For a theoretically perfect gas,

K

=

k..

.(2 a.)

The specific heat under constant pressure, deduced from the expression for the thermodynamic function in Article 248, equation 1, is as follows:

being simply the real specific heat increased by the work performed by unity of weight of the gas in undergoing, at any constant pressure, the expansion corresponding to one degree of rise of temperature; a quantity of work which is constant for a given perfect gas

d2 v under all circumstances. The quantities and

d2 p d = 2

d-2'

represent

ing the deviation of the laws of the elasticity of actual gases from those of the ideal condition of perfect gas, are so small, that their effects on apparent specific heat, though calculable, fall within the probable limits of errors of observation in the direct experiments hitherto made on the specific heat of the more common gases, such as air and carbonic acid. Referring, therefore, to the detailed papers already cited in the Trans. of the Royal Society of Edinburgh, vol. xx., for computations of the effects of such deviations, it will be sufficient for practical purposes to consider the specific heats of gases as represented by the formulæ 2A and 3A. The specific heats of gases, as expressed in the customary way, by their ratios to that of water, are found by dividing the quantities in these formulæ by Joule's equivalent (J), and may be thus expressed :

Examples of specific heat, stated in both ways, are given in Table II., at the end of the volume. Before the period of M. Regnault's experiments on a great variety of gases and vapours, published in the Comptes Rendus for 1853, no trustworthy direct experimental determination of the specific heat of any gas or vapour existed, except an approximate determination by Mr. Joule, made in 1852, of the specific heat of air; for the results formerly relied upon have been shown to be erroneous. In one of the papers referred to in the preceding Article, however (Edinburgh Transactions, 1850), the

dynamical specific heats of air had been computed from the following data :

Po vo, from M. Regnault's experiments 26214 foot-pounds. 493°-2 Fahrenheit.

Po vo
Το

=

.: K, — K、 = 53.15 foot-pounds per degree of Fahrenheit; being the energy exerted by one pound of air in undergoing, at a constant pressure, the expansion corresponding to one degree of rise of temperature, and the mechanical equivalent of the latent heat of expansion of the air under those circumstances, which (as stated in Article 212) is 0.069 of a British thermal unit, 53.15

772

7 =

=

Kas deduced from the velocity of sound in air, assumed

in the paper referred to as approximately value is 1-408. Consequently,

=

14; but a more exact

foot-pounds per degree of Fahrenheit. Hence is deduced the following ratio of the specific heat of air under constant pressure to

that of water,

e, according to M. Regnault's experiments, published }

Difference,.....

.0.2377.

0.2379

0.0002

In the calculation published in 1850, y was assumed 14, and cp was computed as = 0.24; but the calculation just given being founded on a more accurate value of y, is of course to be preferred as a test of the dynamical theory of heat. Mr. Joule's approximate determination in 1852 was 0.23. According to the dynamical theory of heat, the apparent specific heat of a gas under constant pressure is sensibly the same at all pressures and temperatures, if the gas is nearly perfect. According to the hypothesis of substantial caloric, that specific heat diminishes as the pressure increases, according to a law which is stated in many treatises on physics, even of the most recent dates (in some, indeed, as confidently as if it were an observed fact). The experiments of M. Regnault, by which the specific heat of air under constant pressure was determined at various temperatures from-22° Fahr. up to 437° Fahr., and at various pressures of from one to ten atmospheres, and found to be sensibly the same under all these circumstances, constitute "experimenta crucis" conclusive against

251. Heating and Cooling of Gases and Vapours by Compression and Expansion.-If a substance wholly or partially in the state of gas or vapour be enclosed in a vessel which does not conduct any appreciable amount of heat to or from the substance, then the compression and expansion of the substance through variations of the volume of the vessel will produce respectively heating and cooling, according to a law expressed by the condition, that the thermodynamic function is constant.

The following equations contain two modes of expressing this condition, deduced from the expressions in Articles 246 and 248 respectively :

and each of those is the equation of an adiabatic curve. For a perfect gas, we have

hence let p11 correspond to one given absolute temperature 71, to another given absolute temperature 2; then for a per

and P2 V2

fect gas, or a gas sensibly perfect,

2

or,

=

Τι

(2 1) log

1

....(4.)

=

OF, ()'' - ()

These equations give, for the law of expansion of a perfect gas, without receiving or emitting heat, the following relation between the pressure and the volume,

and this is the simplest form of the equation of an adiabatic curve for a perfect gas. The values of the several exponents in equations 4 and 5 for AIR are,

that "idolon fori," the hypothesis of caloric. Those experiments also afford evidence of the fact, that the scale of the air thermometer sensibly agrees with that of absolute temperatures.

Y

For STEAM in the perfectly gaseous state, taking (as in Article 202, equation 4), po vo 42141, and according to M. Regnault's Po = experiments, K,772 × 0·48 = 371, we find,

In the experiments of MM. Hirn and Cazin, the value of 7-7 -1 ranged from 4.23 to 4:47. (Annales de Chimie, 1867, vol. x.) These values, however, are not so certain as those of the corre sponding quantities for air. From equation 1 is easily deduced the law of the variation of the pressure with the volume of any fluid, whether perfectly gaseous or not, enclosed in a non-conducting vessel, viz. the rate of variation of the pressure with the volume when the fluid is enclosed in a non-conducting vessel, exceeds the rate of variation when the temperature is constant, in the ratio of the apparent specific heat of the fluid at constant pressure to its apparent specific heat at constant volume:-a law expressed symbolically as follows:

as equation 5 also shows. The cooling of air ly expansion has been applied to practical purposes by Dr. Gorrie, Professor Piazzi Smyth, Mr. Kirk, and other inventors.

252. Velocity of Sound in Gases.*-The velocity of sound in any fluid is well known to be equal to that acquired by a heavy body in falling through one-half of the height which represents the variation of the pressure of the fluid with its density during a sudden change of density. That is to say, let a be the velocity of sound in feet per second, g the accelerating force of gravity in a second = 32.2 feet per second, D the weight of one cubic foot of the fluid its elastic pressure in pounds per square

1
"
บ

in pounds = and Ρ foot, then

During the transmission of a wave of sound, the compression and expansion of the particles of a fluid take place so rapidly, that there is not time for any appreciable transmission of heat between different particles, and the variations of the pressure and density are related to each other as they would be in a non-conducting vessel; consequently, if h represents the rate of variation of pressure with density at a constant temperature, then it follows from the principle d p of equation 6, Art. 251, that = yh, and d D

This equation was proved long ago by Laplace and Poisson, for perfect gases, for which

Po vo

h = p v =

T........

Το

.(3.)

but it is true, as we have seen, for all fluids whatsoever.

Applying the formula to air, considered as a sensibly perfect gas,

with the following data:

• In this Article the sounds are supposed to be of moderate intensity, so that there is no sensible acceleration of the sound due to the cause investigated by Mr. Earnshaw: as to which see Proc. Roy. Soc., 1859.

Proved by Prof. G. G. Stokes.