Explain how the sum of heat and work can be a state function, even though heat and work are themselves not state functions.

A state function is a fundamental concept in thermodynamics that refers to a property of a physical system that depends only on the current state of the system, regardless of how that system arrived at its present state. Temperature, pressure, volume, and internal energy are classic examples of state functions. For instance, when you climb to the top of a hill, your potential energy is higher regardless of whether you took a steep direct path or a longer, winding one. The importance of this concept lies in its intrinsic value; it simplifies the otherwise complex analysis of thermodynamic systems by making certain properties predictable and consistent.

In the context of the exercise, understanding that internal energy is a state function helps highlight why, despite the path-dependent nature of heat and work, their sum can reflect a single, conclusive state of the system. This understanding aids in solving problems related to energy changes within a system and reinforces the fundamental principles of thermodynamics.

Path functions distinctly differ from state functions, as they describe quantities whose values depend on the transition, or path, a system takes between two states. Heat (Q) and work (W) are prime examples of path functions. These variables account for the energy transferred between the system and its surroundings due to temperature difference and force, respectively.

To visualize this concept, consider taking a trip from one city to another. The distance (a state function) between the two cities is fixed, but the scenery, the number of turns, or the hills you drive over (analogous to path functions) will differ depending on the route you choose. Hence, while the start and end points remain the same, your experience (heat and work) can vary widely. This variability underscores why path functions are not as straightforward to calculate as state functions, but they are equally critical in understanding the processes occurring within thermodynamic systems.

Internal energy, denoted by the symbol U, is a state function that represents the total energy contained within a system. This includes the kinetic and potential energies of particles within the system as well as all forms of energy except those due to external gravitational fields or the system's overall motion or position. A change in internal energy ((Delta U)) occurs due to transfers of energy as heat, work, or both.

The First Law of Thermodynamics, which equates the change in internal energy to the sum of heat added and work done ((Delta U = Q + W)), fundamentally links the concept of internal energy to path functions. The beauty of this law lies in its assertion that although heat and work are path-dependent, their combined effect on a system is quantifiable and independent of the path taken, thus wrapping up the change in internal energy into a neat, state-dependent package. Moreover, recognizing internal energy as a state function is crucial in thermodynamics, as it allows for the prediction and calculation of state changes without needing detailed knowledge of the system's history.