The standard entropy of vaporization of acetone is approximately \(85 \mathrm{~J} \cdot \mathrm{K}^{-1} \cdot \mathrm{mol}^{-1}\) at its boiling point. (a) Estimate the standard enthalpy of vaporization of acetone at its normal boiling point of \(56.2^{\circ} \mathrm{C}\). (b) What is the entropy change of the surroundings when 10. \(\mathrm{g}\) of acetone, \(\mathrm{CH}_{3} \mathrm{COCH}_{3}\), condenses at its normal boiling point?

The Clausius-Clapeyron Equation is a fundamental principle in thermodynamics that describes the relationship between the pressure and temperature of a substance during a phase transition, such as vaporization. It is expressed as:
\[\frac{dP}{dT} = \frac{\Delta H_{vap}}{T \cdot \Delta V}\]
In simpler terms, it tells us how the vapor pressure of a liquid changes with temperature. Given that the volume change (\(\Delta V\)) from liquid to vapor is significant, this equation can be simplified by assuming ideal gas behavior for the vapor phase and negligible volume for the liquid phase. This provides a way to estimate the enthalpy of vaporization (\(\Delta H_{vap}\)) when the entropy of vaporization (\(\Delta S_{vap}\)) and boiling temperature are known, as done in the provided exercise.

Enthalpy of vaporization (\(\Delta H_{vap}\)) is the amount of energy required to convert a substance from its liquid phase to its gas phase at constant pressure and temperature. This value is crucial in understanding how a substance behaves when it reaches its boiling point. In the context of the exercise, the Clausius-Clapeyron equation is used to estimate \(\Delta H_{vap}\) as the product of the entropy of vaporization and the boiling temperature in Kelvin. As a result, you get the energy needed for the phase change per mole of the substance.

When dealing with thermochemical equations, temperature must always be in Kelvin. This is because Kelvin is the SI unit for thermodynamic temperature and the scales used in these types of calculations. To convert from Celsius to Kelvin, you simply add 273.15 to the Celsius temperature:
\[K = \degree C + 273.15\]
This conversion is vital in thermodynamics calculations, including those involving the Clausius-Clapeyron equation, as absolute temperature (in Kelvin) is a key factor in determining the physical properties of substances at different temperatures.

The entropy change of the surroundings (\(\Delta S_{sur}\)) during a phase transition is directly related to the entropy change of the system but of opposite sign. If a substance condenses (changing from gas to liquid), it releases energy to its surroundings, increasing the entropy of the surroundings. In quantitative terms, suppose you have a certain mass of a substance condensing at a known temperature. First, you calculate the number of moles of the substance. Then, you multiply the moles by the entropy of vaporization \(\Delta S_{vap}\), taking its negative (since it's condensation). Finally, the entropy change of the surroundings is the negative of that value, reflecting the energy dispersion in the surroundings due to the phase transition.