The modern understanding of heat is often partly attributed to Thompson's 1798 mechanical theory of heat (An Experimental Enquiry Concerning the Source of the Heat which is Excited by Friction), postulating a mechanical equivalent of heat. A collaboration between Nicolas Clément and Sadi Carnot (Reflections on the Motive Power of Fire) in the 1820s had some related thinking along similar lines. In 1842, Julius Robert Mayer frictionally generated heat in paper pulp and measured the temperature rise. In 1845, Joule published a paper entitled The Mechanical Equivalent of Heat, in which he specified a numerical value for the amount of mechanical work required to "produce a unit of heat", based on heat production by friction in the passage of electricity through a resistor and in the rotation of a paddle in a vat of water. The theory of classical thermodynamics matured in the 1850s to 1860s.
In 1850, Clausius, responding to Joule's experimental demonstrations of heat production by friction, rejected the caloric doctrine of conservation of heat, writing:
The process function Q was introduced by Rudolf Clausius in 1850. Clausius described it with the German compound Wärmemenge, translated as "amount of heat".
In 1856, Rudolf Clausius, referring to closed systems, in which transfers of matter do not occur, defined the second fundamental theorem (the second law of thermodynamics) in the mechanical theory of heat (thermodynamics): "if two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of heat Q from work at the temperature T, has the equivalence-value:"
In 1865, he came to define the entropy symbolized by S, such that, due to the supply of the amount of heat Q at temperature T the entropy of the system is increased by
In 1907, G.H. Bryan published an investigation of the foundations of thermodynamics, Thermodynamics: an Introductory Treatise dealing mainly with First Principles and their Direct Applications, B.G. Teubner, Leipzig.
Over the years, for example in his 1879 thesis, but particularly in 1926, Planck advocated regarding the generation of heat by rubbing as the most specific way to define heat. Planck criticised Carathéodory for not attending to this. Carathéodory was a mathematician who liked to think in terms of adiabatic processes, and perhaps found friction to tricky to think about, while Planck was a physicist.