Question

20.9a) The maximum efficiency of a Carnot engine is \(100 \%\) since the Carnot cycle is an ideal process. b) The Carnot cycle consists of two isothermal processes and two adiabatic processes. c) The Carnot cycle consists of two isothermal processes and two isentropic processes (constant entropy). d) The efficiency of the Carnot cycle depends solely on the temperatures of the two thermal reservoirs.

Short answer

Question: Evaluate the accuracy of each statement about the Carnot engine. a) Maximum efficiency of a Carnot engine is 100%. b) The Carnot cycle consists of two isothermal processes and two adiabatic processes. c) The Carnot cycle consists of two isothermal processes and two isentropic processes. d) The efficiency of the Carnot cycle depends solely on the temperatures of the two thermal reservoirs. Answer: a) False b) True c) True d) True

Step-by-step solution

Statement a: Maximum efficiency of a Carnot engine is 100%.

False. The ideal "Carnot engine" is a theoretical construct that assumes perfectly reversible processes. Its efficiency is given by the formula: Efficiency = \(1-\frac{T_{cold}}{T_{hot}}\) Where \(T_{cold}\) and \(T_{hot}\) are the cold and hot reservoir temperatures, respectively, in Kelvin. Even in this ideal case, the efficiency can never reach 100% unless the temperature of the cold reservoir is absolute zero, which is impossible.

Statement b: The Carnot cycle consists of two isothermal processes and two adiabatic processes.

True. A Carnot cycle comprises four distinct steps: two isothermal processes (one at the hot reservoir, one at the cold reservoir) and two adiabatic processes (one for expansion, one for compression). During isothermal processes, the temperature remains constant; during adiabatic processes, there is no transfer of heat.

Statement c: The Carnot cycle consists of two isothermal processes and two isentropic processes.

True. The two adiabatic processes in the Carnot cycle are also isentropic (constant entropy) processes. In an ideal Carnot cycle, the two isothermal processes are assumed to be perfectly reversible, so they are also isentropic. As a result, both statements are true: the Carnot cycle consists of two isothermal processes and two adiabatic or isentropic processes.

Statement d: The efficiency of the Carnot cycle depends solely on the temperatures of the two thermal reservoirs.

True. The efficiency of the Carnot cycle relies on the temperature difference between the hot and cold reservoirs. As previously mentioned, the efficiency of a Carnot engine is given by the formula: Efficiency = \(1-\frac{T_{cold}}{T_{hot}}\) Where \(T_{cold}\) and \(T_{hot}\) are the cold and hot reservoir temperatures, respectively, in Kelvin. The larger the temperature difference, the higher the Carnot engine's efficiency.

Key concepts

Efficiency of Carnot Engine

The efficiency of a Carnot engine is an important concept in thermodynamics. It is a measure of how well a heat engine converts heat into work. Unlike real engines, the Carnot engine is an idealized model that operates in a cycle involving reversible processes. The efficiency of this ideal engine depends on the temperatures of the hot and cold reservoirs. For a Carnot engine, the efficiency is determined by the formula:\[ \text{Efficiency} = 1 - \frac{T_{\text{cold}}}{T_{\text{hot}}} \]where \( T_{\text{cold}} \) and \( T_{\text{hot}} \) are the temperatures of the cold and hot reservoirs measured in Kelvin.- It reflects that the efficiency can never reach 100% unless the cold reservoir is at absolute zero temperature, which is not achievable in practice. - The greater the difference in temperature between the hot and cold reservoirs, the higher the potential efficiency of the Carnot engine.Understanding the Carnot efficiency formula helps in setting the upper limit of what is achievable in real heat engines, making it a benchmark for engineers.

Isothermal Process

An isothermal process is a key component of the Carnot cycle and many other thermodynamic cycles. During an isothermal process, the system's temperature remains constant. This means that as the system undergoes changes in pressure and volume, the temperature does not change. - In the context of a Carnot cycle, there are two isothermal processes: one at high temperature (\(T_{\text{hot}}\)) and one at low temperature (\(T_{\text{cold}}\)).- During the isothermal expansion, the system (typically a gas) absorbs heat from the hot reservoir at a constant temperature.- During the isothermal compression, the system releases heat to the cold reservoir, again maintaining a constant temperature.These processes involve careful balancing of heat exchange to ensure that temperature remains constant, and they are characterized by the equation of state for an ideal gas: \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the amount of substance (moles), \( R \) is the ideal gas constant, and \( T \) is temperature.

Adiabatic Process

The adiabatic process is another vital part of both the Carnot cycle and thermodynamic cycles in general. In an adiabatic process, no heat is exchanged with the surroundings. Therefore, any change in the system's internal energy is due to work done by or on the system.- During the Carnot cycle, there are two adiabatic processes: one involves expansion and the other involves compression.- Both adiabatic processes are isentropic, meaning they occur at a constant entropy level, making them reversible processes in the idealized Carnot cycle.During an adiabatic expansion of a gas, the system does work on its surroundings, resulting in a drop in temperature. Conversely, during adiabatic compression, the system's temperature increases due to work being done on it. These processes are characterized by changes in pressure and volume that follow the relation: \( PV^\gamma = \text{constant} \), where \( \gamma \) is the heat capacity ratio \( C_p/C_v \). Understanding adiabatic processes aids in analyzing energy transformations without heat transfer.

Thermodynamic Cycles

Thermodynamic cycles are sequences of processes that return a system to its original state, after transforming heat into work or vice versa. The Carnot cycle is one of the most celebrated thermodynamic cycles due to its theoretical efficiency. - A cycle consists of multiple stages, or processes, often including both expansion and compression of a gas. - In the Carnot cycle, the system undergoes two isothermal and two adiabatic processes, forming a complete cycle that is both reversible and most efficient under given constraints. Different cycles, like the Otto cycle for internal combustion engines or the Rankine cycle for steam engines, all follow the basic principles of thermodynamics but differ in process specifics and efficiencies. Understanding thermodynamic cycles is crucial for engineers to design processes that convert energy forms efficiently and sustainably, bridging the gap between theoretical models like the Carnot cycle and practical applications.