Carnot’s function

Nothing more is assumed about the temperatures, and one result of Carnot s investigation is a rigorous definition of temperature. Further, let there be a cylinder and piston, of an absolute non-conductor of heat, closed at the bottom by a perfect conductor of heat, and containing the working substance—any substance, or mixture of substances, the pressure of which is uniform in all directions at all points and is a continuous function of temperature. Finally, we have a stand formed of a perfect non-conductor of heat (Fig. 7). 

In the operations constituting a Carnot s cycle, changes of Q and T occur separately. In the majority of cases both these changes occur together, so that the temperature of the working substance may be regarded as a function of the time. Equation (4) therefore requires extension, and this was effected by Lord Kelvin in May, 1854, in the following way ... 

Fig. 2 Efficiency as a function of temperature of the energy conversion of a fuel cell and a conversion process limited by the Carnot s factor.

In Chapter 5 we said that Carnot s classic model of an idealized heat engine resulted in the following properties of the entropy function ... 

Now we turn to the task of obtaining the efficiency of reversible heat engines. Since the efficiency of a reversible heat engine is the maximum, all of them must have the same efficiency. Hence obtaining the efficiency of one particular reversible engine will suffice. The following derivation also makes it explicit that the efficiency of Carnot s engine is only a function of temperature. 

The fact that the efficiency of a reversible heat engine is independent of the physical and chemical nature of the engine has an important consequence which was noted by Lord Kelvin, William Thomson (1824-1907). Following Carnot s work, Lord Kelvin introduced the absolute scale of temperature. The efficiency of a reversible heat engine is a function only of the temperatures of the hot and cold reservoirs, independent of the material properties of the engine. Furthermore, the efficiency cannot exceed 1, in accordance with the First Law. These two facts can be used to define an absolute scale of temperature which is independent of any material properties. 

When Gibbs first turned his attention to thermodynamics in the early 1870 s, the subject had already achieved a certain level of maturity. The essential step had been taken in 1850 by Rudolf Clausius, when he argued that two laws are needed to reconcile Carnot s principle about the motive power of heat with the law of energy transformation and conservation. Efforts to understand the second of the two laws finally led Clausius in 1865 to his most concise and ultimately most fruitful analytical formulation. In effect, two basic quantities, internal energy and entropy, are defined by the two laws of thermodynamics. The internal energy U is that function of the state of the system whose differential is given by the equation expressing the first law,... 

The entropy function S also simplifies the graphical depiction of the Carnot cycle. Consider, for example, the form of the Carnot cycle shown in the PV diagram of Fig. 4.4a. The corresponding ST diagram for the same Carnot cycle is shown in Fig. 4.4b. As can be seen, the ST representation of the Carnot cycle is a simple rectangle whose... 

The right side of Eq. (5.4) is the ratio of- rs evaluatedat two thermodynamictemperatures the tA s are to each other as the absolute values of the heats absorbed and rejected by a Carnot engiire operatiirgbetweeir reservoirs at these tenrperahires, quite independent of the properties of arry substairce. Moreover, Eq. (5.4) allows arbitrary choice of the empirical teirrperahire represented by 0 once this choice is made, the function must be detemrined. If 0 is chosen as the kelvin temperature T, thenEq. (5.4)beconres ... 

By analyzing the Carnot cycle description of macroscopic energy transfer processes, Clausius demonstrated that the quantity J(l/T)dqrev is a state function, because its value for any reversible process is independent of the path. Based on this result, Clausius defined the procedure for calculating the entropy change AS = Sf — S for a system between any thermodynamic states i and f as... 

The task of finding a mathematical formulation of the second law of thermodynamics was accomplished by Sadi Carnot and Clausius on the traditional, macroscopic side, and, again, by Boltzmann from a molecular, statistical perspective. What is sought is a state function that quantitatively describes the degree of dispersion in a chemical system. This is entropy, and its symbol is S. It must increase in any irreversible process. 

In his 1854 paper, he acc ts Joule s suggestion that the Carnot function C is simply the value of the temperature measured from -273°C, and demonstrates that ... 

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