For a certain reaction at constant pressure, \(\Delta H=-15 \mathrm{~kJ}\) and \(22 \mathrm{~kJ}\) of expansion work is done on the system. What is \(\Delta U\) for this process?

Enthalpy change, denoted as \(\Delta H\), is a measure of the total heat content change in a system during a process, often a chemical reaction, at constant pressure. It's an important concept in thermodynamics because it gives us a quick way to calculate the energy change without needing to know all details about the system's internal energy states.

This value can be either positive or negative; a negative \(\Delta H\) indicates that heat is released by the system to the surroundings, known as an exothermic process, while a positive \(\Delta H\) means that heat is absorbed from the surroundings, or an endothermic process. In the given exercise, the negative \(\Delta H\) signifies that the reaction releases heat, hence why it is associated with a negative sign.

Internal energy, represented as \(\Delta U\), is the total energy contained within a system. It encompasses all forms of energy, including kinetic and potential energy at the molecular level. In thermodynamics, the change in internal energy is what we often calculate or measure because we're interested in how energy moves in and out of systems.

The First Law of Thermodynamics states that energy can neither be created nor destroyed, only transformed from one form to another or transferred between systems. When dealing with \(\Delta U\), we see this law in action; any increase or decrease in a system's internal energy must be accounted for by energy transfers in the form of work or heat.

Carrying out a process at constant pressure is particularly relevant when discussing enthalpy change because it simplifies the relationship between heat added to the system and the work done by the system. At constant pressure, the work done \(W\) can be easily connected to the system's volume changes.

If, for example, a gas expands against a constant pressure, it does work on its surroundings, and the energy required to do that work comes from the internal energy of the gas. Hence, if we know the enthalpy change and the work done, we can deduce other properties of the system, as shown in our exercise.

Energy conservation is a fundamental principle stating that in an isolated system, the total energy remains constant over time. This is the essence of the First Law of Thermodynamics. In the context of our exercise, where we have quantified both the enthalpy change and the work done, energy conservation allows us to determine the change in internal energy of the system.

The crux of solving thermodynamics problems often lies in recognizing that we must account for all forms of energy transfer, such as heat and work, to honor the principle of energy conservation. Understanding this principle is critical for interpreting thermodynamic processes and predicting the behavior of systems under various conditions.