• Describe the four stages of the Carnot cycle: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression.
• Sketch and interpret the Carnot cycle on a P-V diagram.
• Understand the significance of the Carnot cycle as an ideal reversible cycle.
• State the expression for the efficiency of a Carnot engine and understand its dependence on reservoir temperatures.

• Define Carnot cycle, isothermal process, adiabatic process, reversible process, and engine efficiency using precise scientific terminology.
• Explain each stage of the Carnot cycle, detailing changes in heat, work, and internal energy.
• Discuss the theoretical importance of the Carnot cycle in thermodynamics.

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Focus on understanding the definition and application of each term.

Carnot Cycle

A theoretical thermodynamic cycle proposed by Nicolas Léonard Sadi Carnot. It is the most efficient cycle possible for converting a given amount of thermal energy into work, or conversely, for creating a temperature difference by doing a given amount of work. It consists of two isothermalизотермиялық / изотермических processes and two adiabaticадиабаталық / адиабатических processes.

Translation (Қазақша)

Николя Леонар Сади Карно ұсынған теориялық термодинамикалық цикл. Бұл берілген жылу энергиясының мөлшерін жұмысқа айналдыру үшін немесе, керісінше, берілген жұмыс мөлшерін атқару арқылы температура айырмасын жасау үшін мүмкін болатын ең тиімді цикл. Ол екі изотермиялық процесс пен екі адиабаталық процесстен тұрады.

Isothermal Process

A thermodynamic process in which the temperature of the system remains constant (∆T = 0). For an ideal gas, this means the internal energy also remains constant (∆U = 0). Heat must be transferred slowly to maintain constant temperature.

Translation (Қазақша)

Жүйенің температурасы тұрақты болып қалатын термодинамикалық процесс (∆T = 0). Идеал газ үшін бұл ішкі энергияның да тұрақты болып қалатынын білдіреді (∆U = 0). Тұрақты температураны сақтау үшін жылу баяу тасымалдануы керек.

Adiabatic Process

A thermodynamic process in which no heat is transferred into or out of the system (Q = 0). This usually occurs if the process is very rapid or the system is well-insulated. Temperature, pressure, and volume all change in an adiabatic process.

Translation (Қазақша)

Жүйеге жылу берілмейтін немесе жүйеден жылу шығарылмайтын термодинамикалық процесс (Q = 0). Бұл әдетте процесс өте жылдам болса немесе жүйе жақсы оқшауланған болса орын алады. Адиабаталық процесте температура, қысым және көлем өзгереді.

Reversible Process (in Thermodynamics)

A process that can be reversed by an infinitesimal change in some property of the system, returning both the system and its surroundings to their original states without any change in the universe. Idealized processes like the Carnot cycle are reversible.

Translation (Қазақша)

Жүйенің қандай да бір қасиетінің шексіз аз өзгеруі арқылы кері айналдыруға болатын процесс, бұл кезде жүйе де, оның қоршаған ортасы да әлемде ешқандай өзгеріссіз бастапқы күйлеріне оралады. Карно циклі сияқты идеалдандырылған процестер қайтымды болып табылады.

Efficiency (η) of a Heat Engine

The ratio of the useful work done by the engine to the heat energy supplied to it from the hot reservoir. η = W / Q_H. For a Carnot engine, η_Carnot = 1 — (T_C / T_H), where T_C and T_H are the absolute temperatures of the cold and hot reservoirs, respectively.

Translation (Қазақша)

Қозғалтқыштың атқарған пайдалы жұмысының оған ыстық резервуардан берілген жылу энергиясына қатынасы. η = W / Q_H. Карно қозғалтқышы үшін, η_Карно = 1 — (T_C / T_H), мұндағы T_C және T_H – сәйкесінше суық және ыстық резервуарлардың абсолюттік температуралары.

The Carnot cycle is a theoretical, ideal thermodynamic cycle conceived by French physicist Sadi Carnot in 1824. It describes the most efficient possible way for a heat engine to convert heat into work between two temperature reservoirs. The cycle is reversibleқайтымды / обратимый and consists of four distinct stages, using an ideal gas as the working substance:

The Four Stages of the Carnot Cycle:

Imagine an ideal gas enclosed in a cylinder with a frictionless piston.

