Question

Use the following data to estimate \(\Delta H_{\mathrm{f}}^{\circ}\) for potassium chloride. $$ \mathrm{K}(s)+\frac{1}{2} \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{KCl}(s) $$ Lattice energy \(-690 . \mathrm{kJ} / \mathrm{mol}\) Ionization energy for \(\mathrm{K}\) \(419 \mathrm{~kJ} / \mathrm{mol}\) Electron affinity of \(\mathrm{Cl} \quad-349 \mathrm{~kJ} / \mathrm{mol}\) Bond energy of \(\mathrm{Cl}_{2}\) \(239 \mathrm{~kJ} / \mathrm{mol}\) \(\begin{array}{ll}\text { Enthalpy of sublimation for } \mathrm{K} & \text { 90. } \mathrm{kJ} / \mathrm{mol}\end{array}\)

Short answer

The standard enthalpy of formation for potassium chloride (KCl) can be estimated using the Born-Haber cycle and the given data. By considering the five steps involved in the formation process and summing their respective energies, we find that \(\Delta H_{\mathrm{f}}^{\circ} = (-201)\,\mathrm{kJ/mol}\).

Step-by-step solution

The Born-Haber cycle is an energy cycle used to analyze the formation of an ionic compound from its constituent elements, which are in their standard states. It correlates different types of energies involved in the formation process to the enthalpy of formation \(\Delta H_{\mathrm{f}}^{\circ}\), so we can calculate it using the provided data. #Step 2: Write the equations for all involved processes#

To estimate the enthalpy of formation, we need to consider five steps: 1. Sublimation of solid potassium into gaseous potassium atoms (Enthalpy of sublimation for Potassium) \( \mathrm{K}(s) \longrightarrow \mathrm{K}(g) \) 2. Dissociation of chlorine molecule into chlorine atoms (Bond energy of Cl) \(\frac{1}{2} \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{Cl}(g)\) 3. Ionization of potassium atom (Ionization energy for K) \(\mathrm{K}(g) \longrightarrow \mathrm{K}^{+}(g) + e^{-}\) 4. Formation of chloride ion (Electron affinity of Cl) \(\mathrm{Cl}(g) + e^{-} \longrightarrow \mathrm{Cl}^{-}(g)\) 5. Formation of ionic compound (Lattice energy) \(\mathrm{K}^{+}(g) + \mathrm{Cl}^{-}(g) \longrightarrow \mathrm{KCl}(s)\) #Step 3: Calculate the enthalpy of reaction (the sum of all steps)#

Using the given values, we can now calculate the enthalpy of reaction: \(\Delta H_{\mathrm{f}}^{\circ} = \Delta H_{\text{sublimation}} + \Delta H_{\text{bond}} + \Delta H_{\text{ionization}} + \Delta H_{\text{electron affinity}} + \Delta H_{\text{lattice}} \) #Step 4: Substitute the values and calculate#

Insert the given values into the equation: \(\Delta H_{\mathrm{f}}^{\circ} = 90\,\mathrm{kJ/mol} + 239\,\mathrm{kJ/mol} + 419\,\mathrm{kJ/mol} - 349\,\mathrm{kJ/mol} - 690\,\mathrm{kJ/mol} \) Now, calculate the enthalpy of formation: \(\Delta H_{\mathrm{f}}^{\circ} = (-201)\,\mathrm{kJ/mol}\) Finally, the standard enthalpy of formation for potassium chloride is estimated to be -201 kJ/mol.

Key concepts

Enthalpy of Formation

Enthalpy of formation, symbolized as \( \Delta H_{\mathrm{f}}^{\circ} \), is the change in enthalpy during the formation of 1 mole of a compound from its elements in their standard states. This value gives insights into the stability of a compound.
In the Born-Haber cycle context, the enthalpy of formation is calculated by summing various forms of energy required or released in forming an ionic compound. This includes sublimation, bond dissociation, ionization, electron affinity, and lattice energy.
For our example of potassium chloride (KCl), the enthalpy of formation is calculated by applying these energy values to see what total energy change occurs as the elements form the compound. It's a vital metric in thermochemistry as it helps chemists predict whether a reaction will be endothermic or exothermic.

Lattice Energy

Lattice energy is the energy released when oppositely charged gaseous ions come together to form an ionic solid. It reflects the strength of the bonds in the ionic lattice.
In the Born-Haber cycle, lattice energy is a crucial component. It essentially represents the energy required to reverse the compound back into gaseous ions, which makes it a negative value, indicating energy release.

• A higher magnitude of lattice energy suggests a more stable compound.
• For potassium chloride, the given lattice energy of \(-690\,\text{kJ/mol}\) indicates substantial energy is released during the crystal formation.

Ionization Energy

Ionization energy is the energy required to remove an electron from a gaseous atom or ion. It's a measure of how strongly an atom holds onto its electrons.
In the formation of KCl, potassium (K) undergoes ionization to become \(\text{K}^+\). The ionization energy value for potassium provided, \(419\,\text{kJ/mol}\), is needed to remove one electron from each atom of potassium.
The ionization energy is a positive value since energy must be supplied to overcome the attraction between the electron and the nucleus.

Electron Affinity

Electron affinity is the energy change that occurs when an electron is added to a gaseous atom, forming an ion. It indicates how much an atom "wants" an additional electron.
For \(\text{Cl}\) in the Born-Haber cycle, it gains an electron to become \(\text{Cl}^-\). The electron affinity is negative, reflecting the energy released when the electron is accepted.

• In our example, the electron affinity of chlorine is \(-349\,\text{kJ/mol}\).
• This energy release contributes to the overall stability of the ionic bond formed between \(\text{K}^+\) and \(\text{Cl}^-\).

Bond Energy

Bond energy, sometimes known as bond dissociation energy, is the energy required to break a bond between two atoms in a molecule. It is crucial in understanding the energy changes when bonds are formed or broken.
In the Born-Haber cycle for KCl, only half of a chlorine molecule (\(\text{Cl}_2\)) dissociates into chlorine atoms, demanding half the bond energy.

• The given bond energy for \(\text{Cl}_2\) is \(239\,\text{kJ/mol}\), representing the energy to dissociate one mole of the bonds.
• Therefore, for half a mole (\(\frac{1}{2}\text{Cl}_2\)), only \(119.5\,\text{kJ/mol}\) would be required.

Sublimation

Sublimation entails the phase change where a solid converts directly into a gas without passing through the liquid state. For metals like potassium, this is the crucial first step in the Born-Haber cycle.
Sublimation energy, or enthalpy of sublimation, is the energy needed for this process.

• In our exercise, the enthalpy of sublimation for potassium is \(90\,\text{kJ/mol}\).
• This figure represents the energy invested to transition the solid potassium to gaseous potassium atoms.

This process is necessary to have isolated gaseous atoms ready for ionization and subsequent reactions leading to compound formation.