The solubility of \(\mathrm{NaCl}(\mathrm{s})\) in water at \(298 \mathrm{~K}\) is about 6 moles per litre. Suppose you add 1 mole of \(\mathrm{NaCl}(\mathrm{s})\) to a litre of water. For the reaction: \(\mathrm{NaCl}(\mathrm{s})+\mathrm{H}_{2} \mathrm{O}\) \(\rightarrow \mathrm{NaCl}(\mathrm{aq})\) (a) \(\Delta G>0, \Delta S>0\) (b) \(\Delta G<0, \Delta S>0\) (c) \(\Delta G>0, \Delta S<0\) (d) \(\Delta G<0, \Delta S<0\)

The concept of solubility is pivotal in understanding how substances interact in solutions. Solubility calculations involve determining how much of a substance (the solute) can be dissolved in a given amount of solvent (like water) at a specific temperature before the solution becomes saturated. For example, the solubility of table salt (\r\(\mathrm{NaCl}(\mathrm{s})\)) in water at room temperature (298 K) is approximately 6 moles per liter.\r

\rThis means that you can dissolve up to 6 moles of salt in a liter of water before the solution can't hold any more, also known as the point of saturation. If you add less than 6 moles, as in our exercise where we add just 1 mole, all of the salt should dissolve completely. In practice, solubility calculations also account for other factors like the presence of other solutes and the intermolecular forces at play.

Gibbs Free Energy (\r\(\Delta G\)) is an invaluable term in thermodynamics that helps predict the spontaneity of a process. A negative \r\(\Delta G\) indicates that the process can occur spontaneously under constant pressure and temperature. This quantity combines the system's enthalpy (\r\(\Delta H\)) and entropy (\r\(\Delta S\)), along with temperature (\r\(T\)), in the following relationship: \r\(\Delta G = \Delta H - T\Delta S\).

\rIn the context of dissolving a solid like \r\(\mathrm{NaCl}\) in water, we're considering the Gibbs free energy of solvation. If this value is negative, it means that the solute will dissolve spontaneously, as it is thermodynamically favorable. This has direct practical implications, for example in predicting how salts will behave when mixed with water, whether in cooking or in industrial processes.

Entropy (\r\(\Delta S\)) reflects the disorder or randomness of a system. When a solid dissolves in a liquid like water, the ordered structure of the solid breaks down as the ions disperse throughout the solvent, leading to an increase in entropy (\r\(\Delta S > 0\)).

\rThe higher entropy reflects the increased freedom of movement for the individual particles once they are not part of the solid crystal lattice. The significance of entropy changes lies in their predictive power for spontaneous reactions. Reactions that increase the disorder of the universe (positive \r\(\Delta S\)) are often, though not always, spontaneous—especially if they result in a negative Gibbs free energy when taking temperature into account.\r

\rUnderstanding entropy can help predict and explain the behavior of dissolving processes, which is crucial for fields ranging from pharmaceuticals to environmental science.

A saturated solution represents a point of equilibrium between the dissolved ions and the undissolved solute. This occurs when the maximum amount of solute has been dissolved in the solvent at a specific temperature, and any additional solute will not dissolve. At this point, the rate at which the solute particles dissolve equals the rate at which they crystallize out of solution.\r

\rThe concept is essential in many applications such as cooking, where saturation determines the taste and texture of food, or in pharmaceuticals, where it influences the effectiveness of drugs. In our exercise, since we're adding less solute (1 mole of \r\(\mathrm{NaCl}\)) than the solubility limit (6 moles per liter), the created solution is unsaturated—a scenario where it could further dissolve more salt should it be added.