tube communicating with the atmosphere under the diminished pressure p. on the other side.

or

But if the gas does not obey that law, then will

will be the expression for the quantity of heat absorbed in the change of volume due to the escape of the gas into the air. And if the internal work caused by compression and expansion in passing through the porous plug be nothing, then should there be no change of temperature.

But if this be not rigorously true for the gas operated upon.

S"Q = ASU

will express the quantity of heat requisite to this internal change. Substituting these values in equation (56), we obtain

&Q = d'Q + 8′′ Q = A§ (U + S),

for the variation of heat measured in thermal units, which the thermometers should indicate.

The experiments showed for hydrogen a thermal variation which was scarcely appreciable, a very small change for atmospheric air, but a very considerable reduction of temperature for carbonic acid.

They gave the general result, that for each gas the ratio of the reduction of temperature to the difference of pressures is a constant factor,

t― to = a (p − po) ;

hence, if we denote by c the specific heat of the gas,

and

SU = Eac (p − Po) + pv — Povo •

For an elementary variation of the normal pressure and volume

and by equation (98), the formula of Regnault,

From these equations, omitting infinitesimals of the second order,

But in this equation the common factor of the second member, Po dv, is the variation of external work. Dividing by it, we get for the ratio of the change of internal energy to that of external work,

an expression in which all the quantities in the second member have been experimentally determined; c by the velocity of sound; A' and by Regnault; and a and E by Joule and Thomson for atmospheric air, hydrogen, and carbonic acid.

These three gases gave the following values of the ratio of internal to external work:

Atmospheric air

Hydrogen

Carbonic acid

0.0020

0.0008

0.0080

Hence, we see that when air at the usual barometric pressure expands slightly, the internal work is only 0.002 of the external. While for hydrogen it is but two-fifths of that amount, and in fact scarcely appreciable. But for carbonic acid, a liquefiable gas, it is four times greater than for air, and amounts to nearly one per cent. The effect of elevated temperature was ascertained for air and carbonic acid, and found to be a great diminution of the constant a, or of the rate of cooling to variation of pressure on the different sides of the porous plug. For air, at temperatures of 15° to 20° centigrade, the mean value of the rate a was 0.262; but at 91°.5 it was only 0.206; for carbonic acid at 20° it was 1.151; and at 91.5 it was reduced to 0.703; comparing these numbers, we see that for carbonic acid the rate reduces from five to three and a half times that of air; thus again manifesting its tendency to approach at high temperatures to the character of a permanent gas.

88. From the above, it is evident that the second law of Joule, like those of Mariotte and Charles, constitutes a limit to which the action of real gases tends to approach when they are highly rarefied; and it is rigorously true only for a theoretically perfect gas.

To comprehend fully its meaning, the analysis of the total action of heat upon an expanding body, already given in § 49, must be borne in mind. That action divides itself into three distinct effects: 1°, change of temperature, rendering the body hotter; 2°, change of internal molecular structure, or variation

of the potential of molecular action, which may be called internal work; 3°, external work upon the enveloping surface, or vessel, as that upon a piston pushed by steam, or by expanding air, in the cylinder of an engine. For an elementary thermal change, the last of these three effects is expressed by pdv; and the first is denoted by cdt in the formula

or by the first term of the second member of the equation

It is, therefore, only the second term of the last member of this expression, denoting internal work done against cohesion or molecular action, which Joule found to be insensible by his first method of experiment; and which the far superior method of the porous plug proves to be very small, though not inappreciable for air and hydrogen, but quite large for carbonic acid.

Hence it appears that molecular attraction, or the force of cohesion, must be very small in permanent gases, but is quite sensible in a liquefiable gas, such as carbonic acid, and should be nothing for the perfectly gaseous state.

From the equations just used and the second law of Joule, we have already deduced equations (78) for perfect gases,

the last being a very simple relation between the mechanical equivalent of heat, the latent heat of expansion and the pressure, from which any one of these three quantities may be determined when the other two are known.

By substituting for its value Ap, and for dt its equivalent dr, the variation of absolute temperature, § 71, we have

At the absolute zero of heat, both dQ and cdr are nothing; and therefore p is so also. Consequently, at that zero, a perfect gas would be without external pressure, without internal motion or temperature, and without molecular attraction or repulsion; or it would be in a state of utter dynamic inaction and indifference, both within itself, and relatively to an enveloping surface.

LAW OF DULONG AND PETIT.

89. The laws of Charles and Mariotte and the second law of Joule being for real gases only limits, or approximations, the question naturally arises, whether the two remaining experimental postulates, from which we have deduced the properties of the perfectly gaseous state, are more rigorously true; or whether they too need to be corrected for perturbations when applied to gases.

To the analyses chiefly of Berzelius we owe the establishment of the great fundamental law of chemistry, that bodies combine in definite proportions by weight; and to Gay Lussac the equally simple law, that gases unite by volumes which are in very simple ratios to each other; water, for example, being composed of one volume of oxygen and two of hydrogen, and nitric acid consisting of two volumes of nitrogen united with five of oxygen.

The postulate that the product of the density by the specific heat of a gas is constant, see equation (83), is a consequence of the laws of Berzelius and Gay Lussac, coupled with the discovery of Dulong and Petit, that the product of the specific heat of