Question

Crystallization of sodium acetate from a supersaturated solution occurs spontaneously (see Figure 13.2 ). Based on this, what can you deduce about the signs of \(\Delta S\) and \(\Delta H ?\)

Short answer

ΔS is negative and ΔH is negative.

Step-by-step solution

Understand the Process

Crystallization involves the formation of a solid crystal structure from a supersaturated solution. In a supersaturated solution, there's more solute (sodium acetate) than the solvent can normally accommodate. When crystallization occurs, the system moves from a higher energy state (disturbed and disordered) to a lower energy state (ordered crystal formation). This is a spontaneous process.

Analyze the Entropy Change ( ΔS )

Entropy ( ΔS ) is a measure of the disorder in a system. Since the crystallization process involves going from a disordered solution to an ordered crystal, the entropy decreases. Therefore, ΔS for the process would be negative.

Consider the Enthalpy Change ( ΔH )

Enthalpy ( ΔH ) reflects the heat absorbed or released during a process. Since the process is spontaneous even though it decreases entropy, it implies heat is released, which compensates for the decrease in entropy. Thus, ΔH is negative because the process is exothermic.

Apply the Gibbs Free Energy Equation

Using the Gibbs free energy equation ΔG = ΔH - TΔS , for a spontaneous process, ΔG must be negative. Given ΔH is negative and ΔS is negative, ΔG will be negative if the magnitude of ΔH is large enough to compensate for the (- TΔS) term.

Key concepts

Supersaturated Solution

A supersaturated solution is like a ticking time bomb of crystals. It has more solute than the solvent can usually dissolve. Imagine trying to fit extra sugar into your tea when it’s already sweet enough. This solution is unstable. It's ready to release the extra solute in the form of crystals. This transformation can happen quickly when triggered, such as by a dust particle or a seed crystal. When sodium acetate crystallizes from this supersaturated state, it moves from chaos towards order. The solute particles leave the solution to form a neat crystal lattice. This process, going from high energy disorder to low energy order, is called crystallization.

Entropy Change

Entropy measures disorder in a system. Think of it like the messiness of your room. The more cluttered it is, the higher the entropy. During crystallization, things get ordered. The messy, mixed solution of sodium acetate becomes a tidy crystal structure. As a result, entropy decreases during this process. Mathematically, the change in entropy \((\Delta S)\) for such a process becomes negative, since the order in the system increases and the disorder decreases.

Enthalpy Change

Enthalpy is about heat exchange during a process. It's like whether your room gets warmer or cooler. The crystallization of sodium acetate releases heat, making the surrounding space warmer. This release of heat occurs because the bonds forming in the crystal release energy. Hence, this is an exothermic process where the change in enthalpy \((\Delta H)\) is negative. The system loses energy, transitioning into a more stable state.

Gibbs Free Energy

Gibbs free energy helps predict the spontaneity of a process. Imagine it as the energetic currency for a process to occur without external help. The equation \(\Delta G = \Delta H - T\Delta S\) pulls together enthalpy, entropy, and temperature to tell us if a process can just happen. For crystallization, we need \(\Delta G\) to be negative for it to be "free." If \(\Delta H\) is negative (energy is released), and \(\Delta S\) is also negative (system gets ordered), the process can still be spontaneous if the energy released is greater than the energy needed to increase order (\(T\Delta S\) term).

Spontaneous Process

A spontaneous process is like a ball rolling down a hill, happening on its own without a push. When the energy balance favors the shift towards order or stability, it happens naturally. In the case of sodium acetate crystallizing, the process occurs spontaneously. Although entropy decreases, the large amount of heat released (negative \(\Delta H\)) more than compensates. Together, these factors ensure a negative Gibbs free energy \(\Delta G\), signaling that the crystallization takes place without external intervention.