy needed is 109 kJ/mol. Since our exercise involves two sodium atoms, the total energy is 218 kJ/mol.

• Solid to gas conversion is key to understanding formations in the Born-Haber Cycle.
• Sublimation is typically an endothermic process, meaning it requires the input of energy.
• Understanding sublimation helps us see how solid elements transform into reactive gas forms.

This concept is fundamental to grasping the complete picture of ionic compound formation.