Question

A Carnot engine whose high-temperature reservoir is at 620 K takes in 550 J of heat at this temperature in each cycle and gives up 335 J to the low- temperature reservoir. (a) How much mechanical work does the engine perform during each cycle? What is (b) the temperature of the low-temperature reservoir; (c) the thermal efficiency of the cycle?

Short answer

(a) 215 J of work; (b) 377.6 K; (c) 39.09% efficiency.

Step-by-step solution

Calculate Work Done by the Engine

To find the work done by the engine, use the formula for work done in terms of heat input and heat rejected: \[ W = Q_{H} - Q_{C} \]where \( Q_{H} = 550 \text{ J} \) is the heat absorbed from the high-temperature reservoir and \( Q_{C} = 335 \text{ J} \) is the heat rejected to the low-temperature reservoir.Substituting the values, we get:\[ W = 550 - 335 = 215 \text{ J} \]Thus, the mechanical work performed is 215 Joules per cycle.

Calculate the Temperature of the Low-Temperature Reservoir

The efficiency \( \eta \) of a Carnot engine is given by:\[ \eta = 1 - \frac{T_{L}}{T_{H}} \]where \( T_{L} \) is the temperature of the low-temperature reservoir and \( T_{H} = 620 \text{ K} \) is the temperature of the high-temperature reservoir. Efficiency \( \eta \) can also be calculated using:\[ \eta = \frac{W}{Q_{H}} \]We found that \( W = 215 \text{ J} \) and \( Q_{H} = 550 \text{ J} \), substituting these values:\[ \eta = \frac{215}{550} = 0.3909 \]Now, substituting \( \eta \) back into the first equation to find \( T_{L} \):\[ 0.3909 = 1 - \frac{T_{L}}{620} \]Solving for \( T_{L} \):\[ \frac{T_{L}}{620} = 1 - 0.3909 = 0.6091 \]\[ T_{L} = 0.6091 \times 620 = 377.6 \text{ K} \]Thus, the temperature of the low-temperature reservoir is approximately 377.6 K.

Calculate the Thermal Efficiency of the Cycle

We have already calculated the thermal efficiency in Step 2:\[ \eta = \frac{W}{Q_{H}} = 0.3909 \]Hence, the thermal efficiency of the cycle is approximately 39.09%.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that explores how heat energy is transferred between systems and how it affects systems' states and functions. It's all about energy transformations and how energy moves from one form to another. In thermodynamics, we often discuss concepts such as work, heat, and energy conservation principles. The laws of thermodynamics help us understand how energy flows and how it can be harnessed, which is crucial for developing efficient engines and other energy-related systems. These laws can guide us in designing a more sustainable future by improving how we use and conserve energy.
In our example involving a Carnot engine, thermodynamics principles tell us how energy input as heat from a high-temperature reservoir translates into work done by the engine. This involves understanding energy conservation, where total energy in a closed system (like our engine cycle) remains constant. By comprehending how the energy is taken in and given out, we can better understand the workings of engines and other systems in both practical and theoretical contexts.

Thermal Efficiency

Thermal efficiency is a measure of how well an engine converts heat into work. It provides insight into the engine's performance by comparing the work output to the heat input. Simply put, the higher the thermal efficiency, the better the engine is at converting heat into useful work.
In mathematical terms, thermal efficiency \((\eta)\) is expressed as:

• \( \eta = \frac{W}{Q_H} \)

where \( W \) is the work done by the engine and \( Q_H \) is the heat input from the high-temperature reservoir.
Higher thermal efficiency means less energy is wasted and more is used to do work, which is why it's a significant performance indicator. In the Carnot engine example, we calculated a thermal efficiency of 39.09%, meaning about 39% of the heat energy taken in by the engine is converted into work.
Engineers strive for higher thermal efficiency to reduce energy loss and improve engine performance. However, due to natural laws and real-world complexities, perfect efficiency (100%) is unattainable, but understanding and improving thermal efficiency helps engineers design better and more sustainable systems.

Temperature Reservoirs

Temperature reservoirs are essential components in the operation of a Carnot engine. They are theoretical bodies large enough to absorb or supply heat without changing temperature. In practice, they help define the energy exchange limits in a cycle.
The high-temperature reservoir is the heat source. It provides the energy absorbed by the engine. For example, it might be a hot steam boiler.

• In our exercise, the high-temperature reservoir is at 620 K.

On the other side, we have the low-temperature reservoir, where the engine releases energy. This energy release is part of the inevitable energy dissipation in any real system.

• The temperature of the low-temperature reservoir was calculated to be approximately 377.6 K in our solution.

These reservoirs allow the Carnot engine to operate by absorbing and rejecting heat. The efficiency and work provided by the engine directly relate to the temperatures of these reservoirs. The greater the difference in temperature between the two reservoirs, the more potential there is to convert heat energy into work.
Understanding temperature reservoirs helps learners grasp why certain processes are more efficient in specific temperature conditions and aids in the design of more effective thermal management systems.