Derivation : Carnot Engine and it's efficiency ?
Carnot Engine: Definition, Working, Carnot Cycle and Efficiency | Class 11 Physics Notes
The Carnot engine is one of the most important concepts in thermodynamics. It is an ideal heat engine that operates on a completely reversible cycle called the Carnot cycle. The Carnot engine establishes the maximum possible efficiency that any heat engine can achieve while operating between two thermal reservoirs. Since no real engine can be completely reversible, the Carnot engine serves as the benchmark against which the performance of all practical heat engines is compared.
In this article, we will study the definition, history, construction, working, Carnot cycle, detailed derivation of efficiency, Carnot theorem, universal relation, applications, limitations, important formulas, FAQs, and MCQs.
History of Carnot Engine
In 1824, the French engineer Sadi Carnot investigated an important question:
What is the maximum possible efficiency of a heat engine operating between two thermal reservoirs?
Surprisingly, Carnot answered this question even before the laws of thermodynamics were fully established. He showed that the most efficient heat engine must operate through a completely reversible cycle. This ideal engine is now known as the Carnot engine, and the cycle followed by it is called the Carnot cycle.
Today, the Carnot engine forms the basis of the Second Law of Thermodynamics and the thermodynamic temperature scale.
What is a Carnot Engine?
A Carnot engine is a theoretical heat engine that operates on a completely reversible cycle consisting of two reversible isothermal processes and two reversible adiabatic processes. It gives the maximum possible efficiency for any heat engine working between two fixed temperatures.
Definition:
A Carnot engine is a reversible heat engine operating between a hot reservoir at temperature T1 and a cold reservoir at temperature T2, which works on the Carnot cycle and has the maximum possible efficiency.
Why is the Carnot Engine an Ideal Engine?
The efficiency of a heat engine decreases whenever there is friction, turbulence, sudden expansion, heat leakage, or finite temperature difference during heat transfer. These effects make the process irreversible.
To obtain the maximum possible efficiency, every process of the engine must be completely reversible.
Therefore, the Carnot engine is considered an ideal engine because:
- All processes are reversible.
- There is no friction.
- There is no heat loss.
- Heat transfer occurs through an infinitesimally small temperature difference.
- No energy is dissipated.
Since these conditions cannot be achieved in practice, the Carnot engine is only a theoretical model.
Reversible and Irreversible Processes
Reversible Process
A reversible process is one that can be reversed by an infinitesimally small change in external conditions without leaving any permanent change in the system or surroundings.
A reversible process must satisfy the following conditions:
- It must be quasi-static.
- It must be frictionless.
- There should be no dissipative effects.
- Heat transfer must occur through an infinitesimally small temperature difference.
Irreversible Process
An irreversible process cannot be completely reversed because of friction, turbulence, viscosity, finite temperature difference, rapid compression or expansion, and other dissipative effects.
All real engines operate through irreversible processes; therefore, their efficiency is always less than that of a Carnot engine.
Why Does the Carnot Cycle Consist of Two Isothermal and Two Adiabatic Processes?
Consider a heat engine operating between a hot reservoir at temperature T1 and a cold reservoir at temperature T2.
To absorb heat from the hot reservoir without any temperature difference, the working substance must remain at the same temperature T1. Hence, heat absorption must occur through an isothermal expansion.
Similarly, heat must be rejected to the cold reservoir while the working substance remains at temperature T2. Therefore, heat rejection occurs through an isothermal compression.
After absorbing heat at T1, the temperature of the gas must decrease to T2. Likewise, after rejecting heat, its temperature must increase again from T2 to T1.
This temperature change cannot occur through another isothermal process. It is achieved through reversible adiabatic expansion and reversible adiabatic compression, during which no heat is exchanged with the surroundings.
Therefore, one complete Carnot cycle consists of:
- Two reversible isothermal processes.
- Two reversible adiabatic processes.
Carnot Cycle
The Carnot cycle is a reversible thermodynamic cycle consisting of four successive reversible processes performed by an ideal gas.
- Isothermal expansion at temperature T1.
- Adiabatic expansion from T1 to T2.
- Isothermal compression at temperature T2.
- Adiabatic compression from T2 to T1.
After completing these four processes, the working substance returns to its initial state, and the cycle repeats continuously.
P–V Diagram of Carnot Cycle
The pressure-volume (P–V) diagram of the Carnot cycle consists of two isothermal curves connected by two adiabatic curves. The enclosed area represents the net work done by the engine in one complete cycle.
