For the vaporization of one mole of water at \(100^{\circ} \mathrm{C}\) determine (a) \(\Delta H\) (Table 8.3) (b) \(\Delta P V\) (in kilojoules) (c) \(\Delta E\)

Thermodynamics is a core subject in physics and chemistry that deals with heat, work, temperature, and the laws governing their interactions. It describes how thermal energy is converted to and from other forms of energy and how it affects matter. The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed.

When we talk about the vaporization of water, we're looking at an endothermic process, where heat is absorbed to convert liquid water into a gas. This is where the concept of enthalpy (∆H) shines, as it quantifies the total heat content of a system at constant pressure. The enthalpy of vaporization is defined as the amount of heat required to turn one mole of a substance from its liquid phase into vapor without changing its temperature.

In the example given, the enthalpy of vaporization provides insight into the energy needed for the phase transition of water at its boiling point. This ties into the broader thermodynamic view, where understanding energy changes is critical in predicting the behavior of a system under various conditions.

The ideal gas law is a fundamental equation in thermodynamics that describes the relationship between pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas, under the assumption that the gas behaves ideally. This relationship is mathematically expressed as PV = nRT, where R is the ideal gas constant.

In the context of the problem, after water vaporizes, it behaves as a gas and thus, the ideal gas law can be applied to determine the change in the product of pressure and volume (∆PV) during the phase change. It's important to note that the ideal gas law assumes no interactions between gas molecules and that they occupy no space, which is a good approximation for gases at high temperatures and low pressures.

Additionally, ∆PV provides a bridge between the thermodynamic property changes of a system since the work done during a phase change at constant pressure can be estimated using this value. It is key for understanding other thermodynamic processes at play during the vaporization of water.

Internal energy is the total energy contained within a system due to the kinetic and potential energies of its particles. In thermodynamics, changes in internal energy (∆E) are of particular interest because they represent how the energy of a system changes in response to heat transfer and work done on or by the system.

The change in internal energy can be calculated from the enthalpy of vaporization and the work associated with the change in pressure and volume (∆PV), using the relationship ∆E = ∆H - ∆PV. This equation is derived from the first law of thermodynamics and is crucial in understanding how energy is conserved during a phase change.

Applying this concept to our current scenario, where water is vaporizing, enables us to quantify the change in the kinetic and potential energies of the molecules as they transition from a liquid to a gas phase. The subtraction of the work term (∆PV) from the enthalpy (∆H) gives us the net change in internal energy (∆E), offering us a sober look into the intrinsic energy dynamics of the phase change.