Page:Scientific Papers of Josiah Willard Gibbs.djvu/38

These are subject to the relations expressed by the following differential equations:—

$dW=\alpha pdv,$

(a)

$d\epsilon =\beta dH-dW,$

(b)

$d\eta ={\frac {dH}{t}},$[1]

(c)

where $\alpha$ and $\beta$ are constants depending upon the units by which $v$, $p$, $W$ and $H$ are measured. We may suppose our units so chosen that $\alpha =1$ and $\beta =1$,^[2] and write our equations in the simpler form,

$d\epsilon =dH-dW,$

(1)

$dW=pdv,$

(2)

$dH=td\eta .$

(3)

Eliminating $dW$ and $dH$, we have

$d\epsilon =td\eta -pdv$.

(4)

The quantities $v$, $p$, $t$, $\epsilon$ and $\eta$ are determined when the state of the body is given, and it may be permitted to call them functions of the state of the body. The state of a body, in the sense in which the term is used in the thermodynamics of fluids, is capable of two independent variations, so that between the five quantities $v$, $p$, $t$, $\epsilon$ and $\eta$ there exist relations expressible by three finite equations, different in general for different substances, but always such as to be in harmony with the differential equation (4). This equation evidently signifies that if $\epsilon$ be expressed as a function of $v$ and $\eta$, the partial differential co-efficients of this function taken with respect to $v$ and to $\eta$ will be equal to $-p$ and to $t$ respectively.^[3]

1. Equation (a) may be derived from simple mechanical considerations. Equations (b) and (c) may be considered as defining the energy and entropy of any state of the body or more strictly as defining the differentials $d\epsilon$ and $d\eta$. That functions of the state of the body exist, the differentials of which satisfy these equations, may easily be deduced from the first and second laws of thermodynamics. The term entropy, it will be observed, is here used in accordance with the original suggestion of Clausius, and not in the sense in which it has been employed by Professor Tait and others after his suggestion. The same quantity has been called by Professor Rankine the Thermodynamic function. See Clausius, Mechanische Wärmetheorie, Abhnd. ix. § 14; or Pogg. Ann., Bd. cxxv. (1865), p. 390; and Rankine, Phil. Trans., vol. 144, p. 126.
2. For example, we may choose as the unit of volume, the cube of the unit of length,—as the unit of pressure the unit of force acting upon the square of the unit of length,—as the unit of work the unit of force acting through the unit of length,—and as the unit of heat the thermal equivalent of the unit of work. The units of length and of force would still be arbitrary as well as the unit of temperature.
3. An equation giving $\epsilon$ in terms of $\eta$ and $v$, or more generally any finite equation between $\epsilon$, $\eta$ and $v$ for a definite quantity of any fluid, may be considered as the fundamental thermodynamic equation of that fluid, as from it by aid of equations (2), (3) and (4) may be derived all the thermodynamic properties of the fluid (so far as reversible processes are concerned), viz.: the fundamental equation with equation (4) gives the three relations existing between $v$, $p$, $t$, $\epsilon$ and $\eta$, and these relations being known, equations (2) and (3) give the work $W$ and heat $H$ for any change of state of the fluid.