Question

Born-Haber cycles were used to obtain the first reliable values for electron affinity by considering the EA value as the unknown and using a theoretically calculated value for the lattice energy. Use a Born-Haber cycle for \(\mathrm{KF}\) and the following values to calculate a value for the electron affinity of fluorine: \(\begin{array}{ll}\mathrm{K}(s) \longrightarrow \mathrm{K}(g) & \Delta H^{\circ}=90 \mathrm{~kJ} \\ \mathrm{~K}(g) \longrightarrow \mathrm{K}^{+}(g)+\mathrm{e}^{-} & \Delta H^{\circ}=419 \mathrm{~kJ} \\\ \mathrm{~F}_{2}(g) \longrightarrow 2 \mathrm{~F}(g) & \Delta H^{\circ}=159 \mathrm{~kJ} \\ \mathrm{~K}(s)+\frac{1}{2} \mathrm{~F}_{2}(g) \longrightarrow \mathrm{KF}(s) & \Delta H^{\circ}=-569 \mathrm{~kJ} \\\ \mathrm{~K}^{+}(g)+\mathrm{F}^{-}(g) \longrightarrow \mathrm{KF}(s) & \Delta H^{\circ}=-821 \mathrm{~kJ}\end{array}\)

Short answer

The electron affinity of fluorine is calculated to be \(-336.5 \text{ kJ}\).

Step-by-step solution

Write down the given values

List all the given thermodynamic values:- Sublimation energy of potassium: \(\text{K}(s) \rightarrow \text{K}(g)\) \, \(\Delta H^{\circ}=90 \text{ kJ}\)- Ionization energy of potassium: \(\text{K}(g) \rightarrow \text{K}^{+}(g)+\text{e}^{-} \) \, \(\Delta H^{\circ}=419 \text{ kJ}\)- Dissociation energy of fluorine: \(\text{F}_2(g) \rightarrow 2\text{F}(g) \) \, \(\Delta H^{\circ}=159 \text{ kJ}\)- Formation of potassium fluoride: \(\text{K}(s)+\dfrac{1}{2} \text{F}_2(g) \rightarrow \text{KF}(s)\) \, \(\Delta H^{\circ}=-569 \text{ kJ}\)- Lattice energy of potassium fluoride: \(\text{K}^{+}(g)+\text{F}^{-}(g) \rightarrow \text{KF}(s)\) \, \(\Delta H^{\circ}=-821 \text{ kJ}\)

Set up the Born-Haber cycle equation

The Born-Haber cycle involves multiple steps to find the electron affinity (EA) of fluorine. The overall reaction can be written as:\[\Delta H_{\text{f}}^{\circ} = \Delta H_{\text{sub}} + IE + \dfrac{1}{2} \Delta H_{\text{diss}} + EA + \Delta H_{\text{lattice}}\]Re-arranging to find EA:\[EA = \Delta H_{\text{f}}^{\circ} - (\Delta H_{\text{sub}} + IE + \dfrac{1}{2} \Delta H_{\text{diss}} + \Delta H_{\text{lattice}})\]

Substitute the values

Substitute the given values into the equation:\[EA = -569 \text{ kJ} - (90 \text{ kJ} + 419 \text{ kJ} + \dfrac{1}{2} \times 159 \text{ kJ} - 821 \text{ kJ})\]Calculate each term inside the parenthesis first.

Simplify and solve

Calculate the inside terms:\[90 + 419 + \dfrac{1}{2} \times 159 - 821 = 90 + 419 + 79.5 - 821 = -232.5 \text{ kJ}\]Then,\[EA = -569 \text{ kJ} - (-232.5 \text{ kJ}) = -569 \text{ kJ} + 232.5 \text{ kJ} = -336.5 \text{ kJ}\]

Key concepts

Born-Haber cycle calculation overview

The Born-Haber cycle is an approach to analyze the formation of ionic compounds. It connects lattice energy, ionization energy, electron affinity, sublimation energy, and bond dissociation energy. This cycle helps in calculating an unknown thermodynamic value when other values are given. For instance, the electron affinity of fluorine can be calculated using the Born-Haber cycle involving the creation of potassium fluoride (KF). Knowing all the energies allows students to understand the energetic changes during the formation of ionic compounds.

Born-Haber cycle

The Born-Haber cycle is a thermochemical cycle that breaks down the formation of an ionic compound into simpler steps. It uses Hess's Law to link these steps, ensuring that the total energy change is the same regardless of the route taken. This cycle includes several key steps:
- Sublimation of the metal
- Ionization of the metal atom
- Dissociation of the non-metal molecule
- Formation of the non-metal ion
- Combination of ions to form the ionic lattice
By summing the energies for these steps, we obtain the enthalpy change of formation for the ionic compound.

Electron affinity

Electron affinity is defined as the energy change when an electron is added to a neutral atom in the gaseous state to form a negative ion. It generally releases energy, making it exothermic. For fluorine, the electron affinity is significant due to its high electronegativity and small atomic radius. The Born-Haber cycle allows us to determine this value indirectly by considering the other energies involved in forming the compound.

Lattice energy

Lattice energy is the energy released when oppositely charged ions in the gas phase come together to form a solid ionic lattice. It is a measure of the strength of bonds in an ionic compound. The more negative the lattice energy, the more stable the ionic solid. Lattice energy is crucial in the Born-Haber cycle because it connects the formation of individual ions with the overall ionic compound.

Thermodynamic values

Thermodynamic values refer to the measurable quantities of energy changes in chemical processes, including enthalpy changes. In the Born-Haber cycle, these values include:

- Sublimation energy
- Ionization energy
- Dissociation energy
- Electron affinity
- Lattice energy
Understanding these values helps in calculating unknowns, like electron affinity, and analyzing the energetics of compound formation.

Sublimation energy

Sublimation energy is the energy required to convert a solid element directly into its gas phase. For example, for potassium, the sublimation energy of 90 kJ is the energy needed to turn solid potassium (K) into potassium gas (K(g)). This step is the beginning of the Born-Haber cycle as it prepares the metal for subsequent ionization.

Ionization energy

Ionization energy is the energy required to remove an electron from a neutral atom in the gaseous state to form a positive ion. In the case of potassium, the ionization energy is 419 kJ, converting potassium gas (\text{K(g)}) to its ion (\text{K\(^+\)(g)}). This energy input is necessary to form a positively charged ion, which later combines with a negatively charged ion to create the ionic compound.

Dissociation energy

Dissociation energy is the energy needed to break a molecule into its constituent atoms. For fluorine, the dissociation energy is 159 kJ for converting \text{F\(_2\)(g)} into 2F(g). This value, divided by two for the Born-Haber cycle, represents the energy required to obtain a single fluorine atom in the gas phase, which then gains an electron to form a fluoride ion.

Enthalpy of formation

Enthalpy of formation is the change in enthalpy when one mole of a compound forms from its elements in their standard state. For potassium fluoride (KF), this value is -569 kJ. This negative value indicates an exothermic process, releasing energy as the ionic compound forms. The Born-Haber cycle uses this value to determine the steps and energies involved in forming an ionic solid from its elements.