Problem 81 - Chemistry and Chemical Reactivity

Question

Explain why the entropy of the system increases on dissolving solid NaCl in water \(| S^{\circ}[\mathrm{NaCl}(\mathrm{s})]=72.1 \mathrm{J} / \mathrm{K} \cdot\) mol and \(\left.S^{0}[\mathrm{NaCl}(\mathrm{aq})]=115.5 \mathrm{J} / \mathrm{K} \cdot \mathrm{mol}\right]\)

Step-by-Step Solution

Answer

The entropy increases by 43.4 J/K·mol when NaCl dissolves, indicating greater disorder in the system.

Step 1: Understanding Entropy

Entropy, denoted as \( S \), is a measure of the disorder or randomness of a system. The greater the randomness, the higher the entropy of the system.

Step 2: Identifying Initial and Final States

In this problem, we consider the system before and after the dissolution of NaCl. Initially, the system contains solid NaCl, which has an entropy of \( S^{ullet}[ ext{NaCl(s)}] = 72.1 ext{ J/K·mol} \). When NaCl is dissolved in water, it becomes aqueous NaCl, with an entropy of \( S^{ullet}[ ext{NaCl(aq)}] = 115.5 ext{ J/K·mol} \).

Step 3: Calculating Change in Entropy

The change in entropy, \( \Delta S \), is calculated as the difference between the final entropy and the initial entropy. Mathematically, this is expressed as \( \Delta S = S^{ullet}[ ext{NaCl(aq)}] - S^{ullet}[ ext{NaCl(s)}] = 115.5 ext{ J/K·mol} - 72.1 ext{ J/K·mol} \).

Step 4: Result of the Calculation

Perform the subtraction: \( 115.5 ext{ J/K·mol} - 72.1 ext{ J/K·mol} = 43.4 ext{ J/K·mol} \). This result indicates the change in entropy.

Step 5: Interpretation of Increased Entropy

The positive change in entropy, \( 43.4 ext{ J/K·mol} \), indicates that the system has increased in disorder. The dissociation of NaCl into ions increases the number of possible microstates, hence increasing the entropy of the system. Thus, dissolving a solid in a solvent generally leads to an increase in entropy due to increased freedom of movement for the dissolved particles.

Key Concepts

Dissolution

Dissolution refers to the process by which a solute, such as sodium chloride (NaCl), dissolves in a solvent, like water, forming a solution. This is a physical change where the ionic bonds in NaCl are broken and individual sodium (Na⁺) and chloride (Cl^-) ions are surrounded by water molecules.

During dissolution, interactions occur between the solute particles and solvent molecules. In the case of NaCl, water molecules play a critical role due to their polar nature, effectively interacting with and stabilizing the ions in the solution.

Understanding dissolution enables us to better grasp the changes occurring on a molecular level, such as increased dispersal of energy within the system, which is a key aspect of how entropy is affected during this process.

Entropy Change

Entropy change is a crucial concept in thermodynamics, reflecting the measure of disorder or randomness introduced in a system during a process. When discussing the dissolution of NaCl, we observe an increase in entropy.

This can be calculated using the formula:

• \[ \Delta S = S^{0}[\text{NaCl(aq)}] - S^{0}[\text{NaCl(s)}] \]
• Substituting the given values, \( 115.5 \, \text{J/K·mol} - 72.1 \, \text{J/K·mol} = 43.4 \, \text{J/K·mol} \).

This positive value of entropy change signifies the increase in disorder. More microstates are available for the system as NaCl dissociates into sodium and chloride ions, leading to greater energetic freedom and more possibilities for particle arrangement.

This increase in entropy is vital for understanding natural processes where systems tend to move toward states with higher entropy. It is often a driving force behind spontaneous reactions.

Sodium Chloride Dissolution

Sodium chloride dissolution is a key example used to explain the concept of entropy changes in chemical processes. In its solid form, NaCl is well-ordered, with ions positioned in a rigid lattice structure.

Upon dissolving in water, these ions are freed from the lattice, moving independently throughout the solution. This transition from an ordered solid to a disordered solution increases the randomness or entropy of the system.

• Solid NaCl: High structure, low entropy ( \[ S^{0}[\text{NaCl(s)}] = 72.1 \, \text{J/K·mol} \])
• Aqueous NaCl: More freedom, higher entropy \( \Delta S = 43.4 \, \text{J/K·mol} \), indicating increased randomness.

The process exemplifies how dissolution impacts entropy, showcasing how systems naturally evolve to maximize disorder. This concept is central to predicting the behavior of solutes and solvents in chemical reactions and solutions. It helps us appreciate the tendency of systems to disperse energy and achieve equilibrium states with maximum entropy.