Part 30

in which v is this volume, t the temperature, p the pressure, and c a constant quantity depending on the weight of the vapor and on the units chosen. We give here the table of the volumes occupied by a gramme of vapor formed at different temperatures, and consequently under different pressures.

───────────────────────┬───────────────────────┬─────────────────────── t │ p │ v or degrees Centigrade. │or tension of the vapor│ or volume of a gramme │ expressed in │ of vapor expressed in │millimetres of mercury.│ litres. ───────────────────────┼───────────────────────┼─────────────────────── ° │ mm. │ lit. 0│ 5.060 │ 185.0 20│ 17.32 │ 58.2 40│ 53.00 │ 20.4 60│ 144.6 │ 7.96 80│ 352.1 │ 3.47 100│ 760.0 │ 1.70 ───────────────────────┴───────────────────────┴───────────────────────

The first two columns of this table are taken from the Traité de Physique of M. Biot (vol. i., p. 272 and 531). The third is calculated by means of the above formula, and in accordance with the result of experiment, indicating that water vaporized under atmospheric pressure occupies a space 1700 times as great as in the liquid state.

By using three numbers of the first column and three corresponding numbers of the third column, we can easily determine the constants of our equation

t = (A + B log v)/(A′ + B′ log v).

We will not enter into the details of the calculation necessary to determine these quantities. It is sufficient to say that the following values,

A = 2268, A′ = 19.64, B = −1000, B′ = 3.30,

satisfy fairly well the prescribed conditions, so that the equation

t = (2268 − 1000 log v)/(19.64 + 3.30 log v)

expresses very nearly the relation which exists between the volume of the vapor and its temperature. We may remark here that the quantity B′ is positive and very small, which tends to confirm this proposition—that the specific heat of an elastic fluid increases with the volume, but follows a slow progression.

NOTE E.—Were we to admit the constancy of the specific heat of a gas when its volume does not change, but when its temperature varies, analysis would show a relation between the motive power and the thermometric degree. We will show how this is, and this will also give us occasion to show how some of the propositions established above should be expressed in algebraic language.

Let r be the quantity of motive power produced by the expansion of a given quantity of air passing from the volume of one litre to the volume of v litres under constant temperature. If v increases by the infinitely small quantity dv, r will increase by the quantity dr, which, according to the nature of motive power, will be equal to the increase dv of volume multiplied by the expansive force which the elastic fluid then possesses; let p be this expansive force. We should have the equation

dr = pdv. (1)

Let us suppose the constant temperature under which the dilatation takes place equal to t degrees Centigrade. If we call q the elastic force of the air occupying the volume 1 litre at the same temperature t, we shall have, according to the law of Mariotte,

(v)/(1) = (q)/(p), whence p = (q)/(v).

If now P is the elastic force of this same air at the constant volume 1, but at the temperature zero, we shall have, according to the rule of M. Gay-Lussac,

q = P + P (t)/(267) = (P)/(267)(267 + t);

whence

q = p = (P)/(267) (267 + t)/(v).

If, to abridge, we call N the quantity (P)/(267), the equation would become

p = N (t + 267)/(v);

whence we deduce, according to equation (1),

dr = N (t + 267)/(v)dv.

Regarding t as constant, and taking the integral of the two numbers, we shall have

r = N(t + 267) log v + C.

If we suppose r = 0 when v = 1, we shall have C = 0; whence

r = N(t + 267) log v. (2)

This is the motive power produced by the expansion of the air which, under the temperature t, has passed from the volume 1 to the volume v. If instead of working at the temperature t we work in precisely the same manner at the temperature t + dt, the power developed will be