Question

The element gallium (Ga) freezes at \(29.8^{\circ} \mathrm{C},\) and its molar enthalpy of fusion is \(\Delta H_{\text { fus }}=5.59 \mathrm{k} \mathrm{k} / \mathrm{mol}\) . (a) When molten gallium solidifies to Ga(s) at its normal melting point, is \(\Delta S\) positive or negative? (b) Calculate the value of \(\Delta S\) when 60.0 g of Ga(l) solidifies at \(29.8^{\circ} \mathrm{C}\) .

Short answer

(a) When gallium solidifies, the entropy change \(\Delta S\) is negative because it transitions from a more disordered state (liquid) to a more ordered state (solid). (b) The total entropy change when 60.0 g of Ga solidifies at \(29.8^{\circ} \mathrm{C}\) is \(\Delta S_{total} = -15.9\, \text{J/K}\).

Step-by-step solution

Determine the sign of \(\Delta S\) during solidification

When a substance undergoes a phase transition from a more disordered state (such as liquid) to a more ordered state (such as solid), the entropy change \(\Delta S\) is generally negative. This is because moving from a more disordered to a more ordered state means that there is a decrease in the number of microstates or possible arrangements of the particles, leading to a decrease in entropy. In the case of gallium that solidifies, the phase transition is from liquid to solid, so we can say that \(\Delta S\) is negative.

Use the relationship between \(\Delta G\), \(\Delta H\), and \(\Delta S\)

At the equilibrium point for the phase transition (freezing point of the gallium in this case), the Gibbs free energy change \(\Delta G\) is 0. We have the equation: \[\Delta G = \Delta H - T\Delta S\] Since \(\Delta G = 0\) at equilibrium, the equation becomes: \[0 = \Delta H - T\Delta S\]

Solve for \(\Delta S\)

Now, we can solve for \(\Delta S\). Rearrange the equation: \[\Delta S = \frac{\Delta H}{T}\] We are given the molar enthalpy of fusion \(\Delta H_{\text {fus}} = 5.59 \,\text{kJ/mol}\) and the freezing temperature \(T = 29.8^{\circ} \mathrm{C} = 302.8\, \mathrm{K}\) (converted to Kelvin using \( K = degree \,Celsius + 273.15 \)). Plug these values into the equation: \[\Delta S = \frac{5.59 \,\text{kJ/mol}}{302.8\, \mathrm{K}}\] Now we convert kJ to J, 1 kJ = 1000 J: \[\Delta S = \frac{5.59 \times 1000\, \text{J/mol}}{302.8\, \mathrm{K}} = 18.5 \,\text{J/mol}\, \mathrm{K}\]

Calculate the total entropy change for 60.0 g of Ga

We have calculated the molar entropy change \(\Delta S\). Now, we can calculate the total entropy change for 60.0 g of Ga. First, we need to convert the mass to moles using the molar mass of Ga, which is approximately 69.72 g/mol. Moles of Ga: \[\text{moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{60.0\, \text{g}}{69.72\, \text{g/mol}} = 0.860\, \text{mol}\] Now, multiply the moles of Ga by the molar entropy change to get the total entropy change: \[\text{Total}\, \Delta S = \Delta S \times \text{moles} = 18.5\, \text{J/mol}\, \mathrm{K} \times 0.860\, \text{mol} = 15.9\, \text{J/K}\] The total entropy change when 60.0 g of Ga solidifies at \(29.8^{\circ} \mathrm{C}\) is 15.9 J/K. Since we determined that \(\Delta S\) is negative during solidification, the final answer for the entropy change is: \[\Delta S_{total} = -15.9\, \text{J/K}\]

Key concepts

Entropy

Entropy is a measure of disorder or randomness within a system. When gallium transitions from a liquid to a solid, the amount of disorder in the system decreases. This is because the particles in the solid are more organized compared to those in the liquid state.
The formula used to calculate the change in entropy, \(\Delta S\), involves the change in enthalpy, \(\Delta H\), and the temperature, \(T\). Since the process we're looking at is solidification, the entropy change is expected to be negative.
This negative value reflects the decrease in disorder as the gallium forms a more ordered crystalline structure during freezing. The formula for calculating entropy change during a phase transition is:
\[\Delta S = \frac{\Delta H}{T}\]
Where

• \(\Delta H\) is the enthalpy change,
• and \(T\) is the absolute temperature in Kelvin.

In summary, moving from liquid to a solid state decreases the number of possible arrangements and thus reduces entropy.

Phase Transition

A phase transition occurs when a substance changes from one state of matter to another, such as from liquid to solid. This change involves energy transfer, like the enthalpy of fusion in our gallium example.
During the freezing phase transition, gallium releases energy to the surroundings, allowing it to arrange into a fixed lattice structure, typical of solids. This transition is crucial because it describes the conditions under which a material maintains equilibrium between different phases.
At the melting point, both the liquid and solid phases coexist. This is where the concept of Gibbs Free Energy becomes vital. The equilibrium between two states is maintained when the Gibbs Free Energy difference is zero. Simply put, a phase transition tunes the balance of energy within a system, shifting it from one state to another by either absorbing or releasing energy in the form of heat.

Gibbs Free Energy

Gibbs Free Energy is a central concept in thermodynamics that helps predict whether a process will occur spontaneously. It combines enthalpy, temperature, and entropy into a single value. The formula for Gibbs Free Energy is:
\[\Delta G = \Delta H - T\Delta S\]
At the equilibrium point during a phase transition, such as the freezing of gallium, the change in Gibbs Free Energy, \(\Delta G\), becomes zero. This means there is no net change in energy and the system is perfectly balanced between the two phases.
Understanding \(\Delta G\) helps determine the direction of the phase change. If \(\Delta G\) is negative, the process occurs spontaneously. If zero, as in our gallium case, the phases are in equilibrium. This equilibrium ensures that gallium maintains its state at the freezing point until additional energy prompts further transition.