Question

A textbook on chemical thermodynamics states, "The heat of solution represents the difference between the lattice energy of the crystalline solid and the solvation energy of the gaseous ions." (a) Draw a simple energy diagram to illustrate this statement. (b) A salt such as NaBr is insoluble in most polar nonaqueous solvents such as acetonitrile \(\left(\mathrm{CH}_{3} \mathrm{CN}\right)\) or nitromethane \(\left(\mathrm{CH}_{3} \mathrm{NO}_{2}\right),\) but salts of large cations, such as tetramethylammonium bromide \(\left[\left(\mathrm{CH}_{3}\right)_{4} \mathrm{NBr}\right],\) are generally more soluble. Use the thermochemical cycle you drew in part (a) and the factors that determine the lattice energy (Section 8.2 ) to explain this fact.

Short answer

The energy diagram illustrates the heat of solution as the difference between lattice energy and solvation energy. Lattice energy depends on the charges of ions and the distance between them. In the case of NaBr and \(\ce{[(CH3)4NBr]}\), both salts have the same anion and charge, but different cations. The larger cation (\(\ce{(CH3)4N+}\)) in \(\ce{[(CH3)4NBr]}\) results in a lower lattice energy and a more favorable solvation energy, making it more soluble in polar nonaqueous solvents like acetonitrile or nitromethane.

Step-by-step solution

(a) Energy diagram illustration

The energy diagram should show that the heat of solution comes from the difference between the lattice energy, causing the crystals to separate into ions, and the solvation energy, which is the energy released when a gaseous ion is solvated (surrounded by solvent molecules). To draw the diagram: 1. Draw an x-axis with labels indicating the two processes: lattice energy and solvation energy. Mark two points on the x-axis for these two processes. 2. Draw an arrow from the base horizontal line (lattice energy) to a point above it, and label the difference as "Heat of Solution." 3. Draw another arrow from the end point of the lattice energy arrow to a position right above "Solvation Energy." Label the difference as "-ΔH_sol," which is the negative of the solvation energy. 4. The overall arrow (combination of the lattice energy and solvation energy arrows) represents the heat of solution for the process.

(b) Solubility explanation using thermochemical cycle and lattice energy factors

The lattice energy of a compound depends on the charges on the ions and the distance between them (Section 8.2). For this case, we have two salts: NaBr and \(\ce{[(CH3)4NBr]}\). Both salts have the same anion (Br-) but different cations (Na+ and \(\ce{(CH3)4N+}\)). The factors determining lattice energy are: 1. Charge of the ions: Since both salts have the same charge, this factor remains constant. 2. Distance between ions: The size of the cations is the major difference between these two salts. The \(\ce{(CH3)4N+}\) ion is larger than the Na+ ion. The larger distance between the ions in \(\ce{[(CH3)4NBr]}\) results in a lower lattice energy compared to NaBr, making it easier to separate into ions. Now, consider the solvation process. For solubility to increase, solvation energy should be favorable (negative and large in magnitude). Large cations generally have a greater solvation energy because they can interact better with polar solvents. Comparing these two factors, \(\ce{[(CH3)4NBr]}\) will have a smaller lattice energy and a more favorable solvation energy compared to NaBr, making it more soluble in polar nonaqueous solvents such as acetonitrile or nitromethane.

Key concepts

Chemical Thermodynamics

Chemical thermodynamics is a branch of physical chemistry that deals with the relationship between heat, work, temperature, and chemical reactions or physical changes. It allows scientists and engineers to calculate the energy changes that accompany these processes.

Understanding the heat of solution is part of this field and involves a balance between energy absorbed and released. The absorbed energy is needed to break the lattice structure of a solid (lattice energy), while energy is released when the resulting ions are surrounded by solvent molecules (solvation energy). Thus, the heat of solution reflects the interplay of these opposing energy transformations.

Lattice Energy

Lattice energy is the energy required to separate a mole of an ionic solid into its constituent ions in the gaseous state. It's a crucial concept when considering ionic compounds because it reflects the strength of the bonds holding the ions in the solid lattice structure.

The higher the charge and the smaller the size of the ions, the greater the lattice energy. This is because smaller, highly charged ions can get closer together, increasing the electrostatic attraction between them. For our NaBr versus \(\ce{[(CH3)4NBr]}\) comparison, although both have ions with the same charge, the larger size of \(\ce{(CH3)4N+}\) results in a lower lattice energy.

Solvation Energy

Solvation energy, sometimes referred to as hydration energy when water is the solvent, is the energy change associated with the dispersal and interaction of a solute's ions with the solvent molecules. A crucial factor determining solvation energy is the solvent's polarity, which must align well with the solute's ionic character for solvation to be energetically favorable.

In our case, large cations like \(\ce{(CH3)4N+}\) have lower lattice energies and can more readily interact with polar solvents, leading to significant solvation energy—which helps us understand the solubility behavior of the salts in question.

Energy Diagram

An energy diagram is a visual representation that helps us understand the energy changes during a chemical reaction or process. For the heat of solution, the diagram would showcase the initial energy state of the solid (indicating high lattice energy), then the end state where the ions are solvated (showing solvation energy), and the path between these states.

The overall change along this path represents the heat of solution. It is essential to depict such processes clearly to help students visualize and grasp the underpinning thermodynamics.

Thermochemical Cycle

A thermochemical cycle involves a sequence of steps that describe the breaking and formation of bonds in a chemical reaction, which allows for the indirect determination of energy changes. It's akin to a 'chemical equation' for energy. For instance, when calculating the heat of solution, one can envisage a cycle that includes the sublimation of the ionic solid, the dissociation of the gaseous ions, and their solvation.

This approach offers a comprehensive view of the energy exchanges involved in dissolving an ionic compound and can explain variations in solubility based on lattice and solvation energies.

Ionic Solubility

Ionic solubility is the degree to which an ionic compound can dissolve in a solvent, producing a homogeneous mixture or solution. This concept is directly influenced by the compound's lattice energy and the solvation energy provided by the solvent.

A lower lattice energy and a high solvation energy favor dissolution, as we've seen with tetramethylammonium bromide compared to sodium bromide. Thus, understanding these underlying thermochemical concepts helps predict and rationalize the solubility of ionic compounds in different solvents.