In this paper, we first analyze the difference between the second law of thermodynamics and the laws in other disciplines. There are some phenomena in other disciplines similar to the Clausius Statement of the second law, but none of them has been accepted as the statement of a certain law. Clausius’ mechanical theory of heat, published in the nineteenth century, is then introduced and discussed in detail, from which it is found that Clausius himself regarded “Theorem of the equivalence of the transformation of heat to work, and the transformation of heat at a higher temperature to a lower temperature”, rather than “Heat can never pass from a colder to a warmer body without some other change”, as the statement of the second law of thermodynamics. The latter is only laid down as the fundamental principle for deriving the theorem of the equivalence of transformations. Finally, based on the theorem of the equivalence of transformations and the average temperature method, a general quantitative relation among the heat, the work, and the temperatures is obtained for arbitrary cycles, which is thus recommended as an alternative mathematic expression of the second law. Hence, the theorem of the equivalence of transformations is the real Clausius Statement of the second law of thermodynamics.

In 2019, Schilling et al. claimed that they achieved the supercooling of a body without external intervention in their thermoelectric experiments, thus arguing that the second law of thermodynamics was bent. Kostic suggested that their claim lacked full comprehension of the second law of thermodynamics. A review of history shows that what Clausius referred to as the second law of thermodynamics is the theorem of the equivalence of transformations (unfairly ignored historically) in a reversible heat–work cycle, rather than “heat can never pass from a cold to a hot body without some other change” that was only viewed by Clausius as a natural phenomenon. Here, we propose the theorem of the equivalence of transformations for reversible thermoelectric cycles. The analysis shows that the supercooling phenomenon Schilling et al. observed is achieved by a reversible combined power–refrigeration cycle. According to the theorem of equivalence of transformations in reversible thermoelectric cycles, the reduction in body temperature to below the ambient temperature requires the body itself to have a higher initial temperature than ambient as compensation. Not only does the supercooling phenomenon not bend the second law, but it provides strong evidence of the second law.

A general and simple equation of state (EOS) is a rich source for research and discussion. Experience shows that high-density matter exhibits simple regularities and some “universal” EOSs have been established. Here, we present an ideal dense matter EOS that is symmetric to the ideal gas EOS by means of Oettingen's dual approach. The ideal dense matter EOS makes no assumptions about the structure of matter and is, therefore, general. Thermodynamic symmetry requires that the ideal dense matter EOS has better agreements with matter at higher densities, which is supported by some empirical equations and actual property data of various substances. Two derivative variables that have historically been overlooked, i.e., the specific work at constant entropy and the specific work at constant temperature, are rejuvenated in the ideal dense matter EOS. Additionally, we find another physical constant related to thermodynamic property of matter symmetric to the ideal gas constant and further give its estimated values for several common substances. Thermodynamic symmetry suggests that the ideal dense matter EOS has the same theoretical location as the ideal gas EOS. It is another basis for the EOS theory and enriches the dual framework of thermodynamics.

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A reversed thermodynamic cycle is a cycle which consumes mechnical work and moves heat energy from a source with lower temperature to a source with higher temperature, and the coefficient of performance (COP) is generally applied to describe its performance. It has been widely accepted that within a certain temperature range, the reversed Carnot cycle has the best performance, i.e., it has the biggest COP. However, in this article it has been concluded that the COP of the reversed Carnot cycle is not the biggest within a certain temperature range from three different methods. Besides, a new reversed cycle is constructed in this article, which is called the reversed p-V cycle inspired by the derivation of Carnot cycle. The analysis presents that within the same temperature the COP of reversed p-V cycle is far larger than the COP of the reversed Carnot cycle. From this cycle it can be seen that the mechanical work also has the concept of quality, which is reflected by the pressure of the work source in the cycle. thermodynamic cycle, reversed Carnot cycle, coefficient of performance, Stirling cycle