Question

State the second law of thermodynamics. According to this law, what formula gives the maximum fraction of heat that can be transformed into electricity?

Short answer

The second law states that entropy increases, and the Carnot efficiency formula is \( \eta = 1 - \frac{T_c}{T_h} \).

Step-by-step solution

Understanding the Second Law of Thermodynamics

The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. It may remain constant in ideal cases where the system is in a state of thermodynamic equilibrium or undergoing a reversible process. However, in all real processes, the total entropy of a system and its surroundings always increases. This often implies that energy transformations are not completely efficient, and some energy is always converted to a less useful form, such as heat.

Carnot Efficiency Formula

The maximum efficiency of a heat engine operating between two thermal reservoirs is given by the Carnot efficiency formula. The engine can only convert a portion of the heat energy into work due to the second law of thermodynamics.

Applying the Formula

The formula for the maximum efficiency or fraction of heat that can be converted into work by a heat engine is given by \( \eta = 1 - \frac{T_c}{T_h} \), where \( \eta \) is the efficiency, \( T_c \) is the absolute temperature (in Kelvin) of the cold reservoir, and \( T_h \) is the absolute temperature of the hot reservoir. This formula states that efficiency depends on the temperatures of the heat reservoirs.

Key concepts

Thermodynamic Equilibrium

Thermodynamic equilibrium is a state where a system's macroscopic properties, such as pressure and temperature, remain constant over time. It means that there is no net change occurring in the system. This equilibrium can be achieved when the system is isolated from its environment or if it has reached a stable condition via exchanges with its surroundings.

For a system in thermodynamic equilibrium, there is balance between energy entering and leaving the system. When this balance is achieved, no energy flow or matter flow happens within the system, allowing for a stable state.

Reaching thermodynamic equilibrium is crucial in understanding the second law of thermodynamics, as it defines how energy transformations approach efficiency limits in ideal conditions.

Entropy

Entropy is a fundamental concept in thermodynamics, representing the degree of disorder or randomness in a system. According to the second law of thermodynamics, the total entropy of an isolated system always increases over time. This principle implies that energy transformations are never 100% efficient because some energy is irreversibly lost to disorder.

Entropy is a measure of the portion of energy unavailable for doing work, often associated with "waste energy." As energy transformations happen—like fuel burning in an engine—some energy transforms into entropy and is lost as unusable heat, leading to an overall increase in disorder.

Understanding entropy helps in analyzing the efficiency of systems, especially in engineering, where minimizing unnecessary energy dissipation is essential.

Carnot Efficiency

The Carnot efficiency represents the maximum theoretical efficiency that a heat engine can achieve between two temperatures. It is expressed by the formula \[ \eta = 1 - \frac{T_c}{T_h} \], where \( \eta \) is the efficiency, \( T_c \) is the cold reservoir temperature, and \( T_h \) is the hot reservoir temperature.

This efficiency sets a limit on how much heat energy can be converted into useful work. The formula shows that the efficiency increases when the temperature difference between the hot and cold reservoirs increases. However, it can never reach 100% due to the inevitable entropy increase in real processes.

The Carnot cycle exemplifies idealised heat engine performance, and while real engines can never achieve this perfect scenario, it serves as a benchmark for optimizing designs.

Heat Engine

A heat engine is a device that transforms heat energy into mechanical work. It operates on the principle derived from the second law of thermodynamics, utilizing the flow of heat from a high-temperature source to a low-temperature sink to perform work.

Heat engines can be found in various applications, such as in car engines and power plants. These engines involve cycles of heating and cooling, where working fluids absorb heat energy, convert some of it into useful work, and release the remaining energy as waste heat.

The efficiency of a heat engine is limited by the Carnot efficiency, meaning not all the absorbed heat can be transformed into work. Enhancements in heat engine technology focus on minimizing losses and increasing practical efficiency.

Energy Transformation

Energy transformation is the process of changing energy from one form to another. In the context of thermodynamics, it often involves converting heat energy into mechanical energy using devices like heat engines.

According to the second law of thermodynamics, any energy transformation involves some level of inefficiency, as energy will partly be lost to entropy. This lost energy usually takes the form of unusable heat.

Understanding energy transformations is essential for designing systems that maximize useful energy output, while minimizing losses. Engineers strive to improve efficiency by harnessing energy more effectively and reducing waste, while always acknowledging the constraints imposed by thermodynamic principles.