Problem 83 - Chemistry The Central Science

Question

Construct a Born-Haber cycle for the formation of the hypothetical compound \(\mathrm{NaCl}_{2}\), where the sodium ion has a \(2+\) charge (the second ionization energy for sodium is given in Table 7.2). (a) How large would the lattice energy need to be for the formation of \(\mathrm{NaCl}_{2}\) to be exothermic? (b) If we were to estimate the lattice energy of \(\mathrm{NaCl}_{2}\) to be roughly equal to that of \(\mathrm{MgCl}_{2}\) ( \(2326 \mathrm{~kJ} / \mathrm{mol}\) from Table 8.2 ), what value would you obtain for the standard enthalpy of formation, \(\Delta H_{j}^{9}\), of \(\mathrm{NaCl}_{2}\) ?

Step-by-Step Solution

Answer

The hypothetical compound NaCl$_2$ has a Born-Haber cycle consisting of the formation of gaseous sodium ions (using the first and second ionization energies of sodium), the formation of gaseous chloride ions (using the electron affinity of chlorine), and the lattice energy for the formation of NaCl$_2$ solid. For the formation to be exothermic, the lattice energy must exceed the sum of the ionization energies and the electron affinity. Assuming NaCl$_2$ has a lattice energy similar to that of MgCl$_2$ (2326 kJ/mol), the standard enthalpy of formation, ΔH$_f^{\circ}$, can be estimated by summing the energy changes for all processes in the Born-Haber cycle.

Step 1: Determine the Ionization Energies of Sodium and Chlorine

For sodium, since it has a 2+ charge, we have to consider its second ionization energy which is provided in Table 7.2. For chlorine, we will only consider its first ionization energy since it has a -1 charge. Find these values and note them down.

Step 2: Determine the Electron Affinity of Chlorine

Next, we have to find the electron affinity of chlorine which will be the energy released when it gains an electron. This value can be found using reference tables. Note down the electron affinity value.

Step 3: Write the Formation Reaction for NaCl_2

Write down the complete formation reaction for NaCl_2 from its constituent elements in their standard states: Na(s) + 2 Cl(g) -> Na^2+(g) + 2 Cl^-(g) -> NaCl_2(s)

Step 4: Write the Born-Haber Cycle for NaCl_2

With all the information gathered from the previous steps, we can now construct the Born-Haber cycle for NaCl_2: (a) Formation of gaseous sodium ions: Na(s) -> Na^+(g) + e^- (first ionization energy) Na^+(g) -> Na^2+(g) + e^- (second ionization energy) (b) Formation of gaseous chloride ions: Cl(g) + e^- -> Cl^-(g) (Electron Affinity) (c) Lattice Energy: Formation of NaCl_2(s): Na^2+(g) + 2 Cl^-(g) -> NaCl_2(s) (Lattice Energy)

Step 5: Calculate the Energy Required for the Formation of NaCl_2 to be Exothermic

For the formation of NaCl_2 to be exothermic, the lattice energy must be larger than the sum of the ionization energies of sodium and the electron affinity of chlorine. Calculate the net energy for these processes and determine the minimum value for the lattice energy for NaCl_2 formation to be exothermic.

Step 6: Estimate the Standard Enthalpy of Formation Using the Approximation

Estimate the lattice energy of NaCl_2 by assuming it to be equal to that of MgCl_2, which is 2326 kJ/mol. Use this value to calculate the standard enthalpy of formation, ΔH_f°, of NaCl_2 by summing up the energy changes for all processes in the Born-Haber cycle.

Key Concepts

Ionization Energy

Ionization energy is a cornerstone concept in chemistry, representing the energy required to remove an electron from an atom or ion in its gaseous state. It's a process that lies at the heart of the formation of ions – crucial for creating ionic compounds like salts.

Consider sodium (Na), for example. To transition from a neutral atom to a positively charged ion, sodium must lose one or more electrons – that's where ionization energy comes into play. For sodium to form Na^2+, you must account for its first and second ionization energies. The first ionization energy involves removing the initial electron, while the second is required to remove the next. Typically, the second ionization energy is higher, as it's harder to remove an electron from a positively charged ion.

To put it succinctly, sodium's leap toward a 2+ charge in the formation of NaCl_2 is a two-step process, involving two distinct bouts of energy to bid farewell to its electrons.

Electron Affinity

Electron affinity complements the picture of ion formation, but in reverse. It represents the energy released when an atom in its gaseous state gains an electron to become an anion. It's the atom's 'affinity' for electrons – the more an atom 'wants' an additional electron, the more energy is released in the process.

In our example with chlorine (Cl), it's eager to gain an electron to achieve a stable electronic configuration, forming Cl^-. Thus, the electron affinity of chlorine becomes a pivotal factor in the formation of NaCl_2. This addition of an electron not only stabilizes chlorine but also contributes energy back into the system which counterbalances part of the ionization energy of sodium.

Lattice Energy

Lattice energy is the energy released when ions bind together to form a crystalline lattice – it is the glue that holds an ionic solid together. This energy is a direct reflection of the strength of the bonds between the ions in the solid state. Higher lattice energy correlates with a more stable, lower-energy crystal structure.

For NaCl_2 to be energetically viable, or exothermic, the lattice formation must release enough energy to overcome the input required to ionize the sodium and to account for the electron affinity of chlorine. Think of it as a battle of energies, where the lattice energy must triumph over the sum of the endothermic processes for NaCl_2 to come together in harmony without requiring external energy.

Enthalpy of Formation

The enthalpy of formation, denoted as ΔH_f° , is a measure of the total change in enthalpy that occurs when one mole of a compound is formed from its elements in their standard states. It's integral to understanding the overall energy dynamics of a compound's genesis.

In the case of hypothetical NaCl_2, by assuming its lattice energy is akin to that of MgCl_2, we can predict the enthalpy of formation. This calculation would weave together the ionization energies, electron affinity, and lattice energy from the Born-Haber cycle into a single value. If the resulting ΔH_f° is negative, the formation of NaCl_2 is exothermic, essentially indicating the compound forms spontaneously and releases energy in the process.