Question

A proposal is submitted for a novel engine that will operate between \(400 . \mathrm{K}\) and \(300 . \mathrm{K}\) a) What is the theoretical maximum efficiency of the engine? b) What is the total entropy change per cycle if the engine operates at maximum efficiency?

Short answer

Answer: The theoretical maximum efficiency of the engine is 25%, and the total entropy change per cycle if the engine operates at maximum efficiency is 0.

Step-by-step solution

Calculate the theoretical maximum efficiency of the engine

For an engine operating between two temperatures, the theoretical maximum efficiency is given by the Carnot efficiency formula: Efficiency = \(1 - \frac{T_{cold}}{T_{hot}}\) Where \(T_{cold}\) and \(T_{hot}\) are the cold and hot reservoir temperatures, respectively. In this case, we have: \(T_{cold} = 300 \,\mathrm{K}\) and \(T_{hot} = 400 \,\mathrm{K}\) Now we will insert these values into the Carnot efficiency formula: Efficiency = \(1 - \frac{300}{400}\)

Simplify the expression

We will simplify the expression to find the efficiency: Efficiency = \(1 - \frac{300}{400} = 1 - \frac{3}{4} = \frac{1}{4}\) So, the theoretical maximum efficiency of the engine is \(\frac{1}{4}\) or 25%.

Calculate the total entropy change per cycle

When the engine operates at its maximum efficiency (Carnot efficiency), we know that the total entropy change per cycle is zero: Total entropy change per cycle = 0

Write down the answers

a) The theoretical maximum efficiency of the engine is 25%. b) The total entropy change per cycle if the engine operates at maximum efficiency is 0.

Key concepts

Understanding Entropy Change

Entropy is a fundamental concept in thermodynamics that measures the disorder or randomness in a system. When analyzing thermal processes, like those occurring in a heat engine, understanding entropy change is crucial. Entropy can tell us how efficient a process is and whether it adheres to the laws of thermodynamics. For a reversible process, such as that in a Carnot engine, the total entropy change is zero. This is because the system returns to its original state without any net increase in disorder. This concept is key when evaluating the maximum efficiency of an engine — a scenario where no entropy is gained or lost. Overall, entropy change in an engine cycle gives insight into the irreversibility present. A zero change indicates a theoretically perfect process, while positive entropy change reflects real-world inefficiencies encountered.

The Basics of Thermodynamics

Thermodynamics is the branch of physics that deals with heat and temperature and their relation to energy and work. It provides the necessary groundwork for understanding how heat engines, such as the proposed novel engine, operate. Thermodynamics has several laws, but two are crucial in this context:

• The First Law (Conservation of Energy): It states that energy cannot be created or destroyed, only transformed from one form to another.
• The Second Law: It implies that heat naturally flows from a hot object to a cooler one, not the other way around, unless work is done on the system. It also introduces the concept of entropy.

The role of these laws is significant in determining the performance limits of heat engines. They help us calculate the maximum efficiency an engine can theoretically achieve and understand why all systems have inherent inefficiencies.

Evaluating Heat Engine Efficiency

Heat engine efficiency is a measure of how well an engine converts heat energy into useful work. The Carnot cycle serves as a model for the most efficient engine possible, setting an ideal efficiency limit. The Carnot efficiency depends solely on the temperatures of the hot and cold reservoirs. The formula is given by: \[ \text{Efficiency} = 1 - \frac{T_{\text{cold}}}{T_{\text{hot}}} \]In this formula, \(T_{\text{cold}}\) and \(T_{\text{hot}}\) are the absolute temperatures of the cold and hot reservoirs, respectively. The Carnot efficiency represents the maximum efficiency that no real engine can exceed without violating the second law of thermodynamics.By calculating this theoretical efficiency, we can evaluate the performance of real-world engines and identify the areas for improvement. However, it is essential to remember that this is an idealized scenario, and actual engines always operate below this maximum efficiency.