Sidebilder
PDF
ePub

and which leads to a very simple determination of the external work, viz. the case where the only external force acting on the body, or at least the only force which needs to be referred to in the determination of the work, is a pressure acting on the exterior surface of the body; and in which this pressure (as is always the case with liquid and gaseous bodies, provided no other forces are acting, and which may be the case even with solid bodies) is the same at all points of the surface, and everywhere normal to it. In this case there is no need, in order to determine the external work, that we should consider the body's alterations in form and its expansion in particular directions, but only its total alteration in volume.

[ocr errors]

As an illustrative case, let us take a cylinder, as shewn in Fig. 1, closed by an easily moving piston P, and containing some expansible substance, e.g. a gas, under a pressure per unit-area represented by p. The section of the cylinder, or the area of the piston, we may call a. Then the total pressure which acts on the piston, and which must be overcome in raising it, is pa. Now if the piston stands originally at a height h above the bottom of the cylinder, and is then lifted through an indefinitely small distance dh, the external work performed in the lifting will be expressed by the equation

dW= padh.

Fig. 1.

But if v be the volume of the gas we have v=ah, and therefore dv=adh; whence the above equation becomes

[blocks in formation]

This same simple form is assumed by the differential of the

external work for any form of the body, and any kind of expansion whatever, as may be easily shewn as follows. Let the full line in Fig. 2 represent the surface of the body in its original condition, and the dotted line its surface after an indefi

Fig. 2.

nitely small change of form and volume. Let us consider any element do of the original surface at the point A. Let a normal drawn to this element of surface cut the second surface at a distance du from the first, where du is taken as positive if the position of the second surface is outside the space contained within the original surface, and negative if it is inside. Now let us suppose an indefinite number of such normals to be drawn through every point in the perimeter of the surface-element do to the second surface; there will then be marked out an indefinitely small prismatic space, which has do as its base, and du as its height, and whose volume is therefore expressed by dadu. This indefinitely small volume forms the part of the increase of volume of the body corresponding to the element of surface do. If then we integrate the expression do du all over the surface of the body, we shall obtain the whole increase in volume, dv, of the body, and if we agree to express integration over the surface by an integral sign with suffix w, we may write

dv=

= [_dudw

........ (7).

Now if, as before, we denote the pressure per unit of surface by p, the pressure on the element do will be pdw. Therefore the part of the external work, which corresponds to the element do, and is described by saying that the element under the action of the external force pdw is pushed outwards at right angles through the distance du, will be expressed by the product pdw du. Integrating this over the whole surface, we obtain for the total external work,

aw = [ pdudo.

As p is equal over the whole surface, the equation may be written:

[blocks in formation]

which is the same as equation (6) given above.

Adopting this equation, we may give to equation (III), for the case in which the only external force is a uniform pressure normal to the surface, the following form:

[blocks in formation]

This last equation, which forms the mathematical expression most in use for the first main principle of the Mechanical Theory of Heat, we will in the next place apply to a class of bodies, which are distinguished for the simplicity of their laws, and for which the equation takes accordingly a peculiarly simple form, so that the required calculations can be easily performed.

CHAPTER II.

ON PERFECT GASES.

§ 1. The Gaseous condition of bodies.

Among the laws which characterize bodies in the gaseous condition the foremost place must be given to those of Mariotte and Gay Lussac, which may be expressed together in a single equation as follows. Given a unit-weight of a gas, which at freezing temperature, and under any standard pressure p. (e.g. that of the atmosphere) has the volume v; then if p and v be its pressure and volume at any temperature t (in Centigrade measure) the following equation will hold:

[blocks in formation]

wherein the quantity a, which is usually termed the coefficient of expansion, although it really relates to the change of pressure as well as the change of volume, has one and the same value for every kind of gas.

Regnault has indeed recently proved by careful experiment that these laws are not strictly accurate; but the deviations are for permanent gases very small, and become of importance only for gases which are capable of condensation. It seems to follow that the laws are the more nearly exact, the further a gas is removed, as to pressure and temperature, from its point of condensation. Since for permanent gases under ordinary conditions the exactness of the law is already so great, that for most purposes of research it may be taken as perfect, we may imagine for every gas an ultimate condition, in which the exactness is really perfect; and in what follows we will assume this ideal condition to be actually

reached, calling for brevity's sake all gases, in which this is assumed to hold, Perfect Gases.

As however the quantity a, according to Regnault's determinations, is not absolutely the same for all the gases which have been examined, and has also somewhat different values for one and the same gas under different conditions, the question arises, what value we are to assign to a in the case of perfect gases, in which such differences can no longer appear. Here we must refer to the values of a which have been found to be correct for various permanent gases. By experiments made on the system of increasing the pressure while keeping the volume constant, Regnault found the following numbers to be correct for various permanent gases:

[blocks in formation]

The differences here are so small, that it is of little importance what choice we make; but as it was with atmospheric air that Regnault made the greatest number of experiments, and as Magnus was led in his researches to a precisely similar result, it appears most fitting to select the number 0.003665.

Regnault, however, by experiments made on the other system of keeping the pressure constant and increasing the volume, has obtained a somewhat different value for a in the case of atmospheric air, viz. 0003670. He has further observed that rarefied air gives a somewhat smaller, and compressed air a somewhat larger, coefficient of expansion than air of ordinary density. This latter circumstance has led some physicists to the conclusion that, as rarefied air is nearer to the perfect gaseous condition than air of ordinary density, we ought to assume for perfect gases a smaller value than 0003665. Against this it may be urged, that Regnault observed no such dependence of the coefficient of expansion on the density in the case of hydrogen, but after increasing the density threefold obtained exactly the same value as before; and that he also found that hydrogen, in its deviation from the laws of Mariotte and Gay Lussac, acts altogether differently, and for the most part in exactly the opposite

« ForrigeFortsett »