Question

Write the Born-Haber cycle for the formation of crystalline sodium fluoride \(\left(\mathrm{Na}^{+} \mathrm{F}^{-}\right)\), starting with solid \(\mathrm{Na}\) and gaseous F. Then, using the thermochemical data supplied below, determine its heat of formation: (1) \(\mathrm{Na}(\mathrm{s}) \quad \rightarrow \mathrm{Na}(\mathrm{g}) \quad \Delta \mathrm{H}=+26.0 \mathrm{Kcal}:\) sublimation (2) \(\mathrm{F}_{2}(\mathrm{~g}) \quad \rightarrow 2 \mathrm{~F}(\mathrm{~g})\) \(\Delta \mathrm{H}=+36.6 \mathrm{~K}\) cal : dissociation (3) \(\mathrm{Na}(\mathrm{g}) \rightarrow \mathrm{Na}^{+}(\mathrm{g})+\mathrm{e}^{-} \quad \Delta \mathrm{H}=+120.0 \mathrm{~K}\) cal : ionization (4) \(\mathrm{F}(\mathrm{g})+\mathrm{e}^{-} \rightarrow \mathrm{F}^{-}(\mathrm{g})\) \(\Delta \mathrm{H}=-83.5 \mathrm{Kcal}:\) electron addition (5) \(\mathrm{Na}^{+}(\mathrm{g})+\mathrm{F}^{-}(\mathrm{g}) \rightarrow \mathrm{Na}^{+}, \mathrm{F}^{-}(\mathrm{s}) \Delta \mathrm{H}=-216.7 \mathrm{Kcal}\) : lattice formation.

Short answer

The Born-Haber cycle for the formation of crystalline sodium fluoride (Na⁺F⁻) from solid Na and gaseous F is given by the following steps: (1) Na(s) ⟶ Na(g) ΔH = +26.0 kcal/mol (sublimation) (2) ½F₂(g) ⟶ F(g) ΔH = ½(+36.6 kcal/mol) (dissociation) (3) Na(g) ⟶ Na⁺(g) + e⁻ ΔH = +120.0 kcal/mol (ionization) (4) F(g) + e⁻ ⟶ F⁻(g) ΔH = -83.5 kcal/mol (electron addition) (5) Na⁺(g) + F⁻(g) ⟶ Na⁺F⁻(s) ΔH = -216.7 kcal/mol (lattice formation) By summing up the ΔH values, we can calculate the heat of formation for Na⁺F⁻ as follows: ΔH_formation = -117.4 kcal/mol

Step-by-step solution

Thermochemical Equations

Write all the thermochemical equations, as given in the exercise: (1) Na(s) ⟶ Na(g) ΔH = +26.0 kcal/mol (sublimation) (2) F₂(g) ⟶ 2F(g) ΔH = +36.6 kcal/mol (dissociation) (3) Na(g) ⟶ Na⁺(g) + e⁻ ΔH = +120.0 kcal/mol (ionization) (4) F(g) + e⁻ ⟶ F⁻(g) ΔH = -83.5 kcal/mol (electron addition) (5) Na⁺(g) + F⁻(g) ⟶ Na⁺F⁻(s) ΔH = -216.7 kcal/mol (lattice formation) Step 2: Write the desired equation for the formation of Na⁺F⁻

Desired Equation

We want to determine the heat of formation for Na⁺F⁻(s) from Na(s) and F(g). The desired equation can be written as: Na(s) + ½F₂(g) ⟶ Na⁺F⁻(s) Step 3: Sum the thermochemical equations to obtain the desired equation

Sum the Equations

Combining equations (1), (2), (3), (4), and (5) should give the desired equation: (1) Na(s) ⟶ Na(g) (2) ½F₂(g) ⟶ F(g) (3) Na(g) ⟶ Na⁺(g) + e⁻ (4) F(g) + e⁻ ⟶ F⁻(g) (5) Na⁺(g) + F⁻(g) ⟶ Na⁺F⁻(s) Add all the equations to get: Na(s) + ½F₂(g) ⟶ Na⁺F⁻(s) Step 4: Calculate the heat of formation

