I just took a read of Clausius' original writing from 1867 on the second law. He refers to the "unit of heat" as "that amount of heat which is required to increase the temperature of unit-weight of water by one degree". This makes sense to me as a unit of heat, but I am using my modern understanding of heat, in terms of kinetic energy of particles, to make sense of this. What would Clausius have meant by "heat" itself? What would he have meant by "heat required to raise something by a temperature"?

They apparently understood that "there must be an equivalence between mechanical work and heat, as between cause and effect" (wikipedia), but this doesn't explain what they actually thought heat was, or what it means to "put heat into something to increase its temperature".


Wikipedia page on Theory of heat

On the Second Fundamental Theorem of the Mechanical Theory of Heat (Clausius, 1867)


Until 1850s the dominant theory was that the heat is caloric fluid, and the phrasing reflected that. The caloric theory was a revision by Lavoisier of the earlier phlogiston theory, and Carnot and others still worked with it. Only after the work of Mayer and Joule in 1840s, promoted by Kelvin and interpreted by Clausius, that the paradigm began to change in 1850s. While Clausius reinterpreted heat as mechanical energy of particle motion, he continued to use the established terminology from Carnot, as we still do. It is the same metaphorical phrasing as in "put energy into work", or "energy drain" see What are the major flaws of the “caloric” theory of heat?

Clausius was a firm, if at first secretive, believer in the kinetic theory. Since the temperature was a measure of caloric density, to him it was a measure of energy density. The vague idea about temperature as measuring intensity of micromotion, which in turn stems from analogizing spread of commotion in a particle medium to spread of fluid, goes back to Rumford and his 1798 inconclusive experiments on the mechanical equivalent of heat. The work of Joule and Mayer was much more definitive. Fortunately, Clausius did not need to define temperature, unlike entropy or heat it was a directly measurable quantity, and he used it as a given to define entropy instead. This was in line with his general preference for quantities that directly depended on measurements rather than speculative molecular assumptions.

While Clausius was a champion of kinetic theory, his attitude towards the use of statistical methods was much more ambivalent. He preferred algebraic approaches that were to become a staple thermodynamics, in particular, because they depended much less on specific hypotheses about parameters of molecules and the nature of intermolecular forces, that could not be verified at the time.

Nonetheless, Clausius's 1857 paper inspired Maxwell to apply statistical methods to the theory of heat. Clausius only introduced mean speeds and mean free paths of particles, while Maxwell worked directly with a velocity distribution, and hence could distinguish between squares of mean speeds and means of squared speeds. The flip side was that Maxwell had to introduce very specialized hypotheses to derive his (normal) distribution function, which Clausius criticized him for. Ultimately, it led Clausius to abandon statistical approach altogether, and even Maxwell himself had to reboot, taking inspiration from Boltzmann's work, see Clausius and Maxwell's Kinetic Theory of Gases by Garber. A clear statistical interpretation of temperature as mean kinetic energy of molecules is due to Boltzmann. Here is from Grabiner:

"By 1857, when he first began to publish papers on gases, Clausius had already established a reputation as a leading theoretical physicist with his clear, incisive papers on thermodynamics. These latter papers reveal his ability to separate the essential physical principles from the special matter-theory assumptions that he had used in his development of the subject (all of his thermodynamics papers published after 1860 were based upon his concept of the structure of matter). In 1857 Clausius was still cautious about revealing these assumptions.

[...] While Clausius regarded Maxwell's theory as a serious attempt to solve the problems of gaseous behavior, he criticized it on basic physical grounds. He considered Maxwell's use of a spherically symmetric distribution function in analyzing transport properties as incorrect. He developed his own theory, taking into account the additional vis viva associated with motion in the direction of the gradient within the gas. The factor, ignored by Maxwell, was the cause of the phenomenon, and, in the case of thermal conduction, the gas molecules possessed additional momentum in the direction of the temperature gradient; therefore, the distribution of molecular motions could not be spherically symmetric.

[...] The statistical method might be correct in principle, but Maxwell's particular method was inaccurate. Clausius' acceptance of a statistical method was only tentative and he finally rejected it altogether. His hesitancy in using the distribution function reflects the difference between his and Maxwell's concept of matter and gases... Clausius did not use the statistical distribution even after Maxwell derived it independently of the mean free path. He only used the distribution function once, to find the mechanical expression for the heat added to a gas... In 1862 he had revealed the molecular ideas that had guided his thermodynamics, and his later papers were all directed to the problem of expressing disgregation and entropy in mechanical form. Unlike Boltzmann, Clausius never fully appreciated the power of the statistical method, and he continued to work within the traditional, yet sophisticated, field of analytical mechanics."

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