CHAPTER VI.

APPLICATION OF THE MECHANICAL THEORY OF HEAT TO

SATURATED VAPOUR.

§ 1. Fundamental equations for saturated vapour.

Among the equations of the last chapter, those deduced in § 6, which refer to a partial change in the body's state of aggregation, may conveniently be treated first; inasmuch as the circumstance there mentioned, viz. that the pressure is only a function of the temperature, greatly facilitates the treatment of the subject. We will in the first place consider the passage from the liquid to the vaporous condition.

Let a weight M of any given substance be inclosed in an expansible envelope: of this let the part m be in the condition of vapour, and that vapour (as necessarily follows from its contact with the liquid) at its maximum density; and let the remainder M-m be liquid. If the temperature T of the mass is given, the condition of the vaporous part, and at the same time that of the liquid part, is thereby determined. If m be also given and thereby the magnitudes of both parts known, then we know the condition of the whole mass. We will accordingly choose T and m as the independent variables, and will substitute m for x in equations (29), (30), (31) of the last chapter. Then these equations become

We may now denote the specific volume (i.e. the volume of a unit of weight) of the saturated vapour by s, and the specific volume of the liquid by σ. Both these magnitudes bear some relation to the temperature T and its corresponding pressure, and are therefore, like the pressure, functions of the temperature alone. If we further denote by v the total volume of the mass, we may then put

We will substitute for the difference (so), a simpler ex

pression, by putting

The quantity of heat which must be applied to the mass, if a unit of weight of the substance, at temperature T and under the corresponding pressure, is to pass from the liquid into the vaporous condition, and which may be shortly called the vaporizing heat, may be denoted by p; then we have

We will further introduce into the equations the specific heat of the substance in the liquid and vaporous condition. The specific heat here treated of is not however that at constant volume, nor yet that at constant pressure, but belongs to the case in which the pressure increases with the temperature in the same manner as the maximum expansive power of the saturated vapour. This increase of pressure has very little influence on the specific heat of the liquid, since liquids are but slightly compressible by such pressures as are herein considered. We shall hereafter explain how this influence may be calculated, in our researches on the different kinds of specific heat, and a single example will suffice here. For water at boiling-point the difference between the specific

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heat here considered and the specific heat at constant 1 pressure, is only of the latter, a difference which may be neglected. Accordingly, we may for the purposes of calculation take the specific heat of the liquid here considered as being equal to the specific heat at constant pressure, although their meaning is different. We will call this specific

heat C.

With vapour it is otherwise. The specific heat here considered refers, as shewn above, to that quantity of heat which saturated vapour requires to heat it through 1o, if it is at the same time so powerfully compressed that even at the higher temperature it again returns to the saturated condition. As this compression is very considerable, this kind of specific heat is very different from all which we have hitherto treated of. We shall call it the Specific Heat of Saturated Steam, and shall denote it by H.

Bringing in the two symbols C and H, we may now at once write down the quantity of heat which is necessary to give the increase of temperature dT to the quantity of vapour m, and the quantity of liquid M-m. The result will be as follows:

whence

mHdT+ (M−m) CdT,

Substituting in equations (1, 2, 3) the values given in equa

tions (7, 9, 10) we have

These are the fundamental equations of the Mechanical Theory of Heat as relates to the generation of vapour. Equation (11) is a deduction from the first fundamental principle, (12) from the second, and (13) from both together.

If it is desired to use the ordinary and not the mechanical measures for the quantities of heat, we need only divide all the members of the foregoing equations by the mechanical equivalent of heat. In this case we will denote the two specific heats and the heat of vaporization by new symbols, putting

As the foregoing equations (15), (16) and (17), of which however only two are independent, have thus been obtained by means of the Mechanical Theory of Heat, we may make use of them in order to determine more closely two magnitudes, of which one was previously quite unknown and the other only known imperfectly; viz. the magnitude h and the magnitude s contained in u.

If we first apply ourselves to the magnitude h, or the Specific Heat of Saturated Steam, it may be worth while in

the first place to give some account of the views formerly promulgated concerning this magnitude.

The magnitude h is of special importance in the theory of the steam engine, and in fact the first who published any distinct views upon it was the celebrated improver of the steam engine, James Watt. In his treatment of the subject he naturally started from those views which were based on the older theory of heat. To this class belongs especially the idea mentioned in Chapter I., viz. that the so-called total heat, i.e. the total quantity of heat taken in by a body during its passage from a given initial condition to its present condition, depends only on the present condition and not on the way in which the body has been brought into it; and that it accordingly may be expressed as a function of those variables on which the condition of the body depends. According to this view we must in our case, in which the condition of the body composed of liquid and vapour is determined by the quantities T and m, consider this quantity of heat Q as a function of T and m; accordingly we have the equation

If we here substitute for the two second differentials their values given in equations (9) and (10), we have

whence we have, to determine h, the equation,

This was in fact the equation which was formerly used to determine h, though not quite in the same form. To calculate h from this equation we must know the differential co

dr

dT'

efficient or the change of the vaporizing heat for a given change of temperature.