Introduction:

It is been observed that the transfer of energy between a system and its surroundings occurs via heat and work. This is regulated by the first law of thermodynamics which states that the increase in the energy of a system should be accompanied via an equivalent decrease in the energy of the Surroundings and vice-versa. Though, it doesn't tell us anything regarding the feasibility and direction of flow of energy. According to the first law of thermodynamics, all methods in which energy is conserved are possible. For illustration, if a cup of hot tea is left on a table then according to the first law it may be cooled by transferring energy to the surroundings or be heated via absorbing energy from the surroundings. However we all are familiar from daily experience that the cup of tea will forever cool till it needs the temperature of the surroundings. Likewise, if a bottle of perfume is opened in a room, the perfume spreads all through the room. The reverse procedure in which all the perfume vapours are collected in the bottle does not take place. These are illustrations of what are known as spontaneous processes that are irreversible and carry out only in one direction. Again, according to the first law, there is a direct relationship between the heat and work. However, it doesn't state us whether heat can be fully transformed into work and if so, what is the effect on the system and the surroundings.

Reversible, Irreversible and cyclic processes:

As we are familiar that all the thermodynamic properties are state functions and are independent of the path adopted via the system. As well, the internal energy change of a system is represented by the equation ΔU = q + w. Here, ΔU is independent of the path selected however q and w certainly based on it. Therefore for the same ΔU, different values of q and w are possible by bringing about the method in different ways. It was as well illustrated that the work done by a system is maximum if a reversible path is adopted and this maximum work can be found out from the initial and final states of the system. Let us take a reversible and an irreversible procedure in which ΔU is same. No matter how we carry out the process (that is, reversible or irreversible), ΔU based only on the initial and final states of the system. Therefore,

ΔU = q_rev + w_rev for reversible process

And ΔU = q_rev + w_irrev for irreversible process

And therefore q_rev + w_rev = q_rev + w_irrev

We are familiar that the work done by a system under the reversible conditions (-w_rev) is bigger than the work done by a system under irreversible conditions (-w_irrev).

That is, -w_rev > -w_irrev

Therefore, w_rev < w_irrev

This is true whenever we compare the work done on the system under reversible and irreversible conditions. Let us suppose that the driving forces beneath the two conditions illustrated above are fairly different. As well, let us suppose that the processes are not adiabatic in such a way q_rev or q_irrev is not equivalent to zero. By using the equations q_rev + w_rev = q_rev + w_irrev and w_rev < w_irrev, we can represent,

q_rev > q_irrev

This signifies that in a non-adiabatic process, heat absorbed via a system from the surroundings is more under reversible conditions than under irreversible conditions. This relationship will be employed while computing the entropy changes of isolated systems.

Now, suppose ΔU, q and w are the values of a system in a cyclic process. As we know a cyclic process is one in which the system after undergoing any number of processes return to its initial state. This signifies that ΔU = 0; therefore, the work system throughout all such processes must be equivalent to the heat absorbed by the system.

That is, q = q1 + q2 + .... = - (w1 + w2 + ...) = - w

Here q and w are the heat absorbed and work done on the system in the whole cyclic processes comprising of some processes; in the individual processes, q₁, q₂.... and so on, are the heat absorbed via the system and, w₁, w₂ .... and so on, are the work done on the system.

The Carnot Cycle:

Carnot analyzed the functioning of an engine by the given features: 

a) The engine works in cycles.

b) It absorbs heat from the reservoir termed as source.            

c) It does some work out of the heat absorbed.

d) It returns the unused portion of the heat to the other reservoir, termed as sink.

e) Lastly it returns to its original state.

Such an engine is termed as Carnot engine. The temperature of the source (T_H) is higher than that of the sink (T_C). The source and the sink are supposed to be of infinite heat capacity; that is, the temperatures of the source and the sink are not influenced by small amounts of heat exchange.

Carnot illustrated that the whole amount of the heat absorbed can't be converted into work in a cyclic process, no matter how ideal the heat engine is. He expressed that only a fraction of the net heat absorbed is converted into work and this fraction is termed as the efficiency of the Carnot engine. Now we are going to derive an equation helpful in computing its efficiency. 

For the sake of simplicity, let us suppose that the engine comprises of a cylinder and a piston having lone mole of an ideal gas in between the two. The cylinder has perfectly insulated; walls and a perfectly conducting base; the piston is frictionless. This is just for the sake of convenience that we have taken that the engine has ideal gas in reality there can be any appropriate fluid. We make use of the given expressions (for one mole of gas).

w_isothermal = RT ln (V_initial/V_final)

w_adiabatic = C_V‾(T_final - T_initial)

Here, 'w' is the work done on the system and C_V‾ is the molar heat capacity of the gas.

The plot of the pressure-volume graph is as shown below:

Fig: Carnot Cycle graph

The Carnot cycle operation can be explained by the given four steps:

1) Isothermal Expansion

2) Adiabatic expansion

3) Isothermal compression

4) Adiabatic compression

Step I: Isothermal Expansion:

Primarily the gas consists of pressure p₁ and volume V₁. The cylinder is positioned on a heat source maintained at the temperature T_H. The gas is isothermally and reversibly expanded to a volume V₂ and pressure p₂. Let the work done on the gas be w₁ and the heat absorbed from the source be q_H. In the isothermal process, ΔU = 0. 

