Question

A heat engine operates with an efficiency of $0.5.$ What can the temperatures of the high-temperature and lowtemperature reservoirs be? a) $T_{\mathrm{H}}=600\mathrm{K}$ and $T_{\mathrm{L}}=100\mathrm{K}$ b) $T_{\mathrm{H}}=600\mathrm{K}$ and $T_{\mathrm{L}}=200\mathrm{K}$ c) $T_{\mathrm{H}}=500\mathrm{K}$ and $T_{\mathrm{L}}=200\mathrm{K}$ d) $T_{\mathrm{H}}=500\mathrm{K}$ and $T_{\mathrm{L}}=300\mathrm{K}$ e) $T_{\mathrm{H}}=600\mathrm{K}$ and $T_{\mathrm{L}}=300\mathrm{K}$

Short answer

Answer: The high-temperature and low-temperature reservoirs for a heat engine operating with an efficiency of 0.5 are TH = 600 K and TL = 200 K.

Step-by-step solution

Setup the efficiency formula

Write down the formula for efficiency and plug in the given efficiency value: $$0.5=1-\frac{T_L}{T_H}$$

Test each option

Plug in the temperatures mentioned in each option and check if the equation is satisfied. (a) $T_H=600\mathrm{K}$ and $T_L=100\mathrm{K}$: $$0.5=1-\frac{100}{600}=1-\frac{1}{6}=\frac{5}{6}$$ Answer (a) not correct. (b) $T_H=600\mathrm{K}$ and $T_L=200\mathrm{K}$: $$0.5=1-\frac{200}{600}=1-\frac{1}{3}$$ Answer (b) is correct. (c) $T_H=500\mathrm{K}$ and $T_L=200\mathrm{K}$: $$0.5=1-\frac{200}{500}=1-\frac{2}{5}=\frac{3}{5}$$ Answer (c) not correct. (d) $T_H=500\mathrm{K}$ and $T_L=300\mathrm{K}$: $$0.5=1-\frac{300}{500}=1-\frac{3}{5}=\frac{2}{5}$$ Answer (d) not correct. (e) $T_H=600\mathrm{K}$ and $T_L=300\mathrm{K}$: $$0.5=1-\frac{300}{600}=1-\frac{1}{2}$$ Answer (e) not correct.

Conclusion

The heat engine operating with an efficiency of $0.5$ would have the high-temperature and low-temperature reservoirs as $T_H=600\mathrm{K}$ and $T_L=200\mathrm{K}$ (Option b).

Key concepts

Heat Engine

A heat engine is a fascinating device in the realm of thermodynamics. It is designed to convert thermal energy, or heat, into mechanical work. Think of it as a transformation process where energy changes forms to power machines. Here's how it works: - **Thermal Energy Input**: The engine absorbs heat from a high-temperature reservoir, which is like a very hot source. This source provides the energy needed for the engine to do work. - **Mechanical Work Output**: Part of the absorbed heat is transformed into mechanical work. This can power anything from a car to a power plant generator. - **Heat Energy Release**: Not all the absorbed heat turns into work; some of it is expelled to a low-temperature reservoir. This is where the engine dumps the leftover heat. The operation of a heat engine is cyclical. It continuously absorbs heat, performs work, and releases residual heat. Each cycle aims to maximize work while minimizing the energy lost as waste heat. Understanding the basic cycle helps in grasping the importance of efficiency, a key measure of the engine's effectiveness.

Efficiency Formula

The efficiency of a heat engine is a crucial concept that tells us how well the engine converts heat into work. It can be defined simply as the ratio of work output to heat input. Here's the simple mathematical representation:$$\text{Efficiency}=\frac{W}{Q_H}=1-\frac{T_L}{T_H}$$Being a fraction of the absorbed heat turned into work, efficiency explains how effectively a heat engine operates. - **Efficiency Value**: Efficiency is a number between 0 and 1 (or 0% to 100%). A value of 0.5, for instance, implies that the engine converts 50% of the heat from the high-temperature reservoir into work.- **Ideal Conditions**: The equation also shows that efficiency increases when the temperature difference between the high-temperature ($T_H$) and low-temperature (%$T_L$) reservoirs increases.- **Practical Implications**: In real-world scenarios, it's rare for an engine to achieve maximum efficiency due to energy losses through processes like friction.Knowing the efficiency helps engineers and scientists enhance the design and functionality of engines, making them more environmentally friendly and cost-effective.

Temperature Reservoirs

Temperature reservoirs are integral components of a heat engine, represented by the high-temperature ($T_H$) and low-temperature ($T_L$) reservoirs. They essentially function as sources and sinks for heat energy within the engine system.- **High-Temperature Reservoir ($T_H$)**: This is akin to a hot body that supplies heat to the engine. Its temperature is crucial as it determines the amount of energy available to do work. The higher the temperature, the more potential energy.- **Low-Temperature Reservoir ($T_L$)**: This acts like a cooler that receives unwanted heat energy after work has been extracted. Maintaining a lower temperature here is necessary to maximize the engine's efficiency.- **Role in Efficiency**: The difference in temperatures ($T_H-T_L$) dictates the efficiency, as greater variances can potentially allow more work to be extracted from the engine.In practice, selecting appropriate temperature reservoirs is vital for optimizing performance and efficiency. Engineers carefully choose or design conditions that balance the heat flow, ensuring a smooth and effective energy conversion process. Understanding these reservoirs is key to enhancing engine performance in various applications including vehicles, industrial machinery, and power generation.