Problem 61 - University Physics with Modern Physics

Question

Compare the entropy change of the warmer water to that of the colder water during one cycle of the heat engine, assuming an ideal Carnot cycle. (a) The entropy does not change during one cycle in either case. (b) The entropy of both increases, but the entropy of the colder water increases by more because its initial temperature is lower. (c) The entropy of the warmer water decreases by more than the entropy of the colder water increases, because some of the heat removed from the warmer water goes to the work done by the engine. (d) The entropy of the warmer water decreases by the same amount that the entropy of the colder water increases.

Step-by-Step Solution

Answer

(d) The entropy of the warmer water decreases by the same amount that the entropy of the colder water increases.

Step 1: Understand the Carnot Cycle

The Carnot cycle involves two isothermal processes and two adiabatic processes. During the isothermal processes, the system absorbs heat at a high temperature and expels heat at a low temperature, resulting in a change of entropy.

Step 2: Define the Entropy Change in Carnot Cycle

In the Carnot cycle, the entropy change of the warmer reservoir (hot source) that loses heat is negative, while the entropy change of the colder reservoir (cold sink) that gains heat is positive. The entropy exchange is usually described by the equation: \( \Delta S = \frac{Q_H}{T_H} - \frac{Q_C}{T_C} \), where \( Q_H \) is the heat absorbed from the hot reservoir and \( T_H \) and \( T_C \) are the absolute temperatures of the hot and cold reservoirs, respectively.

Step 3: Apply the Second Law of Thermodynamics

For a reversible process like the Carnot cycle, the total entropy change of the system and its surroundings is zero. This is expressed as: \( \Delta S_{total} = \Delta S_{hot} + \Delta S_{cold} = 0 \). Therefore, the entropy decrease of the hot reservoir equals the entropy increase of the cold reservoir.

Step 4: Analyze the Options

Given the zero total entropy change in the ideal Carnot cycle: (a) Incorrect because while the total entropy change is zero, individual changes are not zero. (b) Incorrect because it suggests both entropies increase whereas one decreases. (c) Incorrect because it implies unequal entropy changes. (d) Correct because the entropy decrease of the warmer water matches the entropy increase of the colder water, making the total change zero.

Key Concepts

Entropy Change

Entropy is a fundamental concept in thermodynamics, and it is essential to understand how it behaves in a process such as the Carnot cycle. Entropy is a measure of disorder or randomness in a system, often associated with energy dispersal. In the Carnot cycle, the entropy change of a system can be examined using the process of heat transfer between two thermal reservoirs: a hot source and a cold sink.

- The entropy of the hot source decreases ( because it loses heat to the system).- Conversely, the entropy of the cold sink increases ( due to absorbing heat).
In an ideal Carnot cycle, the change in entropy of the system, when combined with the change in the surroundings, results in a net zero change. This means that the entropy lost by the hot reservoir is exactly equal to the entropy gained by the cold reservoir. The equation governing this is:\[\Delta S = \frac{Q_H}{T_H} - \frac{Q_C}{T_C} = 0\]where:- \( Q_H \) is the heat removed from the hot reservoir.- \( T_H \) is the absolute temperature of the hot reservoir.- \( Q_C \) is the heat added to the cold reservoir.- \( T_C \) is the absolute temperature of the cold reservoir.
In summary, despite the internal exchanges, the overall change keeps the entropy balance between the reservoirs, which showcases the nature of the Carnot cycle as a perfect idealization.

Second Law of Thermodynamics

The Second Law of Thermodynamics is crucial for understanding the behavior of entropy in thermodynamic processes. This law states that in any natural thermodynamic process, the total entropy of a system and its surroundings always increases when the process is irreversible. However, for reversible processes, such as the Carnot cycle, the total change in entropy is zero.

According to the Second Law:- Entropy tends to increase ( in real processes, making them spontaneous and irreversible usually).- Reversible processes, like the Carnot cycle ( are theoretical and highly idealized, achieving no net change in entropy).
The significance of the Second Law in the Carnot cycle is that it underscores a perfect balance. Here, the entropy lost by the warmer body is exactly counterbalanced by the entropy gained by the cooler body. This results in:\\[\Delta S_{total} = \Delta S_{hot} + \Delta S_{cold} = 0.\]This law highlights the directionality of natural processes and sets an unattainable standard for efficiency in heat engines.

Reversible Process

A reversible process is a theoretical concept often used in thermodynamics to represent a process that can be reversed without leaving any trace on the surroundings. In practice, true reversible processes do not exist, but they provide a convenient model that helps to understand the limits of efficiency.
In a reversible process like the Carnot cycle:- There is no increase in total entropy ( though there is an exchange of entropy between the system and its surroundings).- Every step is made_{ such that the system is in near-equilibrium with its environment, minimizing overall energy losses).
This means that for each energy exchange, the changes are infinitesimally small, and thus the process can be reversed by an infinitesimal adjustment. The Carnot cycle is an iconic example of a reversible process, often cited for its balanced entropy changes between the hot and cold reservoirs. This reversible behavior is characterized by idealized cycles, where the total entropy remains constant:\[\Delta S_{total} = 0\]Reversible processes help engineers and scientists define the theoretical maximum operating efficiency of thermal systems, providing a critical benchmark against which real systems can be measured.