Carnot’s Cycle Proof

What is Carnot’s cycle proof or theorem?

Carnot’s heat engine is an ideal heat engine that operates on the Carnot cycle. Nicolas Leonard Sadi Carnot developed the model for this engine in 1824. Its working is similar to the heat engine and is based on the second law of thermodynamics.

Second law of thermodynamics

It is not possible to design a heat engine which works in a cyclic process and whose only result is to take heat from a body at a single temperature and convert it completely into mechanical work.

This statement of the second law is called the Kelvin-Planck statement. 

One can convert mechanical work completely into heat but cannot convert heat completely into mechanical work. In this respect, heat and work are not equivalent. We shall now study some other aspects of the second law of thermodynamics.

Heat engine

The device used to convert heat energy into mechanical energy is called a heat engine. For conversion of heat into work with the help of a heat engine, the following conditions have to be met:

• There should be a body at higher temperature ‘T1‘ from which heat is extracted, called the source. 

• Body of the engine contains a working substance.

• There should be a body at lower temperature ‘T2‘ to which heat can be rejected, called the sink.

Working of heat engine

The schematic diagram of the heat engine is given below.

Engine derives an amount of ‘Q1‘ of heat from the source.

A part of this heat is converted into work ‘W’. Remaining heat ‘Q,’ is rejected to the sink.

Thus   

Q1 = W+Q2

or the work done by the engine is given by

W=Q1-Q2

Efficiency of heat engine

Efficiency of heat engine (𝛈) is defined as the fraction of total heat supplied to the engine which is converted into work. 

Mathematically,

Since 𝛈=  W/Q1

Or 𝛈= Q1-Q2/Q1 = 1-Q2/Q1

Carnot’s heat engine

The Carnot engine is an ideal heat engine that operates on the Carnot cycle. The model for this engine was developed by Nicolas Leonard Sadi Carnot in 1824. Its different parts are shown below.

Source: It is a reservoir of heat energy with a conducting top maintained at a constant temperature T1 K. The source is so big that extraction of any amount of heat from it does not change its temperature.

Body of heat engine : It is a barrel with perfectly insulating walls and conducting bottom. It is fitted with an airtight piston capable of sliding within the barrel without friction. The barrel contains some quantity of an ideal gas.

Sink: It is a huge body at a lower temperature T2having a perfectly conducting top. The size of the sink is so large that any amount of heat rejected to it does not increase its temperature.

Insulating stand: It is a stand made up of perfectly insulating material such that a barrel placed over it becomes thoroughly insulated from the surroundings.

Carnot’s heat engine working and its cycle 

When the Carnot engine works, the working substance of the engine undergoes a different process known as the Carnot cycle; this cycle comprises four different stages.

1. First stage, known as isothermal expansion

The barrel is placed over the source. The piston is gradually pushed back as the gas expands. Fall of temperature, due to expansion, is compensated by the supply of heat from the source and consequently temperature remains constant.

The conditions of the gas change from A(P1, V1) to B(P2, V2). If W1 is the work done during this process, then heat Q1derived from the source is given by,

Q1 =W1 = -nRT1 loge(V2 / V1)

2. Second stage, known as adiabatic expansion

The barrel is removed from the source and placed over the insulating stand. The piston is pushed back so that the gas expands adiabatically, resulting in a fall of temperature from T1 to T2. The conditions of the gas change from B(P2, V2) to C(P3, V3). If W2 is the work done in this case then,

W2 = nCv(T2 – T1)

3. Third stage known as isothermal compression

The barrel is placed over the sink. Piston is pushed down by compressing the gas. The heat generated due to compression flows to the sink, maintaining the temperature of the barrel constant. The state of the gas changes from  C(P3, V3) to D(P4 , V4). If W, is the work done in this process and Q, is the heat rejected to the sink, then

W3 = -nRT2 loge(V4 / V3)

4. Fourth stage, known as adiabatic compression

The barrel is placed over the insulating stand. The piston is moved down, thereby compressing the gas adibatically till the temperature of gas increases from T2to T1The state of gas changes from D(P4 , V4) to   A(P1, V1) . If W4 is the work done in this process, then

W4 =  nCv(T1 – T2)

 Heat converted into work in Carnot’s cycle

Wcycle = W1 + W2 + W3 + W4

⇒ – nRT1 loge(V2 / V1) +  nCv(T2 – T1) – nRT2 loge(V4 / V3) +  nCv(T1 – T2)

⇒ -nR[ T1loge(V2 / V1) + T2 loge(V4 / V3) ] 

For BC, T1V2𝜸 – 1 = T2V3𝜸 – 1

For DA, T1V1𝜸 – 1 = T2V4𝜸 – 1

(V2 / V1)𝜸 – 1 = (V3 / V4)𝜸 – 1 ⇒  V2 / V1 = V3 / V4

Thus, net work done by the engine during one cycle is equal to the area enclosed by the indicator diagram of the cycle. Analytically,

Wcycle = -nR(T1 – T2) loge(V2 / V1) 

Carnot’s cycle theorem

Carnot engine is a reversible engine. It can be proved from the second law of thermodynamics that:

All reversible engines operating between the two temperatures of equal value have equal efficiency and no engine operating between the same two temperatures is able to achieve the efficiency greater than this.

The above theorem is known as the Carnot’s cycle theorem. It is a consequence of the second law and puts a theoretical limit 𝛈 =1-T2/T1to the maximum efficiency of heat engines.

Carnot’s cycle proof

Efficiency (𝛈) of an engine can be described as the ratio of used heat (heat converted into work) to the total heat supplied to the engine. Thus,

𝛈 =| W/Q1|= |(Q1 – Q2)/Q1|

𝛈 = nR(T1 – T2) loge(V2 / V1) / nRT1 loge(V2 / V1)  = (T1 – T2) / T1

𝛈 = 1 – Q2 / Q1 = 1 – T2 / T1

Hence, we can say that the efficiency of the engine depends only on the temperature of the hot and cold bodies between which the engine works and puts a theoretical limit 𝛈 = 1 – T2/T1

to the maximum efficiency of heat engines.

Some important points regarding Carnot’s engine and its theorem

1. Efficiency of an engine depends upon the temperatures between which it operates.
2. 𝛈 is independent of the nature of the working substance.
3. 𝛈 is one only if T2= 0 Since absolute zero is not attainable, hence even an ideal engine cannot be 100% efficient.
4. 𝛈 is one only if Q2 = 0 But 𝛈 = 1 is never possible even for an ideal engine. Hence Q2 ≠ 0.
5. Thus it is impossible to extract heat from a single body and convert the whole of it into work.
6. If T2= T1 , then 𝛈 = 0
7. In actual heat engines, there are many losses due to friction etc. and various processes during each cycle are not quasistatic, so the efficiency of actual engines is much less than that of an ideal engine.

Conclusion

Carnot cycle is an ideal cycle which predicts the maximum possible efficiency an engine can perform. The cycle includes the processes at constant temperature and at constant heat or isothermal and adiabatic processes. We have also learned the work done in an adiabatic process.