Question

The entropy of the working fluid of the ideal Carnot cycle (increases, decreases, remains the same) during the isothermal heat addition process.

Short answer

Answer: During the isothermal heat addition process in an ideal Carnot cycle, the entropy of the working fluid increases.

Step-by-step solution

Understand the Isothermal Heat Addition Process

In an ideal Carnot cycle, the isothermal heat addition process occurs at constant temperature. During this process, the working fluid absorbs heat (Q_in) from the high-temperature reservoir while remaining at constant temperature (T_high). This absorbed heat causes the fluid to expand, performing work on the surroundings.

Define Entropy

Entropy (S) is a measure of the disorder of a system and indicates the degree of thermal energy that is not available to perform useful work. For a reversible process, the change in entropy (∆S) can be calculated as: ∆S = Q/T, where Q is the heat exchanged, and T is the temperature during the process.

Evaluate the Change in Entropy

For the isothermal heat addition process in an ideal Carnot cycle, the working fluid absorbs heat (Q_in) at a constant high temperature (T_high). According to the formula for entropy change: ∆S = Q_in/T_high Since Q_in is positive (heat is absorbed), the change in entropy (∆S) during the isothermal heat addition process in the ideal Carnot cycle is also positive.

Conclusion

During the isothermal heat addition process in an ideal Carnot cycle, the entropy of the working fluid increases because the system absorbs heat from the high-temperature reservoir while remaining at a constant temperature.

Key concepts

Isothermal Heat Addition

Imagine taking a warm bath, where the water temperature remains consistent, providing continuous comfort. This is similar to the isothermal heat addition phase in the Carnot cycle, which occurs at a constant temperature. Here, the working fluid—our bath water in this analogy—absorbs heat from a high-temperature source without any change in its own temperature. This process is akin to adding more hot water to the bath, ensuring the temperature stays the same even though heat is introduced.
This constant-temperature addition of heat to the working fluid enables it to expand, and this expansion allows the fluid to do work on the environment, such as turning the blades of a turbine. It's crucial to understand that although the fluid's temperature doesn't change, its internal energy and volume do, leading to work done on the surroundings.

Real-Life Examples

• Boiling water at sea level: The water remains at 100°C, the boiling point, while it absorbs heat and converts to steam.
• Melting ice: The ice maintains a temperature of 0°C while it absorbs heat and transitions to liquid water.

When heat is added isothermally in a Carnot cycle, the fluid's entropy does change, which is what differentiates it from an adiabatic process, where the system is insulated and no heat is exchanged.

Change in Entropy

Entropy is a rather abstract concept, often described as the measure of disorder in a system. More precisely, in thermodynamics, it quantifies the amount of energy in a system that cannot be used to do work. During the isothermal heat addition of the Carnot cycle, the heat that the system absorbs results in a change in the system's entropy.
The change in entropy, denoted as ∆S, mathematically expressed as ∆S = \(\frac{Q}{T}\), where Q represents the heat exchanged and T the absolute temperature during the process. In isothermal processes, since the temperature (T) stays constant, the change in entropy is directly proportional to the heat added (Q). If the heat is absorbed by the system, the entropy increases; conversely, if the system loses heat, the entropy decreases.

Entropy as a 'Before and After' Picture

Consider the entropy change as a 'before and after' snapshot of energy distribution in a process. Before the heat addition, the system has a certain level of order. Afterward, with heat energy absorbed, the energy is more spread out, which translates to an increase in disorder or entropy. The fact that the entropy increases during the isothermal heat addition of a Carnot cycle signifies that the system becomes less ordered and more energy becomes unavailable for work, echoing the laws of thermodynamics.

Entropy and Thermodynamics

Entropy features prominently in the Second Law of Thermodynamics, which states that the total entropy of an isolated system can never decrease over time. It implies that processes occur in a direction that increases the total entropy of the universe — entropy can be thought of as the arrow of time in thermodynamics.
In the context of the Carnot cycle, the importance of entropy can't be overstated—each step of the cycle has implications for the system's entropy. The isothermal heat addition phase increases entropy because it absorbs heat, adding to the disorder. But, this is only one part of the cycle. There's also heat rejection when entropy decreases, and two adiabatic processes—a compression and an expansion—where entropy remains the same because these processes are designed to prevent heat exchange with the surroundings.

Applications in Real World Machines

The principles of entropy and the Carnot cycle underpin the operation of many real-world systems, like refrigerators and heat engines. They use cycles similar to the Carnot cycle, albeit less efficient due to the irreversible nature of real processes. Engineers use these principles to design more efficient systems, aiming to get as close to the ideal, reversible processes of the Carnot cycle as possible.In summary, entropy is a fundamental concept in thermodynamics that helps us understand energy transformation, system disorder, and the feasibility of processes. It's the gauge that measures the spread of energy and the direction of spontaneous reactions—always towards increased entropy in an isolated system.