Question

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

Short answer

(a) During the solidification process, the entropy (ΔS) decreases, making ΔS negative. (b) When 60.0 g of Ga solidifies at 29.8°C, the value of ΔS is approximately -15.67 J/K.

Step-by-step solution

Determine the sign of ΔS during the solidification process

In a phase transition where a substance changes from a liquid to a solid state (i.e., it solidifies), the substance becomes more organized and its molecules move in a more restricted way. Therefore, the entropy, which represents the degree of disorder in a system, decreases. Thus, since the entropy decreases during the solidification process, ΔS must be negative.

Calculate the amount of moles of Ga

To calculate the value of ΔS, we first need to find the number of moles for the given mass of Ga. We use the molar mass of Ga, which is approximately 69.72 g/mol. Moles of Ga = \( \frac{Mass}{Molar~mass} = \frac{60.0~g}{69.72~g/mol} \) Moles of Ga ≈ 0.860 mol

Apply Gibbs Free Energy equation to find ΔS

The Gibbs Free Energy equation is: ΔG = ΔH - TΔS Since the phase transition takes place at the melting point, the Gibbs Free Energy, ΔG, is equal to zero. Therefore, we can rewrite the equation as: 0 = ΔH - TΔS We know ΔH (the molar enthalpy of fusion) = 5.59 kJ/mol and the melting temperature (T) is 29.8°C, which needs to be converted to Kelvin (K): T(K) = T(°C) + 273.15 = 29.8 + 273.15 = 302.95 K Now, we can solve the equation for ΔS: ΔS = \( \frac{ΔH}{T} \)

Calculate ΔS_value for given mass of Ga

Now that we have all the information needed, we can calculate ΔS for the given amount of gallium: ΔS_value = (0.860 mol) * \( \frac{5.59~kJ/mol}{302.95~K} \) ΔS_value = (0.860 mol) * \( \frac{5500~J/mol}{302.95~K} \) (convert kJ to J by multiplying by 1000) ΔS_value ≈ -15.67 J/K The value of ΔS when 60.0 g of Ga solidifies at 29.8°C is approximately -15.67 J/K.

Key concepts

Entropy

Entropy is a fundamental concept in thermodynamics that measures the disorder or randomness of a system. It provides insight into how energy spreads or disperses in a system. When a liquid, like gallium, solidifies, its molecules arrange into a more ordered structure. This ordering means that the system is moving towards a lower entropy state.
When molecules in a liquid state move freely, they possess higher entropy due to their high disorder. As they transition into a solid state, their movement is restricted, leading to decreased entropy. Thus, during solidification, the change in entropy (\(\Delta S\)) is negative, indicating the system has become more ordered.
In summary:

• Higher disorder = Higher entropy
• Lower disorder (solidification) = Lower entropy (negative \(\Delta S\))

Phase Transition

Phase transitions are essential processes in which a substance changes from one state of matter to another. These states include solid, liquid, and gas. The transition point, such as melting or freezing, is characterized by specific temperature and pressure conditions.
For gallium, which melts and freezes at 29.8°C, the phase transition involves changing from a liquid (\(Ga(l)\)) to a solid (\(Ga(s)\)). This transition is crucial as it involves energy transfer and entropy changes.
During the phase transition, the substance absorbs or releases energy but its temperature remains constant. This is because all the energy changes go into altering the structure of the substance rather than changing its temperature.
Understanding phase transitions helps us grasp how materials behave under different thermal conditions and energy requirements, like the energy needed for gallium to solidify.

Molar Enthalpy of Fusion

The molar enthalpy of fusion (\(\Delta H_{\text{fus}}\)) is the heat energy required to change one mole of a substance from solid to liquid at constant pressure and temperature. It is a critical property for characterizing the thermal behavior of materials during phase transitions.
In the given exercise, gallium has a molar enthalpy of fusion of 5.59 kJ/mol. This specific value indicates the energy needed to break the intermolecular bonds in the solid state, enabling gallium to transition to a liquid.
Although the enthalpy of fusion often refers to melting, it applies similarly during freezing, as it's the energy released when a substance transitions from a liquid to a solid. This implies that the enthalpy change remains the same magnitude but opposite in sign.
In practice, knowing the molar enthalpy of fusion is crucial for processes needing precise thermal management, such as cooling systems, metallurgy, and material fabrication.

Gibbs Free Energy

Gibbs Free Energy (\(\Delta G\)) is a thermodynamic property that combines enthalpy (\(\Delta H\)) and entropy (\(\Delta S\)) to determine the spontaneity of a process at constant pressure and temperature. The equation relating them is:
\[\Delta G = \Delta H - T\Delta S\]
Where \(T\) is the absolute temperature in Kelvin. A negative \(\Delta G\) indicates a spontaneous process, while a positive \(\Delta G\) suggests that external energy is needed for the process to occur.
During a phase transition at a substance's melting (or freezing) point, such as gallium solidifying at its melting point, \(\Delta G\) equals zero. This means the system is in equilibrium, with no net change in free energy. The energy (enthalpy) input needed for phase change equals the energy dispersed or ordered (entropy), confirming the system's balanced nature.
Understanding Gibbs Free Energy helps us predict and control reactions and processes, particularly those involving phase transitions, ensuring efficiency and stability in various scientific and engineering applications.