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The First Law of Thermodynamics is a fundamental principle of physics that relates to energy conservation. It states that energy cannot be created or destroyed, only transferred or changed from one form to another. In the context of a heat engine, the first law manifests itself in the way energy is accounted for in the system.

For a heat engine, the first law is expressed as:

• Input Heat Energy (\[Q_h\]) = Work Done (\[W\]) + Waste Heat (\[Q_c\])

This equation underscores that the energy put into the system as heat (\[Q_h\]) gets converted into two outcomes: useful work done by the engine (\[W\]) and the remaining energy that is wasted or exhausted as heat (\[Q_c\]).

Understanding this law is crucial when calculating heat engine efficiency because it ensures the energy flows within the engine are properly accounted for. Recognizing how the input heat is divided helps engineers optimize engines to convert as much energy as possible into work.

Waste heat is the portion of energy that is not converted into work by a heat engine and is instead transferred to the surroundings. This concept is key when discussing efficiency since waste heat can significantly impact an engine's performance. In our problem, the waste heat (\[Q_c\]) is given as 7800 J.

A large amount of waste heat indicates that a significant portion of input energy is not being utilized efficiently. This can be due to the inevitable energy losses caused by factors such as friction, thermal inefficiencies, and material limitations. Being able to manage and reduce waste heat is a critical goal for engineers.

Efficient systems aim to minimize waste heat, thus allowing more input energy to be converted into useful work. Methods such as improving insulation, designing better heat exchangers, and using high-efficiency materials can help mitigate waste heat.

Input heat energy (\[Q_h\]) refers to the total heat energy supplied to a system, in this case, a heat engine. For analyzing and designing heat engines, it's essential to track the input heat energy, as it determines how much energy can potentially be transformed into work.

From the first law of thermodynamics, we know:

• Total Input Heat Energy (\[Q_h\]) = \[2600 \, \text{J}\] (work done) + \[7800 \, \text{J}\] (waste heat)
• Thus, \[Q_h = 10400 \, \text{J}\]

The efficiency of the engine depends on how effectively this input heat (\[10400 \, \text{J}\]) is converted into work (\[W\]). In our example, the efficiency is 25%, meaning a quarter of the energy is useful for performing work, while the rest is lost as waste heat.

Understanding input heat energy helps in identifying possible inefficiencies and working towards better engine designs that can utilize more of this input energy, ultimately leading to more sustainable and energy-efficient systems.