Lord Kelvin: An account of his scientific life and work

by Andrew Gray

J∂H = ∂e + O

Thus

J (dH − ∂H) = de − ∂e + dW

and the assumption is that de = ∂e, so that dW = J (dH − ∂H); that is, dW = J (C_p − C_v), when the rise of temperature is 1° C. and the mass of air is one gramme. This assumption requires justification, and by an experiment of Joule's, which was repeated in a more sensitive form devised by Thomson, it was shown to be a very close approximation to the truth. Joule's experiment is well known: the explanation given above may serve to make clear the nature of the research undertaken later by Thomson and Joule conjointly.

The inverse process, the conversion of heat into work, required investigation, and it is this that constitutes the science of thermodynamics. It was the subject of the celebrated Réflexions sur la Puissance Motrice du Feu, et sur les Machines Propres à Développer cette Puissance, published in 1824 by Sadi Carnot, an uncle of the late President of the French Republic. Only a few copies of this essay were issued, and its text was known to very few persons twenty-four years later, when it was reprinted by the Academy of Sciences. Its methods and conclusions were set forth by Thomson in 1849 in a memoir which he entitled, "An Account of Carnot's Theory of the Motive Power of Heat." Numerical results deduced from Regnault's experiments on steam were included; and the memoir as a whole led naturally in Thomson's hands to a corrected theory of heat engines, which he published in 1852. Carnot's view of the working of a heat engine was founded on the analogy of the performance of work by a stream of water descending from a higher level to a lower. The same quantity of water flows away in a given time from a water wheel in the tail-race as is received in that time by the wheel from the supply stream. Now a heat engine receives heat from a supplying body, or source, at one temperature and parts with heat to another body (for example, the condenser of a steam engine) at a lower temperature. This body is usually called the refrigerator. According to Carnot these temperatures corresponded to the two levels in the case of the water wheel; the heat was what flowed through the engine. Thus in his theory as much heat was given up by a heat engine to the body at the lower temperature as was received by it from the source. The heat was simply transferred from the body at the higher temperature to the body at the lower; and this transference was supposed to be the source of the work.[17]

The first law of thermodynamics based on Joule's proportionality of heat produced to work expended, and the converse assumed and verified a posteriori, showed that this view is erroneous, and that the heat delivered to the refrigerator must be less in amount than that received from the source, by exactly the amount which is converted into work, together with the heat which, in an imperfect engine, is lost by conduction, etc., from the cylinder or other working chamber. This change was made by Thomson in his second paper: but he found the ideas of Carnot of direct and fruitful application in the new theory. These were the cycle of operations and the ideal reversible engine.

In the Carnot cycle the working substance—which might be a gas or a vapour, or a liquid, or a vapour and its liquid in contact: it did not matter what for the result—was supposed to be put through a succession of changes in which the final state coincided with the initial. Thus the substance having been brought back to the same physical condition as it had when the cycle began, has the same internal energy as it had at the beginning, and in the reckoning of the work done by or against external forces, nothing requires to be set to the account of the working substance. This is the first great advantage of the method of reasoning which Carnot introduced.

The ideal engine was a very simple affair: but the notion of reversibility is difficult to express in a form sufficiently definite and precise. Carnot does not attempt this; he merely contents himself with describing certain cycles of operations which obviously can be carried through in the reverse order. Nor does Thomson go further in his "Account of Carnot's Theory," though he states the criterion of a perfect engine in the words, "A perfect thermodynamic engine is such that, whatever amount of mechanical effect it can derive from a certain thermal agency, if an equal amount be spent in working it backwards, an equal reverse thermal effect will be produced." This proposition was proved by Carnot: and the following formal statement in the essay is made: "La puissance motrice de la chaleur est independante des agents mis en œuvre pour la réaliser: sa quantité est fixée uniquement par les temperatures des corps entre lesquels se fait, en dernier résultat, le transport du calorique." The result involved in each, that the work done in a cycle by an ideal engine depends on the temperatures between which it works and not at all on the working substance, is, as we shall see, of the greatest importance. The proof of the proposition, by supposing a more efficient engine than the ideal one to exist, and to be coupled with the latter, so that the more efficient would perform the cycle forwards and the ideal engine the same cycle backwards, is well known. In Carnot's view the former would do more work by letting down a given quantity of heat from the higher to the lower temperature than was spent on the latter in transferring the same quantity of heat from the lower to the higher temperature, so that no heat would be taken from or given to source or refrigerator, while there would be a gain of work on the whole. This would be equivalent to admitting that useful work could be continually performed without any resulting thermal or other change in the agents performing the work. Even at that time this could not be admitted as possible, and hence the supposition that a more efficient engine than the reversible one could exist was untenable.

Carnot showed that the work done by an ideal engine, in transferring heat from one temperature to another, was to be found by means of a certain function of the temperature, hence called "Carnot's function." The corresponding function in the true dynamical theory is always called Carnot's. A certain assignment of value to it gave, as we shall see, Thomson's famous absolute thermodynamic scale of temperature.

In the light of the facts and theories which now exist, and are almost the commonplaces of physical text books, it is very interesting to review the ideas and difficulties which occurred to the founders of the science of heat sixty years ago. For example, Thomson asks, in his "Account of Carnot's Theory," what becomes of the mechanical effect which might be produced by heat which is transferred from one body to another by conduction. The heat leaves one body and enters another and no mechanical effect results: if it passed from one to the other through a heat engine, mechanical effect would be produced: what is produced in place of the mechanical effect which is lost? This he calls a very "perplexing question," and hopes that it will, before long, be cleared up. He states, further, that the difficulty would be entirely avoided by abandoning Carnot's principle that mechanical effect is obtained by "the transference of heat from one body to another at a lower temperate." Joule urges precisely this solution of the difficulty in his paper, "On the Changes of Temperature produced by the Rarefaction and Condensation of Air" (Phil. Mag., May 1845). Thomson notes this, but adds, "If we do so, however, we meet with innumerable other difficulties—insuperable without further experimental investigation, and an entire reconstruction of the theory of heat from its foundation. It is in reality to experiment that we must look, either for a verification of Carnot's axiom, and an explanation of the difficulty we have been considering, or for an entirely new basis of the Theory of Heat."