Acetone \(\mathrm{CH}_{3} \mathrm{COCH}_{3}\), is a common organic solvent with relatively low melting point \((178 \mathrm{~K})\) and boiling point $(329 \mathrm{~K})\(. The enthalpy of fusion of acetone is \)5.72 \mathrm{~kJ} / \mathrm{mol}\(, and its enthalpy of vaporization is \)29.1 \mathrm{~kJ} / \mathrm{mol}\(. The specific heats of solid and liquid acetone are \)96 \mathrm{~J} / \mathrm{mol}-\mathrm{K}\( and \)125.5 \mathrm{~J} / \mathrm{mol}-\mathrm{K}$ respectively. (a) How much heat is required to convert \(23.0 \mathrm{~g}\) of acetone at \(273 \mathrm{~K}\) to the vapor phase at \(329 \mathrm{~K} ?(\mathbf{b})\) How much heat is required to convert the same amount of acetone at \(77 \mathrm{~K}\) to the vapor phase at $329 \mathrm{~K} ?$

(a) The total heat required to convert 23.0 g of acetone at 273 K to the vapor phase at 329 K can be found by calculating and adding the heat required to heat the solid and liquid acetone, melting, and vaporization: \(q_{total} = q_{solid} + q_{fus} + q_{liquid} + q_{vap} = 0.396 \text{ mol} * 96 \frac{\text{J}}{\text{mol} \cdot \text{K}} * (178 \text{ K} - 273 \text{ K}) + 0.396 \text{ mol} * 5.72 \frac{\text{kJ}}{\text{mol}} + 0.396 \text{ mol} * 125.5 \frac{\text{J}}{\text{mol} \cdot \text{K}} * (329 \text{ K} - 178 \text{ K}) + 0.396 \text{ mol} * 29.1 \frac{\text{kJ}}{\text{mol}}\) (b) To find the heat required to convert the same amount of acetone at 77 K to the vapor phase at 329 K, calculate the heat required to heat the solid acetone from 77 K to the melting point, then follow steps 3-6 from part (a), and finally add all heat values: \(q_{total} = q_{solid} + q_{fus} + q_{liquid} + q_{vap} = 0.396 \text{ mol} * 96 \frac{\text{J}}{\text{mol} \cdot \text{K}} * (178 \text{ K} - 77 \text{ K}) + q_{fus} + q_{liquid} + q_{vap}\)