1. Stage 1: Isothermal Expansion (A → B)
   • The gas is in thermal contact with a hot reservoirыстық резервуар / горячий резервуар at a constant high temperature, T_H.
   • The gas expands slowly, doing work on the surroundings (W_AB > 0).
   • To keep the temperature constant during expansion, heat Q_H is absorbed from the hot reservoir.
   • For an ideal gas, since temperature is constant, the internal energy (U) remains constant (ΔU = 0).
   • From the First Law of Thermodynamics (ΔU = Q — W), we have 0 = Q_H — W_AB, so Q_H = W_AB.
2. Stage 2: Adiabatic Expansion (B → C)
   • The gas is now thermally insulated from both reservoirs.
   • The gas continues to expand, doing work on the surroundings (W_BC > 0).
   • Since the process is adiabaticадиабаталық / адиабатический, no heat is exchanged with the surroundings (Q = 0).
   • As the gas expands and does work, its internal energy decreases (ΔU = -W_BC), causing its temperature to drop from T_H to a lower temperature, T_C.
3. Stage 3: Isothermal Compression (C → D)
   • The gas is placed in thermal contact with a cold reservoirсуық резервуар / холодный резервуар at a constant low temperature, T_C.
   • The gas is compressed slowly by an external force, so work is done on the gas (W_CD < 0, if W is work done *by* gas; work done *on* gas is positive).
   • To keep the temperature constant during compression, heat Q_C is rejected from the gas to the cold reservoir.
   • For an ideal gas, since temperature is constant, the internal energy (U) remains constant (ΔU = 0).
   • From the First Law (ΔU = Q — W), 0 = -Q_C — W_CD (using Q_C as heat rejected, so Q = -Q_C for the system), so W_CD = -Q_C. (Work done *by* gas is W_CD, which is negative).
4. Stage 4: Adiabatic Compression (D → A)
   • The gas is again thermally insulated.
   • The gas is further compressed by an external force, so work is done on the gas (W_DA < 0, if W is work done *by* gas).
   • Since the process is adiabatic, no heat is exchanged (Q = 0).
   • As work is done on the gas, its internal energy increases (ΔU = -W_DA), causing its temperature to rise from T_C back to the initial temperature T_H, completing the cycle.

P-V Diagram for the Carnot Cycle:

The Carnot cycle can be represented on a Pressure-Volume (P-V) diagram:

• A → B: Isothermal expansion (curve, part of an isotherm T_H)
• B → C: Adiabatic expansion (steeper curve than isotherm, gas cools)
• C → D: Isothermal compression (curve, part of an isotherm T_C)
• D → A: Adiabatic compression (steeper curve than isotherm, gas heats up)

The net work done by the engine in one cycle is represented by the area enclosed by the cycleтұйық контурмен қоршалған аудан / площадь, заключенная внутри замкнутого контура on the P-V diagram. W_net = W_AB + W_BC + W_CD + W_DA. This is also equal to Q_H — Q_C.

Significance and Efficiency of the Carnot Cycle:

The Carnot cycle is theoretically important because:

• It is a reversible cycleқайтымды цикл / обратимый цикл. All stages are performed infinitely slowly (quasi-statically) without friction or other dissipative effects.
• Carnot’s Theorem: No heat engine operating between two given temperature reservoirs can be more efficient than a Carnot engine operating between the same two reservoirs. All reversible engines operating between the same two temperatures have the same efficiency (Carnot efficiency).

The thermal efficiency (η) of any heat engine is defined as:

η = (Net Work Done by Engine) / (Heat Input from Hot Reservoir) = W / Q_H

Since W = Q_H — Q_C (from energy conservation for a cycle, ΔU_cycle = 0),

η = (Q_H — Q_C) / Q_H = 1 — (Q_C / Q_H)

For a Carnot cycle (and any reversible cycle operating between two temperatures), it can be shown using the thermodynamic temperature scale that Q_C / Q_H = T_C / T_H, where T_C and T_H are the absolute temperatures (in Kelvin) of the cold and hot reservoirs, respectively.

Thus, the efficiency of a Carnot engine is:

η_Carnot = 1 — (T_C / T_H)

This formula shows that the efficiency of a Carnot engine depends only on the temperatures of the hot and cold reservoirs. To achieve 100% efficiency (η = 1), T_C would have to be 0 Kelvin (absolute zero), which is practically unattainable (Third Law of Thermodynamics). Higher T_H and lower T_C lead to higher efficiency.

Questions on Theory:

1. Easy: Name the four processes that make up the Carnot cycle, in order.
   
   Answer

   1. Isothermal expansion. 2. Adiabatic expansion. 3. Isothermal compression. 4. Adiabatic compression.
2. Medium: During which stages of the Carnot cycle is heat transferred to or from the gas? Specify the direction of heat transfer in each case.
   
