constant pressure, may be denoted by r. This quantity of heat r is usually called latent heat; for it disappears as regards the thermometer, and during the whole process the temperature t remains unaltered.

The whole quantity of heat which, under the suppositions premised, must be supplied to the water at 0°, in order to convert it into steam at t° and of the corresponding elasticity p, is therefore Q-W+r. But according to Regnault's investigations the quantity of heat requisite for the object in question is Q=606.5+0.305t; consequently, since r=Q-W,

r=606.5-0.695t-0.00002t2-0.000000313.

Instead of this value of r, Clausius makes use of the approximate value r=607-0.70st .

.. 1)

according to which the numerical values of the sixth vertical series of the subjoined table are calculated, while the fifth column contains the corresponding values for Q.

If the whole quantity of heat were exclusively expended in external work, it would be easy to determine, by the first law, the volume of the steam formed; for we should have r=Apu, and since r, A, and p are known, we might determine from this equation the volume u of the space K K', (Fig. 10,) which is free in the cylinder under the piston, while one kilogram of water at t° is converted into steam at t°. But the matter here is not so simple.

The quantity of heat r, which must be supplied to the unit weight of water at to, in order to convert it into saturated vapor at t°, is known, indeed, by Regnault's experiments, but this quantity of heat divides into two parts; one part Apu serves to execute the external work pu; it is the other part p which is expended in overcoming the cohesion of the particles of water, and therefore in the performance of an internal work. Whence r=p+Apu. Neither p, nor Apu, nor the proportion of these two magnitudes is directly given; in order to determine them, we must first seek in some way to eliminate p, as it were, that is to say, we must propose some operation with the vapor by which a definite external work is performed, while the internal work performed shall, at the end of the operation, be nothing. A process of this sort is denoted by the name of a circle-process, (Kreisprocess.)

Let us suppose the volume w of the unit weight of water at t°, to be repre

Fig. 11.

sented by the abscissa O A, (Fig. 11,) the pressure p, which is exerted thereon, by the ordinate A a. Let heat now be conveyed to the water in such wise that the vapor which is formed may maintain the constant temperature t. In virtue of this the pressure p also remains constant. The supply of heat is to be continued until all the water is converted into vapor. The volume w will now have been changed into O B=v, and will thus be increased by A B=u; and since the pressure p has in the meantime remained unchanged, the external work thus performed and represented by the rectangle A a b B is equal to pu. The quantity of heat supplied during this formation of vapor is r.

OAD

BC

To this vapor we now allow, without supplying or withdrawing heat, a further small expansion from O B to O C, till the temperature be sunk 1° and the corresponding tension by . The work d thereby performed is represented by the quadrangle BCcb; and we will denote by g the corresponding heat which is disengaged from the vapor. In the fourth column of the table here given will be found the amount of diminution of tension, when the vapor, which is saturated for any one of the temperatures given in the first column of that table, is cooled 1°. The numbers of the table ranged under are found in the following manner:

If we subtract any of the values of p contained in the third column from the following one, we shall learn how much the tension of the saturated vapor is increased by an elevation of temperature of 5°. How much it is diminished by a lowering of temperature of 5°, we learn by subtracting from the same value of p the preceding one. If we now take the mean of these two differences, and of this mean the fifth part, we shall learn (without sensible error) how much the tension of the vapor of water is changed by an elevation or lowering of temperature by 1°. Thus, for example, for 150° C. the first difference is 6897.7; the second difference, 6195.4; the mean of the two is 6546.5; and the fifth part thereof, 1309.3, the number which stands under ø in the horizontal row of 150° C. Let the vapor, which now has the temperature t-1 and the tension p'=p— •, be compressed by the volume u, (C D, Fig. 11,) while the heat is continuously withdrawn from it in such manner that the temperature shall always remain t-1 and the tension p-.

The quantity of heat becoming free during this compression, and withdrawn from the vapor, consists of two parts, namely: of the quantity of heat A p′u, which corresponds to the labor p'u expended for the compression, and represented by the rectangle CD dc, and the quantity of heat p which becomes free by the condensation of a corresponding quantity of vapor.

Let the compression now be finally continued from O D to O A, without the addition or abstraction of heat; the temperature will thereby be raised to to, the

pressure to p, and the vapor will again be fully restored to its original state, (water of t°.) In this last part of the operation the work d, represented by the rectangle Dda A, is expended, and thereby the quantity of heat q is again supplied to the body under experiment, a quantity which it had lost during the expansion from OB to O C.

