Problem 98 - Chemistry

Question

Given the following information, construct a Born-Haber cycle to calculate the lattice energy of \(\mathrm{CrCl}_{2} \mathrm{I}(s)\) : Net energy change for the formation of \(\mathrm{CrCl}_{2} \mathrm{I}(s)=-420 \mathrm{~kJ} / \mathrm{mol}\) Bond dissociation energy for \(\mathrm{Cl}_{2}(g)=+243 \mathrm{~kJ} / \mathrm{mol}\) Bond dissociation energy for \(\mathrm{I}_{2}(s)=+151 \mathrm{~kJ} / \mathrm{mol}\) Heat of sublimation for \(\mathrm{I}_{2}(s)=+62 \mathrm{~kJ} / \mathrm{mol}\) Heat of sublimation for \(\mathrm{Cr}(s)=+397 \mathrm{~kJ} / \mathrm{mol}\) \(E_{\mathrm{il}}\) for \(\mathrm{Cr}(g)=652 \mathrm{~kJ} / \mathrm{mol}\) \(E_{\mathrm{i} 2}\) for \(\mathrm{Cr}(g)=1588 \mathrm{~kJ} / \mathrm{mol}\) \(E_{\mathrm{i} 3}\) for \(\mathrm{Cr}(\mathrm{g})=2882 \mathrm{~kJ} / \mathrm{mol}\) \(E_{\mathrm{ca}}\) for \(\mathrm{Cl}(g)=-349 \mathrm{~kJ} / \mathrm{mol}\) \(E_{e a}\) for \(\mathrm{I}(g)=-295 \mathrm{~kJ} / \mathrm{mol}\)

Step-by-Step Solution

Answer

Calculate lattice energy by solving: \(\Delta H_{\text{lattice}} = -4740 \, \text{kJ/mol}\).

Step 1: Understanding the Problem

We are tasked with finding the lattice energy of \(\mathrm{CrCl}_2 \mathrm{I}(s)\) using a Born-Haber cycle. The information given includes various energies such as bond dissociation energies, sublimation energies, ionization energies, and electron affinities.

Step 2: Applying Hess's Law

The Born-Haber cycle applies Hess's Law, stating that the enthalpy change of a reaction is the same regardless of the pathway. We'll set up an enthalpy cycle for the formation of \(\mathrm{CrCl}_2 \mathrm{I}(s)\) from its constituent elements.

Step 3: Gathering Energy Contributions

- **Formation:** \(-420 \, \text{kJ/mol}\) (net change) - **Bond Dissociation:** Dissociation of \(\mathrm{Cl}_2\) is \(+243 \, \text{kJ/mol}\), implying \(\frac{1}{2} \times 243\), and dissociation of \(\mathrm{I}_2\) is \(+151 \, \text{kJ/mol}\). - **Sublimation:** \(\mathrm{I}_2\) sublimation is \(+62 \, \text{kJ/mol}\), and \(\mathrm{Cr}\) sublimation is \(+397 \, \text{kJ/mol}\).

Step 4: Calculate Ionization Energies and Electron Affinities

- **Ionization Energy:** We need \(\mathrm{E_{i1}}\), \(\mathrm{E_{i2}}\), and part of \(\mathrm{E_{i3}}\) since chromium forms a \(2+\) ion in \(\mathrm{CrCl}_2\mathrm{I}\), hence: \(652 + 1588 = 2240 \, \text{kJ/mol}\).- **Electron Affinity:** Two electrons to chlorine: \(2 \times (-349) = -698 \, \text{kJ/mol}\), and one to iodine: \(-295 \, \text{kJ/mol}\).

Step 5: Completing the Born-Haber Cycle

Set up the cycle: add the sublimation energies, bond dissociation energies, ionization energies, and electron affinities, then equate this to the net energy change minus the unknown lattice energy \(\Delta H_{\text{lattice}}\).

Step 6: Calculate the Lattice Energy

Using the equation from the cycle: \[ -420 = 397 + \frac{1}{2}(243) + 62 + 2240 - 698 - 295 + \Delta H_{\text{lattice}} \] Solve this for \(\Delta H_{\text{lattice}}\). The sum of all terms except the lattice energy is calculated, and by rearranging, \(\Delta H_{\text{lattice}}\) can be isolated.

Key Concepts

Lattice Energy

Lattice energy is a crucial concept when dealing with ionic compounds. It refers to the amount of energy required to completely separate a mole of a solid ionic compound into its gaseous ions. This energy reflects the strength of the forces between the ions in the lattice. The stronger the attraction between the ions, the higher the lattice energy will be.

• Lattice energy is influenced by the charge of the ions and the distance between them. Higher charges and smaller ionic radii lead to larger lattice energies.
• Your calculation of lattice energy in a Born-Haber cycle involves subtracting all known energy changes from the net enthalpy change for the formation of the compound.
• This requires the sum of sublimation energies, bond dissociation energies, ionization energies, and electron affinities.

In the Born-Haber cycle for CrCl$_2$I, identifying and understanding these contributions help us find the missing lattice energy value.

Ionization Energy

Ionization energy is the amount of energy required to remove an electron from an atom or ion in its gaseous state. When forming ionic compounds, metals like chromium lose electrons to form cations.

• Ionization energy is critical because it's one of the major energy contributions in a Born-Haber cycle.
• Chromium's ionization energies for forming a +2 ion are the sum of the first two ionization energies, E_{i1} and E_{i2}.

For CrCl$_2$I, you add 652 ext{kJ} and 1588 ext{kJ} to find the total ionization energy for Cr. This large energy expense represents the difficulty in removing electrons, contributing to the overall energetic makeup of forming an ionic compound.

Bond Dissociation Energy

Bond dissociation energy is the energy required to break a bond in a molecule, transforming it into separate atoms. In the context of the Born-Haber cycle, it's necessary to break bonds of diatomic molecules like Cl$_2$ and I$_2$ to form individual atoms for compound formation.

• Bond dissociation energy is always a positive value, indicating energy absorption.
• For CrCl$_2$I, the breakdown of Cl$_2$ results in the absorption of half the dissociation energy (as only one Cl atom is needed per molecule), and I$_2$ is fully broken down.

This bond-breaking step is vital before any atom can participate in forming new bonds in ionic compounds.

Electron Affinity

Electron affinity is the amount of energy released when an atom gains an electron. In ionic compounds, nonmetals such as chlorine and iodine gain electrons to form anions.

• Electron affinity values are typically negative, reflecting energy release during electron gain.
• In the Born-Haber cycle, chlorine gains two electrons, releasing energy proportionate to twice its electron affinity, whereas iodine gains one electron.

These energy terms compensate partially for the energy needed to ionize metals and dissociate bonds, playing a crucial role in balancing the Born-Haber cycle equation.