CHAPTER VIII.

ON HOMOGENEOUS BODIES.

§ 1. Changes of Condition without Change in the Condition of Aggregation.

We will now return to the general equations of Chapter V. and will apply them to cases, in which a body undergoes changes which do not extend so far as to alter its condition of aggregation, but in which all parts of the body are always in the same condition. We will suppose these changes to be produced by changes in the temperature and in the external pressure. In consequence of these, changes take place in the arrangement of the molecules of the body, which are indicated by changes in form and volume.

With regard to the external force, the simplest case is that in which an uniform normal pressure alone acts on the body; in this case no account need be taken of changes in the body's form, in determining the external work, but only of its alteration in volume. Here we may take the condition of the body as known, if of the three magnitudes, temperature, pressure and volume, which we will denote as before by T, p and v, any two are given. According as we choose for this purpose v and T, or p and T, or v and p, so we obtain one of the three systems of equations, which in Chapter V. are numbered (25), (26) and (27): these equations we will now use to determine the different specific heats and other quantities, related to changes in temperature, pressure, and volume."

§ 2. Improved Denotation for the Differential Coeffi

cients.

If the above-named equations of Chapter V. are referred to a unit of weight of the substance, the differential coeffidQ

cient will denote in equations (25) the specific heat at dT

constant volume, and in equations (26) the specific heat at con

dQ

stant pressure. Similarly has different values in (25)

dQ dp

dv

and (27) and has different values in (26) and (27). Such indeterminate cases always occur where the nature of the question occasions the magnitudes chosen as independent variables to be sometimes interchanged. If we have chosen any two magnitudes as independent variables, it follows that in differentiating according to one we must take the other as constant. But if, whilst keeping the first of these as one independent variable throughout, we then choose for the other different magnitudes in succession, we naturally arrive at a corresponding number of different significations for the differential coefficients taken according to the first variable.

This fact induced the author, in his paper "On various convenient forms of the fundamental equations of the Mechanical Theory of Heat,"* to propose a system of denotation which so far as he knows had not been in use before. This was to subjoin to the differential coefficient as an index the magnitude which was taken as constant in differentiating. For this purpose the differential coefficient was inclosed in brackets and the index written close to it, a line being drawn above the latter, to distinguish it from other indices, which might appear at the same place. The two differential coefficients named above, which represent the specific heat at constant volume and at constant pressure, would thus be (dQ This method was

written respectively (2) and

dQ dT

soon adopted by various writers, but the line was generally

*

Report of the Naturalists' Society of Zurich, 1865, and Pogg. Ann., Vol.

CXXV. p. 353.

left out for the sake of convenience. More recently* the author introduced a simpler form of writing, which yet retained the essential advantage of the method. This consisted in placing the index next to d, the sign of differentiation. The brackets were thus rendered needless and also the horizontal line, because no other index is in general placed in this position. The two above-named differential coeffid.Q d, Q and ; and this method dT

cients would thus be written

will be adopted in what follows.

dT

§ 3. Relations between the Differential Coefficients of Pressure, Volume, and Temperature.

If the condition of the body is determined by any two of the magnitudes, Temperature, Volume, and Pressure, we may consider each of these as a function of the two others, and thus form the following six differential coefficients :

dp dp dv dv dT d.T

In these the suffixes, which shew which magnitude is to be taken as constant, may be omitted, provided we agree once for all that in any differential coefficient that one of the three magnitudes, T, p, v, which does not appear, is to be considered as constant for that occasion. We shall however retain them for the sake of clearness, and because we shall meet with other differential coefficients between the same magnitudes, for which the constant magnitude is not the same as here.

The investigations to be made by help of these six differential coefficients will be facilitated, if the relations which exist between them are laid down beforehand. In the first place it is clear that amongst the six there are three pairs which are the reciprocals of each other. If we take v as constant, T and p will then be so connected that each may be treated as a simple function of the other. The same holds with T and v where p is constant, and with v and p when T

ย

* "On the principle of the Mean Ergal and its application to the molecular motions of Gases." Proceedings of the Niederrhein. Ges. für Naturund Heilkunde, 1874, p. 183.

is constant. Hence we may put

1 d.p 1 dov 1 d,v

d.T

dp

=

dT

#

=

dTdT' dpdp:

.(1).

To examine further the relation between these three pairs, we will by way of example treat p as a function of T and v. Then the complete differential equation for p is

If p is constant, we must put in this equation,

By means of this equation combined with equations (1), we may express each of the six differential coefficients by the product or the quotient of two other differential coefficients.

§ 4. Complete Differential Equations for Q.

We will now return to the consideration of the heat taken in and given out by the body. If we denote the specific heat at constant volume by C,, and at constant pressure by C,, and take the weight of the body as unity, we have

We have also the equations (25) and (26) of Chapter V., which with our present notation will be written as follows:

From these two we easily obtain a third differential equation for Q, which relates to v and p as independent variables. For multiplying the first equation by C, and the second by C,, subtracting, and dividing the result by C, C, we have

C-C

dᎢ

' dᎢ

.(5).

These three equations correspond exactly to those obtained in Chapter II. for perfect gases, except that the latter are simplified by applying the law of Mariotte and Gay-Lussac. The equation expressing this law is

Substituting these values in the above equations, and in the

These equations are the same as (11), (15) and (16) of Chapter II.

The equations (3), (4) and (5) are not immediately integrable, as has been already shewn with respect to the special equations holding for gases. For equations (3) and (4) this