Carnot's Theorem: Proof, Applications, and Limitation

Carnot's Theorem is a basic principle of thermodynamics that tells us how efficient a heat engine can be at its best.

• Carnot’s theorem is also known as Carnot’s rule or Carnot’s principle.
• The theorem states that any heat engine does not have 100% efficiency. For maximum efficiency of the heat engine, the processes involved should be reversible. 
• This theorem is the basis of modern thermodynamics. It is based on the idea that some processes can be reversed. 
• It's used in engineering, physics, and chemistry, among other fields, and it's still a key idea in the study of energy and heat transfer.

Key Terms: Carnot’s theorem, Heat engine, Entropy change, Second law of thermodynamics, Carnot cycle, Efficiency of heat engine, Reversible and irreversible process.

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Carnot’s Theorem

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In 1924, French scientist Nicolas Leonard Sadi Carnot developed a theory that specifies limits on the maximum efficiency that any heat engine can obtain, which is famously known as Carnot’s theorem. He is also known as the "Father of Thermodynamics" because of his work on the Carnot theorem and cycle.

Carnot’s theorem states that

• All heat engines operating between the two heat reservoirs cannot have efficiencies greater than a reversible heat engine operating between the same reservoirs.
• The efficiency of the Carnot engine is independent of the working substance.

All reversible heat engine operating between two temperatures is called Carnot engine and the sequences of steps constituting one cycle are called the Carnot cycle.

The Maximum efficiency of a heat engine, also known as the efficiency of a Carnot engine is given as,

η_max = η_carnot = \(\frac{T_H-T_C}{T_H} = 1-\frac{T_C}{T_H}\)

Where

• ƞ is the ratio of the work done by the engine to the heat drawn out of the hot reservoir.
• T_C is the absolute temperature of the cold reservoir.
• T_H is the absolute temperature of the hot reservoir.

Reversible Process

A reversible process is a process in which at any stage of the process it can be traversed back in the opposite direction in such a way that the system passes through exactly the same conditions at every step in the reverse process as in the direct process.

The basic requirement for a process to be reversible is:

1. The process should take place very slowly.
2. The pressure difference between the working substance and surrounding at any stage of the process should be very small.
3. There should be no friction.
4. There should be no loss of energy.

Irreversible Process

Irreversible process may be defined as the process in which the system cannot be retraced to its initial state.

Thermodynamics processes that occur in nature are all irreversible processes. Some examples of irreversible process are

• Heat generation during the flow of current through a resistor.
• Work done against friction.
• Diffusion of two gases.
• Exchange of heat between bodies at different temperatures.

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Carnot’s Theorem Proof

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Let's say that there is a heat engine X that works between two reservoirs at temperatures T₁ and T₂ respectively. The heat engine X makes work W by moving heat Q₁ from the hot reservoir (source) at T₁ to the cold reservoir (sink) at temperature T₂.

Heat Engine

Also, consider a Carnot engine that works between the same two reservoirs at temperatures T₁ and T₂ respectively. The Carnot engine goes through two processes that are isothermal and two processes that are adiabatic.

The efficiency of the Carnot engine is given by:

ƞ_carnot = 1 – \(\frac{T_2}{T_1}\) ………………(i)

Let ΔS be the net change in universe entropy.

At steady temperature T₁, the source emits heat Q₁, and the change in its entropy is given by

ΔS₁ = – \(\frac{Q_1}{T_1}\)

At steady temperature T₂, the sink receives heat Q₂, and the change in its entropy is given by

ΔS₂ = \(\frac{Q_2}{T_2}\)

Net change in universe’s entropy is given by

ΔS = ΔS₁ + ΔS₂

⇒ ΔS = – \(\frac{Q_1}{T_1}\) + \(\frac{Q_2}{T_2}\)

From Second law of thermodynamics, we have

ΔS ≥ 0

- \(\frac{Q_1}{T_1}\) + \(\frac{Q_2}{T_2}\) ≥ 0

\(\frac{Q_2}{T_2}\) ≥ \(\frac{Q_1}{T_1}\)

\(\frac{Q_2}{Q_1}\) ≥ \(\frac{T_2}{T_1}\)

Subtracting both sides by 1, we get

1 – \(\frac{Q_2}{Q_1}\) ≥ 1 – \(\frac{T_2}{T_1}\)

Left side equation of the above expression shows the efficiency of heat engine X, and from equation (i), right side equation represents the efficiency of Carnot engine. i.e.

ƞ_X ≥ ƞ_carnot

So, we can say that a heat engine running between two heat reservoirs can't be more efficient than a Carnot engine running between the same two reservoirs.

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Limitations of Carnot Theorem

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The Carnot Theorem is crucial to understanding heat engine thermodynamics, but its limits must be considered when applying it to real-world systems.

• Only idealized thermodynamic cycles use the Carnot Theorem. That doesn't work for imperfect systems that lose energy differently.
• The theory assumes thermal equilibrium, which may not be true in practice.
• The Carnot Theorem implies the engine's working fluid is a perfect gas, which many fluids are not.
• The theorem states that heat transfer between the system and reservoirs can always be bidirectional, which is not necessarily true.
• The Carnot Theorem only applies to systems between two reservoirs with fixed temperatures. It's unsuitable for multi-temperature systems.
• The theorem ignores friction, heat leaks, and other energy losses in real-world systems.

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Applications of Carnot Theorem

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The Carnot Theorem has many applications in engineering, physics, and materials research, despite its limits.

• The Carnot Theorem is needed to develop and improve heat engines like steam engines and gas turbines by figuring out how efficient they can be at their theoretical best.
• The Carnot Theorem is a way to figure out how much energy is needed or given off during phase changes like melting and boiling.
• This Theorem can be used to study thermoelectric materials, which directly convert temperature differences into electricity, and to improve thermoelectric devices.

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Things to Remember

• Carnot’s theorem states that no heat engine has greater efficiency than a Carnot engine.
• Carnot's theory gives engineers a theoretical maximum heat engine efficiency to use when constructing and enhancing engines.
• The efficiency of the Carnot engine is given by: η_carnot = 1 – \(\frac{T_2}{T_1}\)
• Entropy is defined as the system’s thermal energy per unit temperature that is unavailable for doing useful work.
• Carnot's theorem calculates thermoelectric devices' maximum efficiency.
• Carnot's theorem can be used to study solid-to-liquid transitions and reservoir temperatures' effects on heat engine efficiency.
• Carnot's theorem can improve solar and wind power efficiency.
• Carnot's theorem can study Earth's climate system's efficiency and temperature changes.
• The efficiency of a heat engine is given by η = 1 – \(\frac{Q_2}{Q_1}\)