Part 10

According to the law that specific heats follow with relation to pressures, it is only necessary to have observed them in two particular cases to deduce them in all possible cases: it is thus that, making use of the experimental result of MM. Delaroche and Bérard which has just been given, we have prepared the following table of the specific heat of air under different pressures:

SPECIFIC HEAT OF AIR. ────────────────────────┬──────────────────────────────────────────── Pressure in Atmospheres.│Specific Heat, that of Air under Atmospheric │ Pressure being 1. ────────────────────────┼──────────────────────────────────────────── ¹⁄₁₀₂₄ │ 1.840 ¹⁄₅₁₂ │ 1.756 ¹⁄₂₅₆ │ 1.672 ¹⁄₁₂₈ │ 1.588 ¹⁄₆₄ │ 1.504 ¹⁄₃₂ │ 1.420 ¹⁄₁₆ │ 1.336 ⅛ │ 1.252 ¼ │ 1.165 ½ │ 1.084 1 │ 1.000 2 │ 0.916 4 │ 0.832 8 │ 0.748 16 │ 0.664 32 │ 0.580 64 │ 0.496 128 │ 0.412 256 │ 0.328 512 │ 0.244 1024 │ 0.160 ────────────────────────┴────────────────────────────────────────────

The first column is, as we see, a geometrical progression, and the second an arithmetical progression.

We have carried out the table to the extremes of compression and rarefaction. It may be believed that air would be liquefied before acquiring a density 1024 times its normal density, that is, before becoming more dense than water. The specific heat would become zero and even negative on extending the table beyond the last term. We think, furthermore, that the figures of the second column here decrease too rapidly. The experiments which serve as a basis for our calculation have been made within too contracted limits for us to expect great exactness in the figures which we have obtained, especially in the outside numbers.

Since we know, on the one hand, the law according to which heat is disengaged in the compression of gases, and on the other, the law according to which specific heat varies with volume, it will be easy for us to calculate the increase of temperature of a gas that has been compressed without being allowed to lose heat. In fact, the compression may be considered as composed of two successive operations: (1) compression at a constant temperature; (2) restoration of the caloric emitted. The temperature will rise through the second operation in inverse ratio with the specific heat acquired by the gas after the reduction of volume,—specific heat that we are able to calculate by means of the law demonstrated above. The heat set free by compression, according to the theorem of page 81, ought to be represented by an expression of the form

s = A + B log v,

s being this heat, v the volume of the gas after compression, A and B arbitrary constants dependent on the primitive volume of the gas, on its pressure, and on the units chosen.

The specific heat varying with the volume according to the law just demonstrated, should be represented by an expression of the form

z = A′ + B′ log v,

A′ and B′ being the different arbitrary constants of A and B.

The increase of temperature acquired by the gas, as the effect of compression, is proportional to the ratio (s)/(z) or to the relation (A + B log v)/(A′ + B′ log v). It can be represented by this ratio itself; thus, calling it t, we shall have

t = (A + B log v)/(A′ + B′ log v).

If the original volume of the gas is 1, and the original temperature zero, we shall have at the same time t = 0, log v = 0, whence A = 0; t will then express not only the increase of temperature, but the temperature itself above the thermometric zero.

We need not consider the formula that we have just given as applicable to very great changes in the volume of gases. We have regarded the elevation of temperature as being in inverse ratio to the specific heat; which tacitly supposes the specific heat to be constant at all temperatures. Great changes of volume lead to great changes of temperature in the gas, and nothing proves the constancy of specific heat at different temperatures, especially at temperatures widely separated. This constancy is only an hypothesis admitted for gases by analogy, to a certain extent verified for solid bodies and liquids throughout a part of the thermometric scale, but of which the experiments of MM. Dulong and Petit have shown the inaccuracy when it is desirable to extend it to temperatures far above 100°.

According to a law of MM. Clement and Desormes, a law established by direct experiment, the vapor of water, under whatever pressure it may be formed, contains always, at equal weights, the same quantity of heat; which leads to the assertion that steam, compressed or expanded mechanically without loss of heat, will always be found in a saturated state if it was so produced in the first place. The vapor of water so made may then be regarded as a permanent gas, and should observe all the laws of one. Consequently the formula

t = (A + B log v)/(A′ + B′ log v)

should be applicable to it, and be found to accord with the table of tensions derived from the direct experiments of M. Dalton.

We may be assured, in fact, that our formula, with a convenient determination of arbitrary constants, represents very closely the results of experiment. The slight irregularities which we find therein do not exceed what we might reasonably attribute to errors of observation.

We will return, however, to our principal subject, from which we have wandered too far—the motive power of heat.

We have shown that the quantity of motive power developed by the transfer of caloric from one body to another depends essentially upon the temperature of the two bodies, but we have not shown the relation between these temperatures and the quantities of motive power produced. It would at first seem natural enough to suppose that for equal differences of temperature the quantities of motive power produced are equal; that is, for example, the passage of a given quantity of caloric from a body, A, maintained at 100°, to a body, B, maintained at 50°, should give rise to a quantity of motive power equal to that which would be developed by the transfer of the same caloric from a body, B, at 50°, to a body, C, at zero. Such a law would doubtless be very remarkable, but we do not see sufficient reason for admitting it à priori. We will investigate its reality by exact reasoning.

Let us imagine that the operations described on p. 70 be conducted successively on two quantities of atmospheric air equal in weight and volume, but taken at different temperatures. Let us suppose, further, the differences of temperature between the bodies A and B equal, so these bodies would have for example, in one of these cases, the temperatures 100° and 100° − h (h being indefinitely small), and in the other 1° and 1° − h. The quantity of motive power produced is, in each case, the difference between that which the gas supplies by its dilatation and that which must be expended to restore its primitive volume. Now this difference is the same in both cases, as any one can prove by simple reasoning, which it seems unnecessary to give here in detail; hence the motive power produced is the same.

Let us now compare the quantities of heat employed in the two cases. In the first, the quantity of heat employed is that which the body A furnishes to the air to maintain it at the temperature of 100° during its expansion. In the second, it is the quantity of heat which this same body should furnish to it, to keep its temperature at one degree during an exactly similar change of volume. If these two quantities of heat were equal, there would evidently result the law that we have already assumed. But nothing proves that it is so, and we shall find that these quantities are not equal.

The air that we shall first consider as occupying the space abcd (Fig. 2), and having 1 degree of temperature, can be made to occupy the space abef, and to acquire the temperature of 100 degrees by two different means:

(1) We may heat it without changing its volume, then expand it, keeping its temperature constant.

(2) We may begin by expanding it, maintaining the temperature constant, then heat it, when it has acquired its greater volume.

Let a and b be the quantities of heat employed successively in the first of the two operations, and let b′ and a′ be the quantities of heat employed successively in the second. As the final result of these two operations is the same, the quantities of heat employed in both should be equal. We have then

a + b = a′ + b′,

whence