Therefore, by using equation ΔU = U_f - U_i = q + w and W = - nRT ln(V₂/V₁) = nRT ln(V₁/V₂)

w₁ = q_H = RTH ln(V₁/V₂)

Or q_H = RT_H ln(V₂/V₁)

Step II: Adiabatic Expansion:

The cylinder is now put on a thermally insulated stand and the gas is adiabatically and reversibly expanded till it attains a pressure p₃, volume V₃ and temperature T_C. Throughout this period, no heat is absorbed by the system. The work done on the gas, w₂, as the gas gets cooled from T_H to T_C is represented by using ΔU = W = nC_V‾ (T₂ - T₁) = nC_V‾ΔT as,

w₂ = C_V‾ (T_C - T_H)

Step III: Isothermal Compression:

The cylinder is now put on a sink at temperature T_C and the gas is isothermally and reversibly compressed to the volume V₄ at pressure p₄. Throughout the process the work done on the gas is w₃ and the heat evolved to the sink is q_c (or q_c is the heat absorbed from the sink).

By using equation ΔU = U_f - U_i = q + w and W = - nRT ln(V₂/V₁) = nRT ln(V₁/V₂)

w₃ = - q_c = RT_C ln (V₃/V₄)

Or q_C = RT_C ln (V₄/V₃)

Step IV: Adiabatic Compression:

In the last step, the cylinder is again put on an insulating stand and the gas is adiabatically and reversibly compressed till it reaches its initial state of volume V₁ and temperature T_H. Therefore the work done on the gas, w₄ is given by the equation ΔU = W = nC_V‾ (T₂ - T₁) = nC_V‾ΔT as,

w₄ = C_V (T_H - T_C)

The total work done on the system is represented by:

w = w₁ + w₂ + w₃ + w₄

Or w = RT_H ln V₁/V₂ + C_V‾ (T_C - T_H) + RT_C ln V₃/V₄ + C_V‾ (T_H - T_C)

That is, w = RT_H ln V₁/V₂ + RT_C ln V₃/V₄

The equation T₂/T₁ = ln (V₁/V₂) ^{γ - 1} can be applied to relate the initial and final values of volume and temperature of the two adiabatic processes illustrated in steps II and IV.

Applying the above equation to step II we obtain,

T_C/T_H = (V₂/V₃)^{γ - 1}

Likewise applying the equationT₂/T₁ = ln (V₁/V₂) ^{γ - 1} to step IV, we obtain:

T_C/T_H = (V₄/V₁)^{γ - 1} or T_C/T_H = (V₁/V₄)^{γ - 1}

Therefore, (V₂/V₃)^{γ - 1} = (V₁/V₄)^{γ - 1}

That is, V₂/V₃ = V₁/V₄

Or V₃/V₄ = V_2/V₁

By using the above in equation w = RT_H ln V₁/V₂ + RT_C ln V₃/V₄, we obtain:

w = RT_H ln V₁/V₂ + RT_C ln V₂/V₁

Therefore, total work done on the system = w = R (T_H - T_C) ln V₁/V₂

Or total work done by the system, = w' = - w = R (T_H - T_C) ln V₂/V₁      

As work done by the system = - (work done on the system)

We are familiar that the heat exchange between the gas and the source or sink occurs only in the isothermal processes (steps 1 and 3); in adiabatic processes (steps 2 and 4), there is no heat exchange. Again, q_H is the heat absorbed from the source in step 1 and q_c is the heat absorbed from the sink in the step 3. The net heat absorbed by the system:

q = q_H + q_C

As predicted, q turns out to be equivalent to - w or w' as for the overall cyclic process, ΔU = 0

Therefore, q= q_H + q_C = w' = - w = R (T_H - T_C) ln V₂/V₁   

It might though be noted that out of the heat q_H, (= RT_H ln V₂/V_l as per equation absorbed from the source, only some of it is converted to helpful work and the rest is lost to the sink. Let us now compute w'/q_H, that is, the ratio between the total work done by the system throughout one cycle and the heat absorbed in the first step. This quantity is known as the efficiency 'η' of a Carnot engine.

Efficiency (η) = Total work done by the system/Heat Absorbed from the source at higher temperature

Efficiency (η) = w'/q_H

By using the equations q_H = RT_H ln (V₂/V₁) and q= q_H + qC = w' = - w = R (T_H - T_C) ln V₂/V₁

(η) = (q_H + q_C)/q_H = [R (T_H - T_C) ln V₂/V₁]/[RT_H ln V₂/V₁]

(η) = (q_H + q_C)/q_H = (T_H - T_C)/T_H

As T_C and T_H are always positive and T_C/T_H is less than one, equation above can be rearranged as:

(η) = 1 T_C/T_H = 1 + (q_C/q_H) = < 1

As q_C is negative and q_H is positive, (q_C/q_H) is a negative quantity; 1 + (q_C/q_H) is as well less than one.

This signifies that the efficiency is for all time less than one, that is, all the heat absorbed at a higher temperature is not converted into work. This is as well clear that efficiency will be more if the ratio T_C/T_H is small. Therefore, for efficient working of the engine, it must absorb heat at as high a temperature as possible and refuse it at as low a temperature as possible. It must as well be noted that the efficiency is independent of the nature of the fluid. This is termed as Carnot theorem which can as well be represented as:

In all the cyclic engines working between the similar temperatures of the source and the sink, the efficiency similar.

It should be pointed out that in the Carnot cycle; all the processes have been taken out reversibly. Therefore, maximum and minimum amount of work are comprised in expansion and compression, correspondingly; this means that there can't be any engine more efficient than Carnot engine. In real engines, there is irreversibility due to unexpected expansion and compression and as well due to the friction of the piston.

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