Suppose that 100. J of energy is taken from a hot source at \(300 .{ }^{\circ} \mathrm{C}\), passes through a turbine that converts some of the energy into work, and then releases the rest of the energy as heat into a cold sink at \(20 .{ }^{\circ} \mathrm{C}\). What is the maximum amount of work that can be produced by this engine if overall it is to operate spontaneously? What is the efficiency of the engine, with work done divided by heat supplied expressed as a percentage? How could the efficiency be increased?

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. It is primarily concerned with the concepts of energy transfer, conversion, and the laws that govern these processes. In the context of our exercise, thermodynamics provides the foundational principles that define the operation of a heat engine, which is a device that converts thermal energy into mechanical work.

Within thermodynamics, there are several laws, but two particularly relevant to our exercise are the first law, which states that energy cannot be created or destroyed, only transformed, and the second law, which addresses the direction of energy transfer and the quality of energy. The exercise relates closely to these laws by demonstrating how heat energy is converted into useful work and highlighting constraints on efficiency due to the temperatures involved.

A heat engine is a system that converts heat or thermal energy into mechanical energy, or work. The performance of a heat engine is determined by its efficiency, which is the ratio of work output to heat input. In the textbook exercise, the engine takes in 100 joules of heat energy from a hot source and then does work by moving a turbine before discharging lower-quality heat to a cold sink. This is a classic example of a heat engine cycle.

Heat engines can be found in many applications such as car engines and power plants. They operate between two temperature reservoirs, absorbing heat from the higher-temperature source and releasing some heat to the lower-temperature sink. The engine's ability to do work stems from this temperature difference. The Carnot efficiency sets a theoretical limit on the efficiency that real engines can achieve, as dictated by the second law of thermodynamics.

Energy conversion is the process of changing energy from one form to another. In our exercise, the engine converts thermal energy (heat) into mechanical energy (work). The conversion of energy is governed by the laws of thermodynamics, which dictate that while energy in a closed system remains constant, its form can change, and the quality of energy can degrade.

During the conversion process, some energy is always lost as waste heat, which is why no practical engine can reach 100% efficiency. The Carnot engine represents an idealized model where the conversion process is reversible and no energy is lost to friction or other processes. It sets the upper limit of efficiency for any real heat engine, although real engines will invariably fall short of this theoretical efficiency due to real-world inefficiencies.

Temperature conversion is necessary to work with thermodynamic equations, which require temperatures to be in an absolute scale. In our step-by-step solution, we converted temperatures from Celsius to Kelvin because the Kelvin scale sets its zero point at absolute zero, the theoretical point where there is no thermal energy in a system.

To convert Celsius to Kelvin, one simply adds 273.15 to the Celsius temperature. This is important because the efficiency calculations for a Carnot engine depend on the absolute temperatures of the hot and cold reservoirs. Using absolute temperatures allows for consistent thermodynamic calculations, as it is a true representation of thermal energy content.

The second law of thermodynamics introduces the concept of entropy, stating that for any spontaneous process, the total entropy of a system and its surroundings must increase. In simpler terms, it means energy tends to spread out or disperse if not hindered. For a heat engine like the one described in our exercise, this law implies that some energy will always be lost to the surroundings and it is impossible to convert all the absorbed heat into work.

This law also tells us why we cannot have a 100% efficient heat engine. As per the Carnot principle, the maximum efficiency depends on the temperature of the hot and cold reservoirs and it shows that to improve the efficiency of an engine, one can increase the temperature of the hot source or decrease that of the cold sink thus, enhancing the temperature difference and, consequently, the potential for work output. This aligns directly with the question of how efficiency could be increased in our original problem.