Part 21

Q = (EPV)/(μ(1 + Et)) log (V)/(V′) = (EP′V′)/(μ(1 + Et)) log (V)/(V′). (10)

From this result we draw the following conclusion:

47. Equal volumes of all elastic fluids, taken at the same temperature and pressure, when compressed to smaller equal volumes, disengage equal quantities of heat.

This extremely remarkable theorem of Carnot’s was independently laid down as a probable experimental law by Dulong, in his “Recherches sur la Chaleur Spécifique des Fluides Élastiques,” and it therefore affords a most powerful confirmation of the theory.

48. In some very remarkable researches made by Mr. Joule upon the heat developed by the compression of air, the quantity of heat produced in different experiments has been ascertained with reference to the amount of work spent in the operation. To compare the results which he has obtained with the indications of theory, let us determine the amount of work necessary actually to produce the compression considered above.

49. In the first place, to compress the gas from the volume v + dv to v, the work required is pdv, or, since

pv = p{0}v{0}(1 + Et), p{0}v{0}(1 + Et)(dv)/(v).

Hence, if we denote by W the total amount of work necessary to produce the compression from V to V′, we obtain, by integration,

W = p{0}v{0}(1 + Et) log (V)/(V′).

Comparing this with the expression above, we find

(W)/(Q) = (μ(1 + Et))/(E). (11)

50. Hence we infer that—

(1) The amount of work necessary to produce a unit of heat by the compression of a gas is the same for all gases at the same temperature;

(2) And that the quantity of heat evolved in all circumstances, when the temperature of the gas is given, is proportional to the amount of work spent in the compression.

51. The expression for the amount of work necessary to produce a unit of heat is

μ(1 + Et)/(E),

and therefore Regnault’s experiments on steam are available to enable us to calculate its value for any temperature. By finding the values of μ at 0°, 10°, 20°, etc., from Table I., and by substituting successively the values 0, 10, 20, etc., for t, the following results have been obtained:

TABLE OF THE VALUES OF (μ(1 + Et))/(E). ───────────────────────────────────┬─────────────────────────────────── Work requisite to produce a unit of│ Temperature of the Gas. Heat by the compression of a Gas. │ ───────────────────────────────────┼─────────────────────────────────── Ft.-pounds. │ ° 1357.1 │ 0 1368.7 │ 10 1379.0 │ 20 1388.0 │ 30 1395.7 │ 40 1401.8 │ 50 1406.7 │ 60 1412.0 │ 70 1417.6 │ 80 1424.0 │ 90 1430.6 │ 100 1438.2 │ 110 1446.4 │ 120 1455.8 │ 130 1465.3 │ 140 1475.8 │ 150 1489.2 │ 160 1499.0 │ 170 1511.3 │ 180 1523.5 │ 190 1536.5 │ 200 1550.2 │ 210 1564.0 │ 220 1577.8 │ 230 ───────────────────────────────────┴───────────────────────────────────

Mr. Joule’s experiments were all conducted at temperatures from 50° to about 60° Fahr., or from 10° to 16° Cent.; and consequently, although some irregular differences in the results, attributable to errors of observation inseparable from experiments of such a very difficult nature, are presented, no regular dependence on the temperature is observable. From three separate series of experiments, Mr. Joule deduces the following numbers for the work, in foot-pounds, necessary to produce a thermic unit Fahrenheit by the compression of a gas.

820, 814, 760.

Multiplying these by 1.8, to get the corresponding number for a thermic unit Centigrade, we

1476, 1465, and 1368.

The largest of these numbers is most nearly conformable with Mr. Joule’s views of the relation between such experimental “equivalents,” and others which he obtained in his electro-magnetic researches; but the smallest agrees almost perfectly with the indications of Carnot’s theory; from which, as exhibited in the preceding table, we should expect, from the temperature in Mr. Joule’s experiments, to find a number between 1369 and 1379 as the result.

III. On the Specific Heats of Gases.

52. The following proposition is proved by Carnot as a deduction from his general theorem regarding the specific heats of gases.

The excess of specific heat under a constant pressure above the specific heat at a constant volume, is the same for all gases at the same temperature and pressure.

53. To prove this proposition, and to determine an expression for the “excess” mentioned in its enunciation, let us suppose a unit of volume of a gas to be elevated in temperature by a small amount, τ. The quantity of heat required to do this will be Aτ, if A denote the specific heat at a constant volume. Let us next allow the gas to expand without going down in temperature, until its pressure becomes reduced to its primitive value. The expansion which will take place will be (Eτ)/(1 + Et), if the temperature be denoted by t; and hence, by (8), the quantity of heat that must be supplied, to prevent any lowering of temperature, will be

(Ep{0}v{0})/(μ) . (Eτ)/(1 + Et), or (E^2p)/(μ(1 + Et)^2)τ.