Zeroth Law of Thermodynamics - History

History

Temperature has long been known as a quality of heat, for example, to Galileo and to Newton. Carnot took it as a presupposition for his work. Thermometers may be described as empirical or absolute. Absolute thermometers are calibrated numerically by the thermodynamic absolute temperature scale. It was not until the middle of the nineteenth century that absolute thermodynamic temperature was recognized, long after the recognition of empirical thermometry.

Empirical thermometry recognizes hotness as a fundamental character of temperature and thermometers. Empirical thermometers are not in general necessarily in exact agreement with each other or with absolute thermometers as to their numerical scale readings, but to qualify as thermometers at all they must agree with absolute thermometers and with each other in the following way: given any two bodies isolated in their separate respective thermodynamic equilibrium states, all thermometers agree as to which of the two has the higher temperature, or that the two have equal temperatures. For any two empirical thermometers, this does not require that the relation between their numerical scale readings be linear, but it does require that relation to be strictly monotonic.

Truesdell reports that Rankine wrote in 1853:

Definition of equal temperatures.

Two portions of matter are said to have equal temperatures, when neither tends to communicate heat to the other.

Discussing the concept of temperature, James Clerk Maxwell in 1872 wrote: "If when two bodies are placed in thermal communication, one of the two bodies loses heat, and the other gains heat, that body which gives out heat is said to have a higher temperature than that which receives heat from it." He drew the corollary "If when two bodies are placed in thermal communication neither of them loses or gains heat, the two bodies are said to have equal temperatures or the same temperature. The two bodies are then said to be in thermal equilibrium."

Further, Maxwell stated, as the "Law of equal temperatures" the following triviality: "Bodies whose temperatures are equal to that of the same body have themselves equal temperatures". Maxwell then offered an argument that this statement was "not a truism". Later in the same text, Maxwell wrote: "Hence the result of the conduction and radiation of heat from one part of a system to another is to diminish the entropy of the system, or the energy, available as work, which can be obtained from the system." This statement was surrounded in Maxwell's text by several others like it that show that it was no slip of the pen. In the same textbook, Maxwell wrote that he was following Tait in re-defining the word entropy that had been introduced by Clausius. In contrast with what Maxwell wrote then, Tait had changed his mind by 1884 when in his text he accepted Clausius's original definition of entropy.

Subsequent writers made statements like Maxwell's. Tait in 1884 wrote "if A is at the same temperature as B and also at the same temperature as C — no transfer of heat takes place between B and C, whatever be these bodies." A similar statement was made by Max Planck in 1897, not labeled as a law but as an important proposition: "If a body, A, be in thermal equilibrium with two other bodies, B and C, then B and C are in thermal equilibrium with one another." Planck repeated this important proposition in the seventh edition of his treatise in 1922.

The title "zeroth law of thermodynamics" began to appear in textbooks to refer to statements of this kind, though now stripped of their explicit reference to heat; their implicit dependence on the notion of heat could not be removed because they rely on the concept of thermal equilibrium which in turn relies on the concept of transfer of heat by conduction or radiation, the presence or absence of which must be empirically recognizable in order to make the concept of thermal equilibrium empirically recognizable. An early example is in the textbook of statistical thermodynamics of Fowler and Guggenheim (1939/1965). Their focus of interest in that book was homogeneous systems (page 1), which they termed 'assemblies'. They dealt with assemblies that were either completely homogeneous or that could be divided into homogeneous parts, called phases (page 58). For their macroscopic thermodynamic account of phenomena, they started by accepting, on empirical physical grounds, the presupposed notions of thermal insulation, thermal contact, and thermal equilibrium (page 56). They emphasized that these notions "can be defined without any reference to temperature" (page 56). Moreover, they gave at this stage of their development of their theory no hint of the notion of heat transfer. On page 56, they wrote:

...we introduce the postulate: If two assemblies are each in thermal equilibrium with a third assembly, they are in thermal equilibrium with each other.

They then proposed that "it may be shown to follow that the condition for thermal equilibrium between several assemblies is the equality of a certain single-valued function of the thermodynamics states of the assemblies, which may be called the temperature t, any one of the assemblies being used as a "thermometer" reading the temperature t on a suitable scale. This postulate of the "Existence of temperature" could with advantage be known as the zeroth law of thermodynamics" (page 56). They did not there state any reason why such a function should have values in a scale consisting of a continuous succession of numbers or that it should have anything to do with heat. Their thinking was apparently conditioned by their asserted belief that "The most logically satisfactory formulation is undoubtedly that of Carathéodory" (page 56). Though they thus apparently professed concern for logicality, they had no apparent compunction about immediately assuming, without apparent justification, that their postulated "temperatures" should exist on a numerical scale, provided for example by "the measured volume of a constant quantity of any chosen substance at constant pressure" (page 56). No worries about pesky but relevant physical realities such as the anomalous behaviour of water that concerned the nineteenth century thermodynamicists, because around 4 C it does not provide a valid empirical temperature. No mention that it would therefore be safer to refer to a permanent gas as a thermometric material. The approach of Fowler and Guggenheim is labeled "mechanical" by Bailyn, who contrasts it with the "thermodynamic" approach of Planck and the founders, who fully recognized the notion of heat transfer as an essential and fundamental presupposition to thermodynamics, without actually labelling it as a numbered law of thermodynamics. Since the time of Fowler and Guggenheim, who believed that the 1909 axiomatic formulation of Carathėodory was the most logically satisfactory, other influential axiomatic formulations of thermodynamics have appeared, some of which do not refer to a zeroth law of thermodynamics.

Sommerfeld in 1951 gave the title the "Zeroth Law" to the statement "Equality of temperature is a condition for thermal equilibrium between two systems or between two parts of a single system"; he wrote that this title followed the suggestion of Fowler, made when he was giving an account of a certain book. Sommerfeld's statement took the existence of temperature for granted, and used it to specify one of the characteristics of thermodynamic equilibrium. This is converse to many statements that are labeled as the zeroth law, which take thermal equilibrium for granted and use it to contribute to the concept of temperature. We may guess that Fowler had made his suggestion because the notion of temperature is in effect a presupposition of thermodynamics that earlier physicists had not felt needed explicit statement as a law of thermodynamics, and because the mood of his time, pursuing a "mechanical" axiomatic approach, wanted such an explicit statement.

Guggenheim in 1966 wrote "If two systems are both in thermal equilibrium with a third system then they are in thermal equilibrium with each other" as the zeroth law of thermodynamics, and followed it with the comment "In other words, systems in thermal equilibrium are said to have the same temperature." This ordinary language statement, although less precise than the statements of Planck and Tait mentioned above, conveys much of the essence of the zeroth law.

The statement of the zeroth law of thermodynamics by Serrin in 1977, though rather mathematically abstract, is more informative for empirical thermometry: "Zeroth Law - There exists a topological line which serves as a coordinate manifold of material behaviour. The points of the manifold are called 'hotness levels', and is called the 'universal hotness manifold'." To this information there needs to be added a sense of greater hotness; this sense can be had, independently of calorimetry, of thermodynamics, and of properties of particular materials, from Wien's displacement law of thermal radiation: the temperature of a bath of thermal radiation is proportional, by a universal constant, to the frequency of the maximum of its frequency spectrum; this frequency is always positive, but can have values that tend to zero.

An unusual statement, written in a context of non-equilibrium thermodynamics, given the label 'zeroth law of thermodynamics' is by I. Müller. His statement of the "zeroth law of thermodynamics", written in 2003, is: "temperature is continuous at an ideal interface between two bodies, typically the interface between a thermometer and a body whose temperature is measured."