While the water has been thus fully restored to its original state, the work pu+d will, during the operation cited, have been performed through the expenditure of the quantities of heat r and q, and thereupon the quantity of heat r'+q have been gained by the expenditure of the work p'u+d. The sum of the work gained is thus,

pu+d-p'u-d=&u,

a work which is represented by the shaded parallelogram a b c d, (Fig. 11.) The heat expended in producing this work is,

r+q-r-q=r-r'.

But the quantity of heat requisite for the performance of the work u is Ag u; we have then the equation

A&u=r-r'.

Now, in this equation the values A and r are already known; only, therefore, is wanting to enable us to determine that of u. And as the rigorous solu- tion of this problem is not possible without the aid of the higher analysis, we must here content ourselves with an elementary process of approximation.

The quantities of heat Apu and Ap'u stand evidently in proportion to the tensions p and p'; and to these we may assign as proportional, since the question regards only slight differences of temperature, the density of the vapor of water at to and at (t-1)°. But to these densities are also proportional the quantities of water which, at the temperature to, are evaporated during the expansion through the volume u, and at the temperature (t-1)° are condensed during the compression through the volume u. Whence, therefore, we have p: p'=p: p', and P+Apu: p'+Ap'u=p: p', or r: r'=p: p'.

But since, within such narrow limits of temperature, the saturated vapor may be assumed as following the law of Mariotte Gay-Lussac; therefore

or, if we divide numerators and denominators by a, and take 1=a,

a

if T denote the absolute temperature* which corresponds to t° C. Thus we have *Let p be the elastic force of a confined mass of air at 0°; then, according to Mariotte Gay-Lussac's law, this elasticity at to C. is equal to p (1+0.00365 t); the elastic force of the enclosed mass of air is thus null, if 1+0.00365 t=0'; that is to say, if t=-273° C. At this temperature, which is 273° below the freezing point of water, the gases lose their power of expansion; and it is this point which we indicate as the absolute zero-point. It is the temperature counted from this point onward, according to the Celsius degrees, T-273+t, (if t be the temperature counted onward from the freezing point of water,) which is denoted as absolute temperature. In the second vertical series of our foregoing table are given the absolute temperatures which correspond to the temperatures of the first column measured by the thermometer of Celsius.

1

f for

A

we place its numerical value 424. From this equation, which is regarded as the second general equation of the mechanical theory of heat, the appropriate value of u at every given temperature admits of being calculated, as r, T, and are known magnitudes. Thus, for example, for t=120° we have

Our table contains in the seventh column the value of u for the temperatures given in the first vertical series.

To these values of u we have only to add 0.001, (the volume of one kilogram of water expressed in cubic metres,) in order to obtain the volume v, which one kilogram of saturated vapor of the corresponding temperature occupies. The numerical values of v are presented in the eighth column of our table. From this we see, for example, that one kilogram of saturated vapor of 100°, of 130°, of 160°, &c., occupies the volume of 1.6459, of 0.6529, of 0.3002, &c., cubic

.metres.

If the saturated vapors of water followed Mariotte Gay-Lussac's law, then the product must be a constant magnitude. But this product is

pv
1+at

46376 for t=100°

37832 for t=130°
32374 for t=160°;

thus for increasing temperatures it becomes continually smaller. From the application of the mechanical theory of heat to saturated vapors it results, therefore, as Clausius first showed, that these do not follow Lussac's law; that rather the elasticity of saturated vapor increases less rapidly with increasing temperature than the density thereof.

Since the quotient indicates the weight of one cubic metre of saturated

vapor,

1

1000 v

v

is the weight of one cubic decimetre, consequently, also, the specific weight or density of the same. The numerical values of the density of saturated vapor are given under 7 in the 9th column of our table.

γ

If we multiply the enlargement of volume u, which ensues from the transformation of one kilogram of water at t° into saturated vapor of to, into the corresponding pressure p, we obtain the external work performed through this operation, while the quantity of heat spent in the performance of this work is Ари. The numerical values of Apu, which stand in the 10th column of our table, are, however, not calculated in the above-cited manner, but according to T the empirical equation proposed by Zeuner: Apu=30.456 · log · whose " 100 results so nearly accord with those computed after the above theory that we may