Question

Calculate a lattice energy for \(\mathrm{CaH}_{2}(s)\) in kilojoules per mole using the following information: \(E_{\mathrm{ea}}\) for \(\mathrm{H}(\mathrm{g})=-72.8 \mathrm{~kJ} / \mathrm{mol}\) \(E_{i 1}\) for \(\mathrm{Ca}(g)=+589.8 \mathrm{~kJ} / \mathrm{mol}\) \(E_{i 2}\) for \(\mathrm{Ca}(g)=+1145 \mathrm{~kJ} / \mathrm{mol}\) Heat of sublimation for \(\mathrm{Ca}(s)=+178.2 \mathrm{~kJ} / \mathrm{mol}\) Bond dissociation energy for \(\mathrm{H}_{2}(g)=+435.9 \mathrm{~kJ} / \mathrm{mol}\) Net energy change for the formation of \(\mathrm{CaH}_{2}(s)\) from its elements \(=-186.2 \mathrm{~kJ} / \mathrm{mol}\)

Short answer

The lattice energy of \(\text{CaH}_2(s)\) is approximately \(-2171.55\, \text{kJ/mol}\).

Step-by-step solution

Write the Born-Haber Cycle Equation

To calculate lattice energy, we can use the Born-Haber cycle, which accounts for all of the energy changes in forming an ionic compound from its elements. The lattice energy can be calculated using the equation:\[ \text{Lattice Energy} = \Delta H_f - (\text{Ionization Energies} + \text{Electron Affinities} + \text{Sublimation Energy} + \text{Bond Energy}) \] Thus, for \( \text{CaH}_{2}(s) \), the lattice energy is:\[ \text{Lattice Energy} = \Delta H_f - (E_{\text{sub}} + E_{i1} + E_{i2} + \frac{1}{2}D + 2E_{\text{ea}}) \]

Identify Each Term in the Equation

Identify each of the terms that will be used in the Born-Haber cycle equation:- \( \Delta H_f = -186.2 \text{ kJ/mol} \) (Net formation energy)- \(E_{\text{sub}} = +178.2 \text{ kJ/mol} \) (Heat of sublimation for \(\text{Ca}\))- \(E_{i1} = +589.8 \text{ kJ/mol} \) (First ionization energy for \(\text{Ca}\))- \(E_{i2} = +1145 \text{ kJ/mol} \) (Second ionization energy for \(\text{Ca}\))- \(D = +435.9 \text{ kJ/mol} \) (Bond dissociation energy for \(\text{H}_{2}\))- \(E_{\text{ea}} = -72.8 \text{ kJ/mol} \) (Electron affinity for \(\text{H}\))

Calculate Individual Energy Contributions

Calculate contributions from the terms:- Bond dissociation energy contribution: \(\frac{1}{2} \times 435.9 = 217.95 \; \text{kJ/mol}\) (Since one mole of \(\text{H}_2\) produces two moles of \(\text{H}\))- Total electron affinity: \(2 \times (-72.8) = -145.6 \; \text{kJ/mol}\) (Since we need two electrons for two \(\text{H}^-\) ions)

Substitute and Calculate the Lattice Energy

Insert these energies into the modified Born-Haber equation to solve for the lattice energy:\[ \text{Lattice Energy} = -186.2 - (178.2 + 589.8 + 1145 + 217.95 - 145.6) \]First, calculate the sum of energies inside the parentheses:\[ 178.2 + 589.8 + 1145 + 217.95 - 145.6 = 1985.35 \; \text{kJ/mol} \]Now, subtract this value from \(\Delta H_f\):\[ \text{Lattice Energy} = -186.2 - 1985.35 = -2171.55 \; \text{kJ/mol} \]

Interpret the Result

The calculated lattice energy corresponds to the amount of energy released when gaseous ions form the solid lattice of \(\text{CaH}_2\).

Key concepts

Born-Haber Cycle

The Born-Haber cycle is a thermodynamic cycle used to analyze the formation of ionic compounds. It's named after the scientists, Max Born and Fritz Haber, who developed it. The cycle combines various steps involving energy changes to derive lattice energy, which is the energy required to form or break an ionic lattice.

When using the Born-Haber cycle for a compound like calcium hydride ( CaH_2(s) ), you begin by considering different energy changes from elemental formation to ionic lattice composition. These changes include sublimation of solid metal to gas, bond dissociation of diatomic gases like hydrogen, ionization energies, and electron affinities. All these steps lead to the final formula for calculating lattice energy using the cycle:

• Lattice Energy = ΔH_f - (E_sub + E_{i1} + E_{i2} + rac{1}{2}D + 2E_{ea})

Applying this formula, you account for the total energy required to form ions and the energy released when an ionic solid is created from gaseous ions.

Ionization Energy

Ionization energy is the energy required to remove electrons from an atom in the gas phase, turning it into a cation. When calculating lattice energy, we consider both the first and the second ionization energies, especially for elements like calcium (Ca) which forms a 2+ cation in its compounds.

Each ionization step has a different energy amount associated with it due to increased nuclear charge on the atom that remains.

• The first ionization energy, E_{i1} , relates to removing the first electron from the neutral atom.
• The second ionization energy, E_{i2} , involves removing the second electron, and it is typically higher due to increased positive charge in the ion.

In the Born-Haber cycle of CaH_2 , these energies add to the total energy requirement before forming Ca^{2+} ions. The necessary addition of ionization energies influences overall thermodynamics in the lattice energy calculation.

Electron Affinity

Electron affinity is the energy change when an electron is added to an atom in the gaseous state, forming an anion. It indicates the tendency of an atom to gain an electron. For non-metals like hydrogen, electron affinity is crucial because hydrogen gains one electron to form H^- ions.

As included in the Born-Haber cycle for CaH_2 , electron affinity (EA) has a negative sign because energy is typically released when electrons are added. The cycle requires the sum of electron affinities for each electron needed.

• In the case of CaH_2 , we calculate the electron affinity for two hydrogen atoms: 2E_{ea} .

Thus, the greater the electron affinity and the more energy released, the less energy is needed overall in lattice formation, driving the reaction forward energetically.

Bond Dissociation Energy

Bond dissociation energy is the energy required to break a chemical bond in a molecule, converting it into its individual atoms in the gaseous state. This concept is essential in evaluating the energy necessary to form ionic compounds from molecular precursors.

Specifically, for CaH_2 formation, we consider H_2(g) dissociation. Since each reaction uses one mole of gaseous hydrogen molecules, splitting them into separate hydrogen atoms requires half of the molecular dissociation energy. The formula requires halving the given bond dissociation energy value:

• Calculate: rac{1}{2}D i.e., half of the H_2 dissociation energy contributes to forming one mole of H gas.

This energy is a critical input in the lattice energy calculation using the Born-Haber cycle, where it represents the energy required to produce separate atoms from the bonded diatomic molecule, H_2.