Calculate the work done (in joules) when 1.0 mole of water is frozen at \(0^{\circ} \mathrm{C}\) and 1.0 atm. The volumes of 1 mole of water and ice at \(0^{\circ} \mathrm{C}\) are \(0.0180 \mathrm{~L}\) and \(0.0196 \mathrm{~L},\) respectively.

Thermodynamics is a branch of physics that deals with the relationships between heat, work, temperature, and energy. In essence, it is the study of how energy is transformed and transferred. At the core of thermodynamics lie four fundamental laws that define how these quantities interact. For instance, the first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed from one form to another. This law is crucial when calculating work done during a process such as the freezing of water.

The work involved in thermodynamic processes is a measure of energy transfer when a force moves its point of application. When we refer to the thermodynamic work calculation in the context of the exercise, we are interested in quantifying the energy necessary to change a system—in this case, from water to ice. Even though this process might not intuitively seem like 'work' as is commonly understood, from a physics perspective, it is.

Enthalpy change, symbolized as \( \Delta H \), refers to the heat content change in a system at constant pressure. It's an essential concept in thermodynamics and is used to describe the heat absorbed or released during a process. For phase transitions, such as melting or freezing, the enthalpy change can be associated with the latent heat, which is the amount of heat required to change the phase of a substance without changing its temperature.

In the given exercise, the enthalpy change could potentially be involved if the process of freezing water is considered under constant pressure. However, since the problem focuses exclusively on work calculation and no temperature change occurs, the role of enthalpy change is not directly addressed. Yet, understanding this concept can be valuable in broader thermodynamic questions where both heat and work are considered.

In thermodynamics, state functions are properties that only depend on the current state of the system, not on the path taken to reach that state. Enthalpy, mentioned previously, is an example of a state function. Other examples include internal energy, temperature, and pressure. These variables are incredibly useful because they simplify the analysis of thermodynamic systems, allowing predictions about the system's behavior without detailed knowledge of its history.

When considering work calculation, we're dealing with a non-state function because work depends on the path taken during the process. The change in volume of water when it freezes, as found in the exercise, is also not a state function, but the volumes themselves at specific states are.

Phase transitions refer to the transformation from one state of matter to another, such as from liquid to solid (freezing), liquid to gas (vaporization), or solid to liquid (melting). These transitions involve energy changes within the material. During a phase transition, properties such as temperature and enthalpy can remain constant while the system absorbs or releases energy. The exercise presents the phase transition of freezing, where water turns into ice.

It is interesting to note that the volume change during a phase transition is an important factor in work calculations. Water is somewhat unusual because it expands upon freezing, which is why ice is less dense than water. This expansion against atmospheric pressure, as seen in the provided solution, constitutes the work done on the system during the freezing transition.