   Answer

   Heat is transferred during the two isothermal stages:
   1. Isothermal Expansion (A → B): Heat Q_H is transferred to the gas from the hot reservoir.
   2. Isothermal Compression (C → D): Heat Q_C is transferred from the gas to the cold reservoir.
   No heat is transferred during the adiabatic stages.
   
3. Medium: How is the net work done by a Carnot engine in one cycle represented on a P-V diagram?
   
   Answer

   The net work done by a Carnot engine in one cycle is represented by the area enclosed by the cycle (the loop A → B → C → D → A) on the P-V diagram.
4. Hard (Critical Thinking): Why is it impossible for a real heat engine operating between a hot reservoir at T_H and a cold reservoir at T_C to be more efficient than a Carnot engine operating between the same two temperatures?
   
   Answer

   This is a consequence of Carnot’s theorem, which itself is derived from the Second Law of Thermodynamics.
   A Carnot engine is, by definition, a reversible engine. Any real engine will have irreversible processes (e.g., friction, heat loss through imperfect insulation, non-quasi-static expansion/compression). These irreversible processes always lead to a loss of ability to do useful work or require more work to achieve the same result in a reversed cycle (like a refrigerator).
   If a real engine were more efficient than a Carnot engine, one could theoretically combine this hypothetical «super-efficient» engine with a Carnot refrigerator (Carnot cycle run in reverse) operating between the same two reservoirs. The super-efficient engine would produce more work from a given amount of heat Q_H than a Carnot engine. This excess work could then be used to drive the Carnot refrigerator, which would pump more heat from the cold reservoir to the hot reservoir than the super-efficient engine took from the hot reservoir to produce that work. The net effect would be a transfer of heat from a cold reservoir to a hot reservoir without any net work input from the surroundings, which violates the Clausius statement of the Second Law of Thermodynamics («No process is possible whose sole result is the transfer of heat from a cooler to a hotter body»).
   Therefore, no real engine can be more efficient than a Carnot engine operating between the same temperature limits. The Carnot efficiency represents the maximum possible theoretical efficiency.
   

Activity 1: Process Identification

For each description, identify the type of thermodynamic process (Isothermal, Adiabatic):

1. A process where the temperature of the gas remains constant. ______________
2. A process where no heat enters or leaves the system. ______________
3. In an ideal gas, ΔU = 0 during this process. ______________
4. This process often occurs very rapidly or in a well-insulated container. ______________

Activity 2: Carnot Cycle Stages

Fill in the blanks regarding the Carnot Cycle stages:

1. During isothermal expansion, heat is ______________ by the gas, and its internal energy ______________ (for an ideal gas).
2. During adiabatic expansion, the temperature of the gas ______________, and work is done ______________ the gas.
3. During isothermal compression, heat is ______________ by the gas, and work is done ______________ the gas.
4. During adiabatic compression, the temperature of the gas ______________, and its internal energy ______________.

This video provides a detailed explanation of the Carnot Cycle:

Further Viewing — Related Topics:

• Thermodynamics: Carnot Cycle and Heat Engines (Crash Course Physics)
• Understanding the Carnot Cycle (AK Lectures)
• Efficiency of a Carnot Engine (Khan Academy)

Example 1: Carnot Engine Efficiency

A Carnot engine operates between a hot reservoir at 527°C and a cold reservoir at 27°C.

(a) Convert the reservoir temperatures to Kelvin.

(b) Calculate the maximum possible efficiency (Carnot efficiency) of this engine.

(c) If the engine absorbs 2000 J of heat from the hot reservoir (Q_H) per cycle, how much work does it perform per cycle?

(d) How much heat (Q_C) is rejected to the cold reservoir per cycle?

While a direct PhET simulation for the complete Carnot cycle might be complex to set up perfectly, you can use the "Gas Properties" simulation to understand its individual components (isothermal and adiabatic processes).

Link to Simulation: PhET Gas Properties Simulation

Instructions & Tasks:

1. Isothermal Expansion (Conceptual):
   • Add particles to the container. Note the temperature.
   • To expand isothermally, you would increase the volume (drag the handle). What would you need to do with the heat control (bottom) to keep the temperature constant as the gas expands and does work? (Hint: Work done by gas tends to cool it).
2. Adiabatic Expansion (Conceptual):
   • Reset or start with particles at a certain temperature.
   • To expand adiabatically, you would increase the volume. For an adiabatic process, Q=0. What would happen to the temperature of the gas if it expands and does work without any heat input? Observe this by rapidly expanding the volume and not adding heat.
3. Isothermal Compression (Conceptual):
   • To compress isothermally, you would decrease the volume. What would you need to do with the heat control to keep the temperature constant as work is done on the gas (which tends to heat it)?
4. Adiabatic Compression (Conceptual):
   • To compress adiabatically, you would decrease the volume with Q=0. What would happen to the temperature of the gas if work is done on it without any heat being removed? Observe this by rapidly compressing the volume.

