Part 38

The density of saturated steam at 100° is taken as ¹⁄₁₆₉₃.5 of that of water at its maximum. Rankine takes it as ¹⁄₁₆₉₆.

Footnote 54:

The part of this expression in the first vinculum (see Regnault, end of ninth memoir) is what is known as “the total heat” of a pound of steam, or the amount of heat necessary to convert a pound of water at 0° into a pound of saturated steam at t°; which, according to “Watt’s law” thus approximately verified, would be constant. The second part, which would consist of the single term t, if the specific heat of water were constant for all temperatures, is the number of thermic units necessary to raise the temperature of a pound of water from 0° to t°, and expresses empirically the results of Regnault’s experiments on the specific heat of water (see end of the tenth memoir), described in the work already referred to.

Footnote 55:

In strictness, the 230th is the last degree for which the experimental data are complete; but the data for the 231st may readily be assumed in a sufficiently satisfactory manner.

Footnote 56:

The numbers here tabulated may also be regarded as the actual values of μ for t = ½, t = 1½, t = 2½, t = 3½, etc.

Footnote 57:

For at the end of the fourth operation the whole mass is liquid, and at the temperature S. Now, this state might be arrived at by first compressing the vapor into water at the temperature t, and then raising the temperature of the liquid to S; and however this state may be arrived at, there cannot, on the whole, be any heat added to or subtracted from the contents of the cylinder, since, during the fourth operation, there is neither gain nor loss of heat. This reasoning is, of course, founded on Carnot’s fundamental principle, which is tacitly assumed in the commonly-received ideas connected with “Watt’s law,” the “latent heat of steam,” and “the total heat of steam.”

Footnote 58:

Thus, from Carnot’s calculations, we find, in the case of alcohol 4.035, and in the case of water 3.648, instead of 3.963 and 3.658, which are Clapeyron’s results in the same cases.

Footnote 59:

A still closer agreement must be expected when more accurate experimental data are afforded with reference to the other media. Mons. Regnault informs me that he is engaged in completing some researches, from which we may expect, possibly before the end of the present year, to be furnished with all the data for five or six different liquids which we possess at present for water. It is therefore to be hoped that, before long, a most important test of the validity of Carnot’s theory will be afforded.

Footnote 60:

The Napierian logarithm of V/V′ is here understood.

Footnote 61:

Carnot varies the statement of his theorem, and illustrates it in a passage, pp. 81, 82, of which the following is translation:

“When a gas varies in volume without any change of temperature, the quantities of heat absorbed or evolved by this gas are in arithmetical progression, if the augmentation or diminutions of volume are in geometrical progression.

“When we compress a litre of air maintained at the temperature 10°, and reduce it to half a litre, it disengages a certain quantity of heat. If, again, the volume be reduced from half a litre to a quarter of a litre, from a quarter to an eighth, and so on the quantities of heat successively evolved will be the same.

“If, in place of compressing the air, we allow it to expand to two litres, four litres, eight litres, etc., it will be necessary to supply equal quantities of heat to maintain the temperature always at the same degree.”

Footnote 62:

The best figure (1896) is J = 778 ft.-lbs. = 1 B.T.U., or J = 426.8 kgm. = 1 calorie, and probably with great accuracy.

Footnote 63:

Or the capacity of a unit of volume for heat.

Footnote 64:

Carnot suggests a combination of the two principles, with air as the medium for receiving the heat at a very high temperature from the furnace; and a second medium, alternately in the state of saturated vapor and liquid water, to receive the heat, discharged at an intermediate temperature from the air, and transmit it to the coldest part of the apparatus. It is possible that a complex arrangement of this kind might be invented which would enable us to take the heat at a higher temperature, and discharge it at a lower temperature than would be practicable in any simple air-engine or simple steam-engine. If so, it would no doubt be equally possible, and perhaps more convenient, to employ steam alone, but to use it at a very high temperature not in contact with water in the hottest part of the apparatus, instead of, as in the steam-engine, always in a saturated state.

Footnote 65:

It is probably this invention to which Carnot alludes in the following passage: “Il a été fait, dit-on, tout récemment en Angleterre des essais heureux sur le développement de la puissance motrice par l’action de la chaleur sur l’air atmosphérique. Nous ignorons entièrement ne quoi ces essais ont consisté, si toutefois ils sont réels.”