Heat of Formation Calculation

To calculate the heat of formation, we sum the ΔH values of each of the thermochemical equations: ΔH_formation = ΔH_sublimation + ΔH_dissociation + ΔH_ionization + ΔH_electron_addition + ΔH_lattice_formation ΔH_formation = (+26.0 kcal/mol) + (0.5 × +36.6 kcal/mol) + (+120.0 kcal/mol) + (-83.5 kcal/mol) + (-216.7 kcal/mol) ΔH_formation = -117.4 kcal/mol The heat of formation for sodium fluoride (Na⁺F⁻) is -117.4 kcal/mol.

Key concepts

Thermochemical Equations

Thermochemical equations are essential tools in understanding chemical reactions involving energy changes. They not only describe the chemical change occurring in a reaction but also provide information about the energy exchanged during the process. In the Born-Haber cycle for sodium fluoride, various thermochemical equations illustrate each step needed to form the ionic compound from its constituent elements.
For example, the transformation of solid sodium into gaseous sodium, sublimation, is described by a thermochemical equation indicating an endothermic process requiring +26.0 kcal of energy. Symbolically, this equation is written as:
\[\text{Na(s)} \rightarrow \text{Na(g)}\, \Delta \text{H} = +26.0 \text{ kcal/mol}\]
By using these kinds of equations, scientists can piece together multiple stages of reactions to calculate overall energy changes in processes like forming ionic solids. They are fundamental when calculating the heat of formation, as they compile all individual energy transitions.

Heat of Formation

The heat of formation is a crucial concept in chemistry. It refers to the change in enthalpy when one mole of a compound is formed from its elements in their standard states.
In the Born-Haber cycle exercise, the goal is to determine the heat of formation for crystalline sodium fluoride from sodium and fluorine gas. This involves summing the ΔH values from each thermochemical equation within the cycle.
The formula utilized here is:

• ΔH_formation = ΔH_sublimation + ΔH_dissociation + ΔH_ionization + ΔH_electron_addition + ΔH_lattice_formation

For sodium fluoride, this results in:
\[\Delta H_{formation} = ( +26.0) + (0.5 \times 36.6) + (120.0) + (-83.5) + (-216.7) = -117.4 \text{ kcal/mol}\]
Understanding this concept is vital because it helps evaluate how much energy is released or absorbed when a compound forms, thus illustrating its stability. A negative heat of formation, as for sodium fluoride, indicates an exothermic process, releasing energy and typically yielding stable products.

Sublimation

Sublimation is the process where a solid transforms directly into a gas without passing through a liquid phase. In the context of the Born-Haber cycle, sublimation is one of the initial steps. It involves extracting sodium from its solid state to become gaseous sodium. This particular step is represented by the equation:
\[\text{Na(s)} \rightarrow \text{Na(g)}\, \Delta \text{H} = +26.0 \text{ kcal/mol}\]
Such a phase transition requires energy to overcome the forces holding the atoms in a solid structure, which explains why the process involves an endothermic reaction.
This step is crucial for understanding the energetics of forming ionic compounds from atomic states. It emphasizes how much energy is invested just in getting the starting materials into a form where further interactions, like ionization, can occur as part of a multi-step formation process.

Lattice Energy

Lattice energy is a significant term in ionic chemistry, reflecting the energy released when gas-phase ions form an ionic solid. It is a measure of the attractive forces between oppositely charged ions in a lattice. This energy release is a driving factor for the formation of many ionic compounds.
In the sodium fluoride cycle, this step is depicted as:
\[\text{Na}^+(g) + \text{F}^-(g) \rightarrow \text{NaF(s)}\, \Delta \text{H} = -216.7 \text{ kcal/mol}\]
The negative sign signifies an exothermic reaction, indicating energy is released as the attractive forces between the ions pull them together into a stable lattice structure. A higher lattice energy generally signifies a more stable ionic compound.
The lattice energy plays a crucial role in determining the feasibility and stability of ionic compounds, heavily influencing properties like melting points and solubility. Understanding lattice energy allows chemists to predict and rationalize the behaviors of salts under various conditions.