There is a simple and unambiguous formulation of the second law of thermodynamics suggested by Sadi Carnot himself (see the quotation below). If heat is to be cyclically converted into work (that is, if a heat engine is to be operative), then
"a cold body is necessary".
In other words, heat cannot be cyclically converted into work (a heat engine will not be operative) unless some temperature gradient is present: a hot body, source of heat, and a cold body, receiver of heat, must be available. Only two-temperature heat engines are possible; isothermal (one-temperature) heat engines don't exist. This false claim is universally taught nowadays.
The problem is that in 1824 Carnot deduced "a cold body is necessary" from a postulate that eventually turned out to be false:
Carnot's (false) postulate: Heat is an indestructible substance (caloric) that cannot be converted into work by the heat engine.
Unpublished notes written in the period 1824-1832 reveal that, after realizing that his postulate was false, Carnot found "a cold body is necessary" implausible:
Sadi Carnot, REFLECTIONS ON THE MOTIVE POWER OF HEAT, p. 225: "Heat is simply motive power, or rather motion which has changed form. It is a movement among the particles of bodies. Wherever there is destruction of motive power there is, at the same time, production of heat in quantity exactly proportional to the quantity of motive power destroyed. Reciprocally, wherever there is destruction of heat, there is production of motive power." p. 222: "Could a motion (that of radiating heat) produce matter (caloric)? No, undoubtedly; it can only produce a motion. Heat is then the result of a motion. Then it is plain that it could be produced by the consumption of motive power, and that it could produce this power. All the other phenomena - composition and decomposition of bodies, passage to the gaseous state, specific heat, equilibrium of heat, its more or less easy transmission, its constancy in experiments with the calorimeter - could be explained by this hypothesis. But it would be DIFFICULT TO EXPLAIN WHY, IN THE DEVELOPMENT OF MOTIVE POWER BY HEAT, A COLD BODY IS NECESSARY; why, in consuming the heat of a warm body, motion cannot be produced." http://www.nd.edu/~powers/ame.20231/carnot1897.pdf
Generally, a cold body is not necessary, which means that the second law of thermodynamics is false. The cold body is only TECHNOLOGICALLY necessary as it makes heat engines fast-working. Heat engines working under isothermal condition (in the absence of a cold body) are commonplace but are too slow and impuissant to be of any technological importance. Except, perhaps, for the case in which a liquid is placed in an electric field. Vigorous jets and flows can appear, able to do mechanical work (e.g. by rotating a waterwheel) at the expense of ambient heat:
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