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 \(29.8^{\circ} \mathrm{C}\) .

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.

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: \[0 = \Delta H - T\Delta S\]

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: \[\Delta S = \frac{5.59 \times 1000\, \text{J/mol}}{302.8\, \mathrm{K}} = 18.5 \,\text{J/mol}\, \mathrm{K}\]

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: \[\Delta S_{total} = -15.9\, \text{J/K}\]