Insert the NCERT P–V diagram here.
Working of Carnot Engine
The Carnot engine works between two heat reservoirs:
- Hot reservoir at temperature T1.
- Cold reservoir at temperature T2.
An ideal gas acts as the working substance. During one complete cycle, it absorbs heat from the hot reservoir, converts part of it into useful work, and rejects the remaining heat to the cold reservoir before returning to its initial state.
The working of the Carnot engine consists of the following four reversible processes:
- Isothermal expansion (A → B)
- Adiabatic expansion (B → C)
- Isothermal compression (C → D)
- Adiabatic compression (D → A)
Each process will be explained in detail with mathematical derivation in the next section.
Detailed Working of Carnot Engine
During one complete Carnot cycle, the working substance (an ideal gas) undergoes four reversible thermodynamic processes. In the first two processes, the gas expands and performs work. In the last two processes, the gas is compressed and returns to its initial state.
Process 1: Isothermal Expansion (A → B)
Initially, the gas is at pressure P1, volume V1, and temperature T1. It is placed in contact with the hot reservoir.
Since the process is isothermal, the temperature remains constant at T1. The gas absorbs heat from the hot reservoir and expands slowly from volume V1 to V2.
Characteristics
- Temperature remains constant (T1).
- Heat is absorbed from the hot reservoir.
- Internal energy remains unchanged.
- Heat absorbed is completely converted into work done by the gas.
Heat absorbed from the source:
$$Q_1=nRT_1\ln\!\left(\frac{V_2}{V_1}\right)$$
Work done by the gas:
$$W_{AB}=nRT_1\ln\!\left(\frac{V_2}{V_1}\right)$$
Since ΔU = 0 for an ideal gas during an isothermal process,
$$Q_1=W_{AB}$$
Process 2: Adiabatic Expansion (B → C)
After the isothermal expansion, the cylinder is thermally insulated from both reservoirs. The gas now expands adiabatically from state B to state C.
No heat enters or leaves the system during this process.
As the gas expands, it performs work at the expense of its internal energy. Consequently, its temperature falls from T1 to T2.
Characteristics
- No heat exchange.
- Temperature decreases from T1 to T2.
- Internal energy decreases.
- The gas continues to perform positive work.
Heat exchanged:
$$Q=0$$
Work done by the gas:
$$W_{BC}=\frac{nR}{\gamma-1}(T_1-T_2)$$
Adiabatic relation:
$$T_1V_2^{\gamma-1}=T_2V_3^{\gamma-1}$$
Process 3: Isothermal Compression (C → D)
At state C, the gas comes in contact with the cold reservoir maintained at temperature T2.
The gas is compressed slowly and reversibly at constant temperature T2. During compression, heat is rejected to the cold reservoir.
Characteristics
- Temperature remains constant (T2).
- Heat is rejected to the sink.
- Internal energy remains constant.
- External work is done on the gas.
Work done by the gas:
$$W_{CD}=nRT_2\ln\!\left(\frac{V_4}{V_3}\right)$$
Since
$$\ln\!\left(\frac{V_4}{V_3}\right)=-\ln\!\left(\frac{V_3}{V_4}\right)$$
the above quantity is negative, indicating that work is done on the gas.
Heat rejected to the sink:
$$Q_2=-nRT_2\ln\!\left(\frac{V_4}{V_3}\right)$$
Process 4: Adiabatic Compression (D → A)
Finally, the cylinder is again thermally insulated. The gas is compressed adiabatically until it regains its original temperature T1.
No heat is exchanged during this process. The external work done on the gas increases its internal energy, causing the temperature to rise from T2 to T1.
Characteristics
- No heat exchange.
- Temperature rises from T2 to T1.
- Internal energy increases.
- External work is done on the gas.
