Part 20

39. At any instant when the temperature of the water and vapor is t, during the fourth operation (see above, § 16, and suppose, for the sake of simplicity, that at the beginning of the first and at the end of the fourth operation the piston is absolutely in contact with the surface of the water), the latent heat of the vapor must be precisely equal to the amount of heat that would be necessary to raise the temperature of the whole mass, if in the liquid state, from t to S. Hence, if v′ denote the volume of the vapor, c the mean capacity for heat of a pound of water between the temperatures S and t, and W the weight of the entire mass, in pounds, we have

kv′ = c(S − t)W.

Again, the circumstances during the second operation are such that the mass of liquid and vapor possesses H units of heat more than during the fourth; and consequently, at the instant of the second operation, when the temperature is t, the volume v of the vapor will exceed v′ by an amount of which the latent heat is H, so that we have

v = v′ + (H)/(k).

40. Now, at any instant, the volume between the piston and its primitive position is less than the actual volume of vapor by the volume of the water evaporated. Hence, if x and x′ denote the abscissæ of the curve at the instants of the second and fourth operations respectively, when the temperature is t, we have

x = v − σv, x′ = v′ − σv′,

and, therefore, by the preceding equations,

x = (1 − σ)/(k){H + c(S − t)W}, (a) x′ = (1 − σ)/(k)c(S − t)W. (b) These equations, along with y = y′ = p, (c)

enable us to calculate, from the data supplied by Regnault, the abscissa and ordinate for each of the curves described above (§ 17) corresponding to any assumed temperature t. After the explanations of §§ 33, 34, 35, 36, it is only necessary to add that c is a quantity of which the value is very nearly unity, and would be exactly so were the capacity of water for heat the same at every temperature as it is between 0° and 1°; and that the value of c(S − t), for any assigned values of S and t, is found, by subtracting the number corresponding to t from the number corresponding to s, in the column headed “Nombre des unités de chaleur abandonnées par un kilogramme d’eau en descendant de T° à 0°,” of the last table (at the end of the tenth memoir) of Regnault’s work. By giving S the value 230°, and by substituting successively 220, 210, 200, etc., for t, values for x, y, x′, y′, have been found, which are exhibited in the table opposite.

─────────────┬─────────────────┬────────────────────────┬───────────── Temperatures.│ Volumes to be │ Volumes from the │Pressures of │described by the │ primitive position of │ saturated │ piston, to │ the piston to those │ steam, in │ complete the │occupied at instants of │pounds on the │fourth operation.│ the second operation. │square foot. t │ x′ │ x │y = y′ = │ │ │ p ─────────────┼─────────────────┼────────────────────────┼───────────── 0°│ 1269. W │x′ + 5.409.H │ 12.832 10│ 639.6. W │x′ + 2.847.H │ 25.567 20│ 337.3. W │x′ + 1.571.H │ 48.514 30│ 185.5. W │x′ + .9062.H │ 88.007 40│ 105.9. W │x′ + .5442.H │ 153.167 50│ 62.62. W │x′ + .3392.H │ 256.595 60│ 38.19. W │x′ + .2188.H │ 415.070 70│ 21.94. W │x′ + .1456.H │ 650.240 80│ 15.38. W │x′ + .09962.H │ 989.318 90│ 10.09. W │x′ + .06994.H │ 1465.80 100│ 6.744. W │x′ + .05026.H │ 2120.11 110│ 4.578. W │x′ + .03688.H │ 2999.87 120│ 3.141. W │x′ + .02758.H │ 4160.10 130│ 2.176. W │x′ + .02098.H │ 5663.70 140│ 1.519. W │x′ + .01625.H │ 7581.15 150│ 1.058. W │x′ + .01271.H │ 9990.26 160│ 0.7369. W │x′ + .01010.H │ 12976.2 170│ 0.5085. W │x′ + .008116.H │ 16630.7 180│ 0.3454. W │x′ + .006592.H │ 21051.5 190│ 0.2267. W │x′ + .005406.H │ 26341.5 200│ 0.1409. W │x′ + .004472.H │ 32607.7 210│ 0.0784. W │x′ + .003729.H │ 39960.7 220│ 0.3310. W │x′ + .003130.H │ 48512.4 230│ 0 │x′ + .002643.H │ 58376.6 ─────────────┴─────────────────┴────────────────────────┴─────────────

Appendix.

(Read April 30, 1849.)

41. In p. 30 some conclusions drawn by Carnot from his general reasoning were noticed; according to which it appears, that if the value of μ for any temperature is known, certain information may be derived with reference to the saturated vapor of any liquid whatever, and, with reference to any gaseous mass, without the necessity of experimenting upon the specific medium considered. Nothing in the whole range of Natural Philosophy is more remarkable than the establishment of general laws by such a process of reasoning. We have seen, however, that doubt may exist with reference to the truth of the axiom on which the entire theory is founded, and it therefore becomes more than a matter of mere curiosity to put the inferences deduced from it to the test of experience. The importance of doing so was clearly appreciated by Carnot; and, with such data as he had from the researches of various experimenters, he tried his conclusions. Some very remarkable propositions which he derives from his theory coincide with Dulong and Petit’s subsequently discovered experimental laws with reference to the heat developed by the compression of a gas; and the experimental verification is therefore in this case (so far as its accuracy could be depended upon) decisive. In other respects, the data from experiment were insufficient, although, so far as they were available as tests, they were confirmatory of the theory.

