Part 8

The quantity of caloric transferred from the body A to the body B is evidently that which is absorbed by the gas in its expansion of volume, or that which this gas relinquishes during compression. We are led, then, to establish the following proposition:

When a gas passes without change of temperature from one definite volume and pressure to another volume and another pressure equally definite, the quantity of caloric absorbed or relinquished is always the same, whatever may be the nature of the gas chosen as the subject of the experiment.

Take, for example, 1 litre of air at the temperature of 100° and under the pressure of one atmosphere. If we double the volume of this air and wish to maintain it at the temperature of 100°, a certain quantity of heat must be supplied to it. Now this quantity will be precisely the same if, instead of operating on the air, we operate upon carbonic-acid gas, upon nitrogen, upon hydrogen, upon vapor of water or of alcohol, that is, if we double the volume of 1 litre of these gases taken at the temperature of 100° and under atmospheric pressure.

It will be the same thing in the inverse sense if, instead of doubling the volume of gas, we reduce it one half by compression. The quantity of heat that the elastic fluids set free or absorb in their changes of volume has never been measured by any direct experiment, and doubtless such an experiment would be very difficult, but there exists a datum which is very nearly its equivalent. This has been furnished by the theory of sound. It deserves much confidence because of the exactness of the conditions which have led to its establishment. It consists in this:

Atmospheric air should rise one degree Centigrade when by sudden compression it experiences a reduction of volume of ¹⁄₁₁₆.

Experiments on the velocity of sound having been made in air under the pressure of 760 millimetres of mercury and at the temperature of 6°, it is only to these two circumstances that our datum has reference. We will, however, for greater facility, refer it to the temperature 0°, which is nearly the same.

Air compressed ¹⁄₁₁₆, and thus heated one degree, differs from air heated directly one degree only in its density. The primitive volume being supposed to be V, the compression of ¹⁄₁₁₆ reduces it to V − ¹⁄₁₁₆ V.

Direct heating under constant pressure should, according to the rule of M. Gay-Lussac, increase the volume of air ¹⁄₂₆₇ above what it would be at 0°: so the air is, on the one hand, reduced to the volume V − ¹⁄₁₁₆ V; on the other, it is increased to V + ¹⁄₂₆₇ V.

The difference between the quantities of heat which the air possesses in both cases is evidently the quantity employed to raise it directly one degree; so then the quantity of heat that the air would absorb in passing from the volume V − ¹⁄₁₁₆ V to the volume V + ¹⁄₂₆₇ V is equal to that which is required to raise it one degree.

Let us suppose now that, instead of heating one degree the air subjected to a constant pressure and able to dilate freely, we inclose it within an invariable space, and that in this condition we cause it to rise one degree in temperature. The air thus heated one degree will differ from the air compressed ¹⁄₁₁₆ only by its ¹⁄₁₁₆ greater volume. So then the quantity of heat that the air would set free by a reduction of volume of ¹⁄₁₁₆ is equal to that which would be required to raise it one degree Centigrade under constant volume. As the differences between the volumes V − ¹⁄₁₁₆ V, V, and V + ¹⁄₂₆₇ V are small relatively to the volumes themselves, we may regard the quantities of heat absorbed by the air in passing from the first of these volumes to the second, and from the first to the third, as sensibly proportional to the changes of volume. We are then led to the establishment of the following relation:

The quantity of heat necessary to raise one degree air under constant pressure is to the quantity of heat necessary to raise one degree the same air under constant volume, in the ratio of the numbers

¹⁄₁₁₆ + ¹⁄₂₆₇ to ¹⁄₁₁₆;

or, multiplying both by 116 × 267, in the ratio of the numbers 267 + 116 to 267.

This, then, is the ratio which exists between the capacity of air for heat under constant pressure and its capacity under constant volume. If the first of these two capacities is expressed by unity, the other will be expressed by the number (267)/(267 + 116), or very nearly 0.700; their difference, 1 − 0.700 or 0.300, will evidently express the quantity of heat which will produce the increase of volume in the air when it is heated one degree under constant pressure.

According to the law of MM. Gay-Lussac and Dalton, this increase of volume would be the same for all other gases; according to the theory demonstrated on page 87, the heat absorbed by these equal increases of volume is the same for all the elastic fluids, which leads to the establishment of the following proposition:

The difference between specific heat under constant pressure and specific heat under constant volume is the same for all gases.

It should be remarked here that all the gases are considered as taken under the same pressure, atmospheric pressure for example, and that the specific heats are also measured with reference to the volumes.