Heat exchanged:
$$Q=0$$
Work done:
$$W_{DA}=\frac{nR}{\gamma-1}(T_2-T_1)$$
Adiabatic relation:
$$T_2V_4^{\gamma-1}=T_1V_1^{\gamma-1}$$
Summary of the Four Processes
| Process | Nature | Heat Transfer | Temperature | Work |
|---|---|---|---|---|
| A → B | Isothermal Expansion | Heat absorbed (Q₁) | Constant (T₁) | Positive |
| B → C | Adiabatic Expansion | Q = 0 | T₁ → T₂ | Positive |
| C → D | Isothermal Compression | Heat rejected (Q₂) | Constant (T₂) | Negative |
| D → A | Adiabatic Compression | Q = 0 | T₂ → T₁ | Negative |
Derivation of Net Work Done in a Carnot Cycle
The total work done during one complete Carnot cycle is the algebraic sum of the work done in all four processes.
$$W_{\text{net}}=W_{AB}+W_{BC}+W_{CD}+W_{DA}$$
Substituting the expressions for work done,
$$W_{\text{net}}=nRT_1\ln\!\left(\frac{V_2}{V_1}\right)+\frac{nR}{\gamma-1}(T_1-T_2)+nRT_2\ln\!\left(\frac{V_4}{V_3}\right)+\frac{nR}{\gamma-1}(T_2-T_1)$$
The adiabatic works cancel each other because
$$W_{BC}+W_{DA}=0$$
Therefore,
$$W_{\text{net}}=nRT_1\ln\!\left(\frac{V_2}{V_1}\right)+nRT_2\ln\!\left(\frac{V_4}{V_3}\right)$$
Using the logarithmic identity,
$$\ln\!\left(\frac{V_4}{V_3}\right)=-\ln\!\left(\frac{V_3}{V_4}\right)$$
we obtain
$$W_{\text{net}}=nRT_1\ln\!\left(\frac{V_2}{V_1}\right)-nRT_2\ln\!\left(\frac{V_3}{V_4}\right)$$
In the next section, we shall use the adiabatic relations to simplify this expression and derive the efficiency formula of the Carnot engine.
Adiabatic Relations Used in the Derivation
To simplify the expression for the net work done, we use the adiabatic relations corresponding to the two adiabatic processes of the Carnot cycle.
For the adiabatic expansion (B → C),
$$T_1V_2^{\gamma-1}=T_2V_3^{\gamma-1}$$
or
$$\frac{T_1}{T_2}=\left(\frac{V_3}{V_2}\right)^{\gamma-1}$$
Similarly, for the adiabatic compression (D → A),
$$T_1V_1^{\gamma-1}=T_2V_4^{\gamma-1}$$
or
$$\frac{T_1}{T_2}=\left(\frac{V_4}{V_1}\right)^{\gamma-1}$$
Comparing the above equations,
$$\left(\frac{V_3}{V_2}\right)^{\gamma-1}=\left(\frac{V_4}{V_1}\right)^{\gamma-1}$$
Therefore,
$$\frac{V_3}{V_2}=\frac{V_4}{V_1}$$
or
$$\frac{V_3}{V_4}=\frac{V_2}{V_1}$$
Hence,
$$\ln\!\left(\frac{V_3}{V_4}\right)=\ln\!\left(\frac{V_2}{V_1}\right)$$
Final Expression for Net Work Done
Substituting the above relation into the expression for net work, we obtain
$$W_{\text{net}}=nR(T_1-T_2)\ln\!\left(\frac{V_2}{V_1}\right)$$
This equation gives the net work done by the Carnot engine in one complete cycle.
Derivation of Carnot Engine Efficiency
The efficiency of a heat engine is defined as the ratio of the net work done in one complete cycle to the heat absorbed from the hot reservoir.
$$\eta=\frac{\text{Net Work Done}}{\text{Heat Absorbed from Source}}$$
or
$$\eta=\frac{W_{\text{net}}}{Q_1}$$
Substituting the expressions for work and heat absorbed,
$$\eta=\frac{nR(T_1-T_2)\ln\!\left(\frac{V_2}{V_1}\right)}{nRT_1\ln\!\left(\frac{V_2}{V_1}\right)}$$
After cancelling the common terms,
$$\boxed{\eta=\frac{T_1-T_2}{T_1}}$$
Therefore, the efficiency of a Carnot engine is
$$\boxed{\eta=1-\frac{T_2}{T_1}}$$
Interpretation of the Efficiency Formula
- The efficiency depends only on the temperatures of the hot and cold reservoirs.
- It does not depend on the working substance.
- The greater the temperature difference, the greater is the efficiency.
- Since absolute zero (0 K) cannot be achieved, no heat engine can attain 100% efficiency.
Carnot Theorem
Carnot's Theorem states that:
No heat engine operating between the same two temperatures can be more efficient than a reversible Carnot engine.
