Determine the lattice energy for CaBr2 if the enthalpy of solution for CaBr2 is -145 kJ/mol, and the heats of hydration for Ca2+ and Br- are -1650 kJ/mol and -292 kJ/mol respectively.

-2162 kJ/mol

To determine the lattice energy for CaBr2, we can use the Born-Haber cycle, which allows us to calculate the lattice energy using the enthalpy of solution and the heats of hydration.

The Born-Haber cycle is a series of steps that represent the formation of an ionic compound from its elements. Here is the breakdown of the steps:

1. Formation of the solid compound:
- Ca(s) + Br2(g) → CaBr2(s)

2. Sublimation of the metal:
- Ca(s) → Ca(g)

3. Dissociation of the diatomic halogen:
- Br2(g) → 2Br(g)

4. Ionization of the metal:
- Ca(g) → Ca+(g) + e-

5. Electron affinity of the non-metal:
- Br(g) + e- → Br-(g)

6. Formation of the gaseous compound:
- Ca+(g) + 2Br-(g) → CaBr2(g)

7. Hydration of the ions:
- Ca+(g) + aq → Ca2+(aq)
- Br-(g) + aq → Br-(aq)

8. Formation of the solid compound from the hydrated ions:
- Ca2+(aq) + 2Br-(aq) → CaBr2(s)

Now, we need to consider the given enthalpies. The enthalpy change for each step in the Born-Haber cycle can be obtained through experimental values or calculated using bond energies and ionization energies.

Given:
- Enthalpy of solution for CaBr2 = -145 kJ/mol
- Heat of hydration for Ca2+ = -1650 kJ/mol
- Heat of hydration for Br- = -292 kJ/mol

To determine the lattice energy, we need to calculate the sum of the enthalpy changes for all the steps. The lattice energy (U) is the negative of the total enthalpy change:

U = -ΔH total

For the steps in the Born-Haber cycle:
ΔH total = ΔH f (CaBr2) + ΔH sub (Ca) + ΔH diss (Br2) + ΔH ion (Ca) + ΔH EA (Br) + ΔH f (CaBr2) + ΔH hyd (Ca2+) + ΔH hyd (Br-) + ΔH f (CaBr2)

Note that the formation of CaBr2 from its elements (step 1) cancels itself out.

Simplifying the equation, we can rewrite it as:
U = - (-ΔH sol - ΔH hyd (Ca2+) - ΔH hyd (Br-))

Plugging in the given values, we get:
U = - (-145 kJ/mol - (-1650 kJ/mol) - (-292 kJ/mol))

Simplifying further:
U = - (-145 kJ/mol + 1650 kJ/mol - 292 kJ/mol)
U = - (-145 + 1650 - 292) kJ/mol
U = - (1213) kJ/mol
U ≈ 1213 kJ/mol

Therefore, the lattice energy for CaBr2 is approximately 1213 kJ/mol.

The lattice energy (ΔHlattice) can be calculated using the Born-Haber cycle, which represents the energy changes involved in the formation of an ionic compound from its constituent elements.

The steps involved in the Born-Haber cycle for CaBr₂ are as follows:

1. Formation of ions:
Ca (s) → Ca²⁺ (g) + 2 e⁻ (sublimation energy) ΔH₁

Br₂ (g) → 2 Br⁻ (g) (bond dissociation energy) ΔH₂

2. Formation of solid compound:
Ca²⁺ (g) + 2 Br⁻ (g) → CaBr₂ (s) (lattice energy) ΔHlattice

3. Hydration of ions:
Ca²⁺ (g) + H₂O (l) → Ca²⁺ (aq) (hydration energy) ΔH₃

Br⁻ (g) + H₂O (l) → Br⁻ (aq) (hydration energy) ΔH₄

4. Dissolving of solid compound:
CaBr₂ (s) → Ca²⁺ (aq) + 2 Br⁻ (aq) (enthalpy of solution) ΔHsolution

According to the Born-Haber cycle, ΔHlattice = -[ΔH₁ + ΔH₂ + ΔH₃ + ΔH₄ - ΔHsolution].

Given:
ΔHsolution = -145 kJ/mol
ΔH₃ = -1650 kJ/mol (hydration energy of Ca²⁺)
ΔH₄ = -292 kJ/mol (hydration energy of Br⁻)

We need to find ΔHlattice.

Let's plug in the values into the formula:

ΔHlattice = -[ΔH₁ + ΔH₂ + ΔH₃ + ΔH₄ - ΔHsolution]

Since we are not given the values for ΔH₁ and ΔH₂, and they are the lattice energy and bond dissociation energy respectively of the constituent elements, we cannot directly calculate them. So, we can assume their values to be zero for simplicity.

Thus, the equation becomes:

ΔHlattice = -[0 + 0 - 1650 - 292 - (-145)]
ΔHlattice = -[-145 + 1650 + 292 + 145]
ΔHlattice = -(1954 + 145)
ΔHlattice = -2099 kJ/mol

Therefore, the lattice energy for CaBr₂ is approximately -2099 kJ/mol.