Discussion: How do these observations relate to the energy changes (ΔU, Q, W) in each stage of the Carnot cycle?

Work with a partner or in a small group.

Task: Energy Flow in a Carnot Cycle

Consider a Carnot engine operating between T_H = 600 K and T_C = 300 K. In one cycle, it absorbs 1200 J of heat from the hot reservoir.

1. Calculate the efficiency of this Carnot engine.
2. Calculate the net work done by the engine per cycle.
3. Calculate the amount of heat rejected to the cold reservoir per cycle.
4. For each of the four stages of the Carnot cycle (Isothermal Expansion, Adiabatic Expansion, Isothermal Compression, Adiabatic Compression), discuss qualitatively:
   • Is heat (Q) added, removed, or zero?
   • Is work (W) done by the gas, on the gas, or zero? (Specify sign of W if W is work done by gas)
   • Does the internal energy (ΔU) of the ideal gas increase, decrease, or remain zero?

You can use a shared document (e.g., Google Docs) or a tool like Lucidchart (for diagrams/flow) to organize your answers and discussion.

Answer the following questions to test your understanding and ability to apply the concepts of the Carnot cycle. Show your working where necessary.

1. A Carnot engine has an efficiency of 30%. The engine absorbs heat from a hot reservoir at a temperature of 600 K.

   (a) Calculate the temperature of the cold reservoir in Kelvin and in Celsius.

   (b) If the engine performs 150 J of work per cycle, how much heat does it absorb from the hot reservoir?

   (c) How much heat is rejected to the cold reservoir per cycle?

2. Explain why a Carnot cycle is described as a «reversible» cycle. What are the practical implications of this reversibility in terms of achieving maximum possible efficiency?

3. Sketch a P-V diagram for a Carnot cycle, clearly labeling the four stages (A→B, B→C, C→D, D→A) and indicating whether each process is isothermal or adiabatic. For each stage, state whether: (i) heat is added to the gas, removed from the gas, or zero; (ii) the internal energy of the ideal gas increases, decreases, or is zero.

4. Two Carnot engines, X and Y, operate between the same hot reservoir temperature T_H. Engine X has a cold reservoir temperature T_CX. Engine Y has a cold reservoir temperature T_CY, where T_CY < T_CX. Which engine has a higher thermal efficiency? Justify your answer using the Carnot efficiency formula.

5. A scientist claims to have built a heat engine that operates between a hot reservoir at 700 K and a cold reservoir at 300 K, and produces 500 J of work for every 1000 J of heat it absorbs from the hot reservoir.

   (a) Calculate the claimed efficiency of this engine.

   (b) Calculate the maximum possible efficiency (Carnot efficiency) for an engine operating between these temperatures.

   (c) Is the scientist’s claim plausible? Explain your reasoning based on Carnot’s theorem.

(Note: These questions are designed for analysis and synthesis. Focus on clear explanations and accurate calculations.)

• Save My Exams — Carnot Cycle & Efficiency: The Carnot Cycle Notes
• PhysicsAndMathsTutor — Carnot Cycle: Search within their A-Level Physics section for «Carnot Cycle» or «Heat Engines». (Direct links vary)
• OpenStax University Physics — Carnot Cycle: Chapter 4.5: Carnot Engine
• HyperPhysics — Carnot Cycle: HyperPhysics — Carnot Cycle
• Chem LibreTexts — Carnot Cycle: The Carnot Cycle (Detailed Explanation)

Take a few moments to reflect on what you have learned in this lesson:

1. What is the primary reason the Carnot cycle is considered theoretically important, even though a perfect Carnot engine cannot be built in practice?
2. If you wanted to increase the efficiency of a Carnot engine, would it be more effective to increase T_H by a certain amount or decrease T_C by the same amount (in Kelvin)? Explain your reasoning using the efficiency formula.
3. How does the concept of a «reversible process» relate to the stages of the Carnot cycle?
4. On a scale of 1 (Not at all understood) to 5 (Very well understood), how would you rate your current understanding of why the Carnot cycle represents the maximum possible efficiency for a heat engine?
5. Can a Carnot cycle be run in reverse? If so, what kind of device would it represent, and what would be its purpose?