The theorem further states that all reversible engines operating between the same two temperatures have the same efficiency, irrespective of the nature of the working substance.
Universal Relation for a Carnot Engine
For a reversible Carnot engine, the ratio of heat exchanged is equal to the ratio of the absolute temperatures of the reservoirs.
$$\boxed{\frac{Q_1}{Q_2}=\frac{T_1}{T_2}}$$
This important relation forms the basis of the thermodynamic temperature scale.
Important Characteristics of a Carnot Engine
- It is an ideal heat engine.
- All four processes are reversible.
- It operates between two thermal reservoirs.
- It has the maximum possible efficiency.
- Its efficiency depends only on reservoir temperatures.
- It is independent of the nature of the working substance.
- The enclosed area of the P–V diagram represents the net work done in one cycle.
Applications of Carnot Engine
- To determine the maximum possible efficiency of heat engines.
- To compare the performance of practical engines.
- To establish the thermodynamic temperature scale.
- To explain the Second Law of Thermodynamics.
- To improve the design of thermal power plants and refrigeration systems.
Limitations of Carnot Engine
- It is only a theoretical engine.
- Perfectly reversible processes cannot be achieved in practice.
- Heat transfer through an infinitesimal temperature difference requires infinite time.
- The engine produces negligible power because the cycle is extremely slow.
- Absolute zero temperature cannot be attained; therefore, 100% efficiency is impossible.
Important Formulae
- $$Q_1=nRT_1\ln\!\left(\frac{V_2}{V_1}\right)$$
- $$Q_2=nRT_2\ln\!\left(\frac{V_3}{V_4}\right)$$
- $$W_{\text{net}}=nR(T_1-T_2)\ln\!\left(\frac{V_2}{V_1}\right)$$
- $$\eta=1-\frac{T_2}{T_1}$$
- $$\frac{Q_1}{Q_2}=\frac{T_1}{T_2}$$
Key Points for CBSE, NEET and JEE
- Carnot engine has the highest possible efficiency.
- Efficiency depends only on the temperatures of the source and sink.
- Efficiency is independent of the working substance.
- Every real engine has an efficiency less than that of a Carnot engine.
- Absolute zero cannot be reached, so no heat engine can have 100% efficiency.
- The Carnot engine is the standard for comparing all practical heat engines.
Conclusion
The Carnot engine is an ideal reversible heat engine that establishes the upper limit of efficiency for all heat engines. It operates on two reversible isothermal and two reversible adiabatic processes. Its efficiency depends only on the temperatures of the hot and cold reservoirs and is completely independent of the working substance. Although it cannot be realized in practice, the Carnot engine remains one of the most important concepts in thermodynamics and forms the foundation of modern heat engine theory.
Frequently Asked Questions (FAQs) on Carnot Engine
1. What is a Carnot engine?
A Carnot engine is an ideal reversible heat engine that operates on the Carnot cycle and provides the maximum possible efficiency for any heat engine working between two thermal reservoirs.
2. What is the Carnot cycle?
The Carnot cycle is a reversible thermodynamic cycle consisting of two isothermal processes and two adiabatic processes.
3. Why is the Carnot engine called an ideal engine?
It is called an ideal engine because all the processes are completely reversible, there is no friction or heat loss, and it has the maximum possible efficiency.
4. What is the efficiency formula of a Carnot engine?
$$\eta=1-\frac{T_2}{T_1}$$
5. Does the efficiency depend on the working substance?
No. The efficiency depends only on the temperatures of the hot and cold reservoirs.
6. Why can't a Carnot engine have 100% efficiency?
Because 100% efficiency requires the sink temperature to be 0 K, and absolute zero cannot be achieved in practice.
7. What is the significance of Carnot's theorem?
It states that no heat engine operating between the same two temperatures can be more efficient than a reversible Carnot engine.
8. What is the working substance of a Carnot engine?
An ideal gas is generally considered as the working substance in a Carnot engine.
Multiple Choice Questions (MCQs) – Carnot Engine
Conclusion
The Carnot engine is the most efficient theoretical heat engine and serves as the standard for comparing all practical heat engines. It operates on two reversible isothermal and two reversible adiabatic processes. The efficiency of a Carnot engine depends only on the temperatures of the source and sink and is independent of the working substance. Although it cannot be realized in practice, the Carnot engine plays a fundamental role in understanding thermodynamics, heat engines, and the Second Law of Thermodynamics.
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