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Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. In essence, thermodynamics looks at how energy changes from one form into another and how these transformations affect the physical properties of the substances involved.

The subject is vast and includes concepts like temperature, entropy, and the very heart of our topic – heat engines. A heat engine is a system that converts heat or thermal energy—and often, but not always, this involves the use of a 'working substance'—into mechanical work. Think of steam engines or car engines; these are everyday examples of heat engines.

In explaining the textbook problem at hand, the engine in question receives a certain amount of thermal energy from a hot source and converts part of this energy into work, with the remainder expelled to a cooler reservoir. This process is a classic scenario studied in thermodynamics.

The first law of thermodynamics, also known as the law of energy conservation, establishes that energy cannot be created or destroyed in an isolated system. Instead, energy can only be transformed from one form to another. This fundamental principle forms the backbone of our understanding of how heat engines operate.

In the context of the heat engine in our exercise, the first law implies that the energy supplied to the engine from the hot reservoir must equal the sum of the work done by the engine and the heat expelled to the cold reservoir. When we demonstrate this concept with numbers \(Q_h = W + Q_c\), we cement our comprehension of how energy balance plays a vital role in engineering and environmental considerations.

The work-energy principle is a fundamental concept in physics that states that the work done on an object is equal to the change in its energy. This principle is closely related to the first law of thermodynamics because it is essentially an application of energy conservation to mechanical systems.

In the heat engine example, the work done by the engine \( W \) is a direct representation of the energy converted from the thermal energy taken in \( Q_h \) to mechanical work. The principle allows us to predict and understand how much work can be extracted from a certain amount of energy, just as we calculated in the exercise. Understanding this relationship not only helps with homework but also lays the foundation for more complex applications in areas such as mechanical design and energy resource management.