42. The recent researches of Regnault add immensely to the experimental data available for this object, by giving us the means of determining with considerable accuracy the values of μ within a very wide range of temperature, and so affording a trustworthy standard for the comparison of isolated results at different temperatures, derived from observations in various branches of physical science.

In the first section of this Appendix the theory is tested, and shown to be confirmed by the comparison of the values of μ found above, with those obtained by Carnot and Clapeyron from the observations of various experimenters on air, and the vapors of different liquids. In the second and third sections some striking confirmations of the theory arising from observations by Dulong, on the specific heat of gases, and from Mr. Joule’s experiments on the heat developed by the compression of air, are pointed out; and in conclusion, the actual methods of obtaining mechanical effect from heat are briefly examined with reference to their economy.

I. On the values of μ derived by Carnot and Clapeyron from observations on Air, and on the Vapors of various liquids.

43. In Carnot’s work, pp. 80–82, the mean value of μ between 0° and 1° is derived from the experiments of Delaroche and Bérard on the specific heat of gases, by a process approximately equivalent to the calculation of the value of (Ep{0}v{0})/(vdq/dv) for the temperature ½°. There are also, in the same work, determinations of the values of μ from observations on the vapors of alcohol and water; but a table given in M. Clapeyron’s paper, of the values of μ derived from the data supplied by various experiments with reference to the vapors of ether, alcohol, water, and oil of turpentine, at the respective boiling-points of these liquids, affords us the means of comparison through a more extensive range of temperature. In the cases of alcohol and water, these results ought of course to agree with those of Carnot. There are, however, slight discrepancies which must be owing to the uncertainty of the experimental data. In the opposite table, Carnot’s results with reference to air, and Clapeyron’s results with reference to the four different liquids, are exhibited, and compared with the values of μ which have been given above (Table I.) for the same temperatures, as derived from Regnault’s observations on the vapor of water.

────────────┬──────────────┬──────────────┬──────────────┬───────────── │ │ │ Values of μ │ │ │ │ deduced from │ Names of the│ │ │ Regnault’s │ Media. │Temperatures. │ Values of μ. │Observations. │Differences. ────────────┼──────────────┼──────────────┼──────────────┼───────────── │ ° │ (Carnot) │ │ Air │ 0.5│ 4.377│ 4.960│ .383 Sulphuric │ (Boil. pt.)│ (Clapeyron)│ │ Ether │ 35.5│ 4.478│ 4.510│ .032 Alcohol │ 78.8│ 3.963│ 4.030│ .071 Water │ 100│ 3.658│ 3.837│ .179 Essence of │ │ │ │ Turpentine│ 156.8│ 3.530│ 3.449│ −.081 ────────────┴──────────────┴──────────────┴──────────────┴─────────────

44. It may be observed that the discrepancies between the results founded on the experimental data supplied by the different observers with reference to water at the boiling-point, are greater than those which are presented between the results deduced from any of the other liquids, and water at the other temperatures; and we may therefore feel perfectly confident that the verification is complete to the extent of accuracy of the observations. The considerable discrepancy presented by Carnot’s result deduced from experiments on air, is not to be wondered at when we consider the very uncertain nature of his data.

45. The fact of the gradual decrease of μ through a very extensive range of temperature, being indicated both by Regnault’s continuous series of experiments and by the very varied experiment on different media, and in different branches of Physical Science, must be considered as a striking verification of the theory.

II. On the Heat developed by the Compression of Air.

46. Let a mass of air, occupying initially a given volume V, under a pressure P, at a temperature t, be compressed to a less volume V′, and allowed to part with heat until it sinks to its primitive temperature t. The quantity of heat which is evolved may be determined, according to Carnot’s theory, when the particular value of μ, corresponding to the temperature t, is known. For, by § 30, equation (6), we have

v(dq)/(dv) = (Ep{0}v{0})/(μ),

where dq is the quantity of heat absorbed, when the volume is allowed to increase from v to v + dv; or the quantity evolved by the reverse operation. Hence we deduce

dq = (Ep{0}v{0})/(μ) (dv)/(v). (8)

Now, (Ep{0}v{0})/(μ) is constant, since the temperature remains unchanged; and therefore we may at once integrate the second number. By taking it between the limits V′ and V, we thus find

Q = (Ep{0}v{0})/(μ) log (V)/(V′), (9)

where Q denotes the required amount of heat evolved by the compression from V to P′. This expression may be modified by employing the equations PV = P′V′ = p{0}v{0}(1 + Et); and we thus obtain