Question

Show that the formation of \(\mathrm{NaCl}_{2}(\mathrm{s})\) is very unfavorable; that is, \(\Delta \mathrm{H}_{\mathrm{f}}^{\circ}\left[\mathrm{NaCl}_{2}(\mathrm{s})\right]\) is a large positive quantity. To do this, use data from Section \(12-7\) and assume that the lattice energy for \(\mathrm{NaCl}_{2}\) would be about the same as that of \(\mathrm{MgCl}_{2},-2.5 \times 10^{3} \mathrm{kJ} \mathrm{mol}^{-1}\)

Short answer

The calculation of the enthalpy change of formation for \(\mathrm{NaCl}_{2}(\mathrm{s})\) shows it to be positive, implying that the formation of \(\mathrm{NaCl}_{2}(\mathrm{s})\) is thermodynamically unfavorable.

Step-by-step solution

Calculate the Sum of Enthalpy Changes

The overall enthalpy change for the synthesis of \(\mathrm{NaCl}_{2}(\mathrm{s})\) from its elements can be found by summing up the enthalpy changes for each step. There are three steps. First, the formation of \(\mathrm{Na(g)}\), second, the formation of \(\mathrm{Cl(g)}\), and third, the formation of \(\mathrm{NaCl}_{2}(\mathrm{s})\). From the data in Section 12-7, these are respectively \(\mathrm{\Delta H_f[Na(g)] = +107.3 kJ/mol}\), \(\mathrm{\Delta H_f[Cl(g)] = +122.0 kJ/mol}\), and \(\mathrm{\Delta H_f[NaCl_{2}(s)]}\) is what we are trying to find.

Calculate the enthalpy change of formation for NaCl_2 (s)

Using the energy associated with lattice formation provided, sum up the changes in enthalpy calculated in Step 1, and the given lattice energy for \(\mathrm{MgCl}_{2}\): \[\Delta \mathrm{H}_{\mathrm{f}}^{\circ}\left[\mathrm{NaCl}_{2}(\mathrm{s})\right] = 2\times(\mathrm{\Delta H_f[Na(g)] + \Delta H_f[Cl(g)]}) + -2.5 \times 10^{3} \mathrm{kJ/mol}\]

Determine Favorability

If the calculated \(\mathrm{\Delta H_{f}^{\circ}[NaCl_{2}(s)]}\) is largely positive, then the formation of \(\mathrm{NaCl_{2}(s)}\) is highly unfavorable.

Key concepts

Enthalpy Change of Formation

The enthalpy change of formation, represented by (\text{\(\Delta H_{f}^\circ\)}), is a measure of the heat change during the formation of one mole of a substance from its constituent elements in their standard states. Understanding this concept is essential because it represents the amount of energy released or absorbed during the formation of a compound.
For example, the formation of water from hydrogen and oxygen would have a negative (\text{\(\Delta H_{f}^\circ\)}) since energy is released when water forms. Conversely, if the formation of a compound requires energy input, like in the hypothetical case of (\text{\(NaCl_{2}(s)\)}), the (\text{\(\Delta H_{f}^\circ\)}) would be positive.

To calculate the (\text{\(\Delta H_{f}^\circ\)}) for (\text{\(NaCl_{2}(s)\)}), we need data from prior thermochemical equations or experiments. For instance, the singled out exercise combines the enthalpy changes for forming sodium and chlorine gas, and the hypothetical lattice energy resembling (\text{\(MgCl_{2}\)}). By summing these values, the enthalpy of formation gives us insight as to whether the compound would form naturally under standard conditions—which in this case, with a positive value, indicates it's unlikely to occur.

Thermochemistry

Thermochemistry is the branch of chemistry that deals with the energy changes associated with chemical reactions. Knowing the basics of thermochemistry is crucial for predicting reaction behavior and understanding energy changes on the molecular level.
Enthalpy, a central concept in thermochemistry, represents the heat content of a system at constant pressure. The enthalpy change (\text{\(\Delta H\)}) can indicate whether a process is exothermic, releasing heat, or endothermic, absorbing heat.

In the context of the aforementioned exercise, we are looking at the hypothetical reaction for (\text{\(NaCl_{2}(s)\)}), which would have a positive enthalpy change and be endothermic, meaning it would require an input of energy. Recognizing these thermochemical patterns allows scientists and students alike to predict the viability of chemical processes merely based on the enthalpy changes involved.

Chemical Bonding

Chemical bonding is the physical phenomenon of chemical substances being held together by attraction of atoms to each other through sharing, as well as exchanging, of electrons. The type of chemical bonding can greatly affect the physical properties and stability of the compounds.
Lattice energy is directly related to ionic bonding and reflects the strength of the bonds in an ionic compound's lattice structure. It is the energy released when the ions bond to form a crystalline lattice. A high lattice energy usually implies a stable ionic compound due to strong electrostatic forces between ions.

However, the exercise highlights an important aspect of this concept. Even though high lattice energy suggests stability, not all ionic compounds with a predicted high lattice energy like (\text{\(NaCl_{2}(s)\)}) will form readily. Other energetic factors and reaction pathways contribute to the overall chemical behavior. This context is vital for students as it shows that chemical stability isn't just about lattice energy or bonding types; it is about looking at the bigger picture that includes enthalpy changes and the laws governing thermochemistry.