It is a very easy matter now for us to prepare a table of the specific heat of gases under constant volume, from the knowledge of their specific heats under constant pressure. Here is the table:

TABLE OF THE SPECIFIC HEAT OF GASES. ───────────────────────┬───────────────────────┬─────────────────────── NAMES OF GASES. │ Specific Heat under │Specific Heat at Const. │ Const. Press. │ Vol. ───────────────────────┼───────────────────────┼─────────────────────── Atmospheric Air, │ 1.000 │ 0.700 Hydrogen Gas, │ 0.903 │ 0.603 Carbonic Acid, │ 1.258 │ 0.958 Oxygen, │ 0.976 │ 0.676 Nitrogen, │ 1.000 │ 0.700 Protoxide of Nitrogen, │ 1.350 │ 1.050 Olefiant Gas, │ 1.553 │ 1.253 Oxide of Carbon, │ 1.034 │ 0.734 ───────────────────────┴───────────────────────┴───────────────────────

The first column is the result of the direct experiments of MM. Delaroche and Bérard on the specific heat of the gas under atmospheric pressure, and the second column is composed of the numbers of the first diminished by 0.300.

The numbers of the first column and those of the second are here referred to the same unit, to the specific heat of atmospheric air under constant pressure.

The difference between each number of the first column and the corresponding number of the second being constant, the relation between these numbers should be variable. Thus the relation between the specific heat of gases under constant pressure and the specific heat at constant volume, varies in different gases.

We have seen that air when it is subjected to a sudden compression of ¹⁄₁₁₆ of its volume rises one degree in temperature. The other gases through a similar compression should also rise in temperature. They should rise, but not equally, in inverse ratio with their specific heat at constant volume. In fact, the reduction of volume being by hypothesis always the same, the quantity of heat due to this reduction should likewise be always the same, and consequently should produce an elevation of temperature dependent only on the specific heat acquired by the gas after its compression, and evidently in inverse ratio with this specific heat. Thus we can easily form the table of the elevations of temperature of the different gases for a compression of ¹⁄₁₁₆.

TABLE OF THE ELEVATION OF TEMPERATURE<BR>OF Gases through the Effect of Compression. ──────────────────────┬──────────────────────────────────────────────── NAMES OF GASES. │ Elevation of Temperature for a Reduction of │ Volume of ¹⁄₁₁₆. ──────────────────────┼──────────────────────────────────────────────── │ ° Atmospheric Air, │ 1.000 Hydrogen Gas, │ 1.160 Carbonic Acid, │ 0.730 Oxygen, │ 1.035 Nitrogen, │ 1.000 Protoxide of Nitrogen,│ 0.667 Olefiant Gas, │ 0.558 Carbonic Oxide, │ 0.955 ──────────────────────┴────────────────────────────────────────────────

A second compression of ¹⁄₁₁₆ (of the altered volume), as we shall presently see, would also raise the temperature of these gases nearly as much as the first; but it would not be the same with a third, a fourth, a hundredth such compression. The capacity of gases for heat changes with their volume. It is not unlikely that it changes also with the temperature.

We shall now deduce from the general proposition stated on page 68 a second theory, which will serve as a corollary to that just demonstrated.

Let us suppose that the gas enclosed in the cylindrical space abcd (Fig. 2) be transported into the space a′b′c′d′ (Fig. 3) of equal height, but of different base and wider. This gas would increase in volume, would diminish in density and in elastic force, in the inverse ratio of the two volumes abcd, a′b′c′d′. As to the total pressure exerted in each piston cd, c′d′, it would be the same from all quarters, for the surface of these pistons is in direct ratio to the volumes.

Let us suppose that we perform on the gas inclosed in a′b′c′d′ the operations described on page 70, and which were taken as having been performed upon the gas inclosed in abcd; that is, let us suppose that we have given to the piston c′d′ motions equal to those of the piston cd, that we have made it occupy successively the positions c′d′ corresponding to cd, and e′f′ corresponding to ef, and that at the same time we have subjected the gas by means of the two bodies A and B to the same variations of temperature as when it was inclosed in abcd. The total effort exercised on the piston would be found to be, in the two cases, always the same at the corresponding instants. This results solely from the law of Mariotte. In fact, the densities of the two gases maintaining always the same ratio for similar positions of the pistons, and the temperatures being always equal in both, the total pressures exercised on the pistons will always maintain the same ratio to each other. If this ratio is, at any instant whatever, unity, the pressures will always be equal.