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

Question: Calculate the theoretical maximum efficiency of an engine with hot and cold temperatures of 400 K and 300 K, respectively. Also, find the total entropy change per cycle when the engine operates at maximum efficiency. Answer: The theoretical maximum efficiency of the engine is 25%, and the total entropy change per cycle when the engine operates at maximum efficiency is zero.

Step-by-step solution

Calculate the Carnot efficiency of the engine.

To calculate the Carnot efficiency, we can use the formula: $$\eta_{Carnot}=1-\frac{T_{cold}}{T_{hot}}$$, where \(\eta_{Carnot}\) is the maximum efficiency of the engine, and \(T_{cold}\) and \(T_{hot}\) are the cold and hot temperatures of the engine in Kelvin, respectively. In this case, we have \(T_{hot} = 400\,\text{K}\) and \(T_{cold} = 300\,\text{K}\).

Substitute the temperature values into the Carnot efficiency formula.

Now, we'll substitute \(T_{cold}\) and \(T_{hot}\) in the Carnot efficiency formula as follows: $$\eta_{Carnot}= 1-\frac{300\,\text{K}}{400\,\text{K}}$$.

Calculate the theoretical maximum efficiency.

Simplify the expression for \(\eta_{Carnot}\) to get the maximum efficiency: $$\eta_{Carnot} = 1-\frac{3}{4} = \frac{1}{4} = 0.25$$ or \(25\%\). Answer to (a): The theoretical maximum efficiency of the engine is \(25\%\).

Determine the total entropy change formula when operating at maximum efficiency.

At maximum efficiency, the total entropy change is zero. This is because the cyclic integral of the heat transfer over the temperature is zero: $$\oint \frac{\delta Q}{T} = 0$$.

Calculate the total entropy change per cycle.

Since we've determined that the total entropy change at maximum efficiency is zero, there is no need for further calculation. Answer to (b): The total entropy change per cycle when the engine operates at maximum efficiency is zero.

Key concepts

Entropy Change

Understanding entropy is crucial when studying thermodynamics, as it is a core concept directly tied to the second law of thermodynamics. Entropy is often considered a measure of disorder or randomness within a system. The more disordered a system, the higher its entropy. Entropy change, specifically, refers to the difference in entropy as a system transitions from one state to another.

In the context of an engine operating between two different temperatures—referred to as a thermal reservoir—the second law stipulates that the entropy of the universe increases over time. When dealing with idealized engines like the Carnot engine, which is a theoretical construct, there is an assumption of reversible processes. In such cases, the entropy change over one cycle is zero because the heat absorbed and released by the engine is done so reversibly and at defined temperatures accounting for the second law's consideration. This makes the Carnot engine a perfect reversible engine—though it's important to note that in real-world applications, some entropy production is inevitable due to irreversibilities.

However, answering part (b) of the exercise draws upon this ideal scenario, concluding that the total entropy change per cycle is zero when the engine operates at its theoretical maximum efficiency. In a real thermodynamic cycle, the calculation of entropy change would require accounting for the specific processes and cannot be assumed to be zero.

Thermodynamic Cycles

A thermodynamic cycle is a series of processes that start and end at the same state, resulting in the system usually returning to its initial temperature and pressure. In such a cycle, an engine can convert heat into work, with the typical example being the internal combustion engine in cars. However, when examining theoretical models such as the Carnot cycle, we're dealing with a series of idealised reversible processes—two isothermal (constant temperature) and two adiabatic (no heat exchange).

During a Carnot cycle, the engine absorbs heat from a hot reservoir, performs work, and then expels the remaining heat to a cold reservoir before returning to its initial state. This cornerstone of thermodynamic cycles is deeply connected to the concept of efficiency, represented by the term Carnot efficiency. It symbolizes the maximum possible efficiency an engine operating between two heat reservoirs can achieve, and no engine can be more efficient than a Carnot engine when operating between the same two reservoirs. The exercise demonstrates calculating Carnot efficiency given two temperatures, which can be used to compare with real-world engines to measure their deviation from the theoretical maximum.

Second Law of Thermodynamics

The second law of thermodynamics is a fundamental principle that provides a direction to thermodynamic processes, such as heat and work interactions. It asserts that in a natural thermodynamic process, the sum of the entropies of the interacting thermodynamic systems increases. In simpler terms, the universe tends toward increased entropy, or disorder, with time.

This law is also what introduces the concept of irreversibility in natural processes. It reveals why, in practice, engines do not reach 100% efficiency—some energy is always dissipated as waste heat. The second law is crucial for determining the best possible efficiency of a heat engine, which is the Carnot efficiency, as illustrated in the textbook exercise. It also imposes the theoretical limit that while an engine like the one proposed between 400 K and 300 K can have a maximum efficiency of 25%, actual machines would have a lower efficiency due to irreversible processes not accounted for in the Carnot cycle.

Coupled with the understanding of entropy change and thermodynamic cycles, these concepts form the foundation of/classical thermodynamics and are immensely useful in the design of systems and understanding the limitations imposed by nature.