Carnot Principle

A temperature scale which is independent of the properties of the substances that are used to measure temperature is called a thermodynamic temperature scale. Its derivation is given below using some Carnot heat engines.

The Carnot principle states that the reversible heat engines have the highest efficiencies when compared to irreversible heat engines working between the same two reservoirs. And the efficiencies of all reversible heat engines are the same if they work between the same two reservoirs. That is, the efficiency of a reversible heat engine is independent on the working fluid used and its properties, the way the cycle operates, and the type of the heat engine. The efficiency of a reversible heat engine is a function of the reservoirs' temperature only.

      η_th = 1 - Q_L/Q_H = g(T_H,T_L)

or

      Q_H/Q_L = f(T_H,T_L)

where
      Q_L = heat transferred to the low-temperature
             reservoir which has a temperature of T_L
      Q_H = heat transferred from the high-temperature
             reservoir which has a temperature of T_H
      g, f = any function

Relations between Reversible
Heat Engines A, B , and C

Consider three reversible heat engines working between a high-temperature reservoir at temperature T₁ and a low-temperature reservoir at temperature T₃. Engine A receives Q₁ from the high-temperature reservoir,and rejects Q₂ to engine B at temperature T₂. Engine B receives Q₂ from engine A at temperature T₂, and rejects Q₃ to the low-temperature reservoir. Engine C receives Q₁ from the high-temperature reservoir,and rejects Q₃ to the low-temperature reservoir. Applying the equation above to the three engines separately yields,

      Engine A:       Q₁/Q₂ = f(T₁,T₂)             (1)
      Engine B:       Q₂/Q₃ = f(T₂,T₃)             (2)
      Engine C:       Q₁/Q₃ = f(T₁,T₃)             (3)

Multiplying equations (1) and (2) gives

             (4)

Comparing (4) and (3) yields,

      f(T₁,T₃) = f(T₁,T₂)f(T₂,T₃)                      5)

The left-hand side of equation (5) is only a function of T₁ and T₃, and thus the right-hand side of equation (5) is also a function of T₁ and T₃ only. This condition will be satisfied only if the function f has the following form:

       and

Substituting them to equation (5) gives,

or,

                                        (6)

Equation (6) provides a basis to define a thermodynamic temperature scale. There are several choices for the function φ. The Kelvin scale is obtained by making a simple choice as

      φ(T) = T

Hence, equation (6) becomes,

      Q₁/Q₃ = T₁/T₃

To bring uniformity to the treatment of heat engines, H is used to denote the high-temperature reservoir, and L to denote the low-temperature reservoir. The above equation becomes,

      Q_H/Q_L = T_H/T_L

Determine Absolute Temperature by
Measuring Heat Transfers Q_H and Q_L

The Kelvin scale temperature is called absolute temperature. At the International Conference on Weights and Measures held in 1954, the triple-point of water was assigned the value 273.16 K. Then, if a reversible cycle is operated between a reservoir at 273.16 K and another reservoir at temperature T, temperature T can be expressed as

      T = 273.16(Q/Q_tp)_{rev. cycle}

where Q and Q_tp are heat transfer between the cycle and the reservoirs at temperature T and at temperature 273.16 K, respectively.

The magnitude of a Kevin scale is defined as 1/273.16 of the temperature interval between the absolute zero and the tripe-point of water. The magnitude of temperature units on Kevin scale is the same as that of Celsius.

      1 K 1 ^oC

When numerical values of the thermodynamic temperature are to be determined, it is not possible to use a reversible cycle since that does not exist in practice. Absolute temperatures can be measured by other means, such as the constant-volume ideal gas thermometer.

A heat engine is called a Carnot heat engine if it operates on the reversible Carnot cycle. The thermal efficiency of heat engine, regardless reversible or irreversible, is given as

      η_th = 1- Q_L/Q_H

where Q_L and Q_H are heat transfer between the cycle and the low-temperature reservoir at temperature T_L and the high-temperature reservoir at temperature T_H, respectively. For reversible heat engines, the heat transfer ratio can be replaced by the absolute temperatures of the two reservoirs.

      η_th = 1- T_L/T_H

This efficiency is called the Carnot efficiency since the Carnot heat engine is one of the well-known reversible engines. It is the highest efficiency that a heat engine working between a high-temperature reservoir at temperature T_H and a low-temperature reservoir at temperature T_L can reach. All irreversible heat engines working between the same two reservoirs have lower efficiencies.