A 2.10 -mole sample of crystalline acetic acid, initially at \(17.0^{\circ} \mathrm{C}\), is allowed to melt at \(17.0^{\circ} \mathrm{C}\) and is then heated to \(118.1^{\circ} \mathrm{C}\) (its normal boiling point) at 1.00 atm. The sample is allowed to vaporize at \(118.1^{\circ} \mathrm{C}\) and is then rapidly quenched to \(17.0^{\circ} \mathrm{C}\), so that it recrystallizes. Calculate \(\Delta H^{\circ}\) for the total process as described.

Enthalpy of fusion, also known as the heat of fusion, is the energy required for a substance to change from a solid to a liquid at its melting point without changing its temperature. This phase change is an endothermic process, meaning that the substance absorbs heat from its surroundings as it melts. The enthalpy of fusion is unique for each substance and is an important factor in determining the enthalpy change during the phase transition in thermodynamic calculations.

For instance, in the provided exercise, acetic acid has an enthalpy of fusion of 11.9 kJ/mol, indicating that each mole of acetic acid requires 11.9 kilojoules of energy to melt at its melting point. To calculate the total enthalpy change during melting, we multiply the enthalpy of fusion by the number of moles of the substance undergoing the phase change.

Specific heat capacity is a measure of the amount of heat energy required to raise the temperature of one gram of a substance by one degree Celsius. This property is essential when calculating the heat exchange involved with warming or cooling a substance without a phase change. Each material has a unique specific heat capacity, which is why different substances require varying amounts of energy to achieve the same temperature change.

In the exercise, the specific heat capacity of liquid acetic acid is given as 2.09 J/g°C. Using this value, the enthalpy change while heating the melted acetic acid from one temperature to another can be determined by multiplying the specific heat capacity by the mass of the substance and the change in temperature. The calculated value provides the amount of energy absorbed by the acetic acid as it is heated from its melting point to its boiling point.

The enthalpy of vaporization is the amount of energy that a substance requires to convert from a liquid to a vapor (or gas) at its boiling point, and it occurs without any temperature change. Similar to the enthalpy of fusion, this phase change is an endothermic process, meaning the substance absorbs heat as it undergoes vaporization. The enthalpy of vaporization is also different for every substance and is a key parameter when calculating the total enthalpy change during the vaporization phase.

In our exercise, we see that the enthalpy of vaporization for acetic acid is 23.5 kJ/mol. To find the total energy absorbed during vaporization of a known amount of acetic acid, this value is multiplied by the number of moles transitioning from liquid to gas. This calculation gives us the total heat required for the vaporization step of the process.

Phase change thermodynamics encompasses the study of energy exchanges involved when a substance transitions between different states of matter: solid, liquid, and gas. The key concepts within this field include the enthalpy of fusion, specific heat capacity, and the enthalpy of vaporization, all of which play critical roles in determining the overall enthalpy change for a phase transition process.

Through these concepts, we can understand how to calculate the energy absorbed or released when a substance undergoes a phase change, always keeping in mind that these processes are conducted at constant pressure and temperature conditions specific to each phase change. In our example with acetic acid, the total \(\Delta H^\circ\) of the combined processes of melting, heating, and vaporization was calculated by adding the enthalpy changes for each part of the phase transition, resulting in a holistic understanding of the energy exchanges from start to finish.