Page:Scientific Papers of Josiah Willard Gibbs.djvu/69

II.

A METHOD OF GEOMETRICAL REPRESENTATION OF THE THERMODYNAMIC PROPERTIES OF SUBSTANCES BY MEANS OF SURFACES.

[Transactions of the Connecticut Academy, II. pp. 382–404, Dec. 1873.]

The leading thermodynamic properties of a fluid are determined by the relations which exist between the volume, pressure, temperature, energy, and entropy of a given mass of the fluid in a state of thermodynamic equilibrium. The same is true of a solid in regard to those properties which it exhibits in processes in which the pressure is the same in every direction about any point of the solid. But all the relations existing between these five quantities for any substance (three independent relations) may be deduced from the single relation existing for that substance between the volume, energy, and entropy. This may be done by means of the general equation,

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

(1)[1]

that is, $p=-\left({\frac {d\epsilon }{dv}}\right)_{\eta },$

(2)

that is, $t=\left({\frac {d\epsilon }{d\eta }}\right)_{v},$

(3)

where $v,p,t,\epsilon ,$ and $\eta$ denote severally the volume, pressure, absolute temperature, energy, and entropy of the body considered. The subscript letter after the differential indicates the quantity which is supposed constant in the differentiation.

Representation of Volume, Entropy, Energy, Pressure, and Temperature.

Now the relation between the volume, entroy, and energy may be represented by a surface, most simply if the rectangular co-ordinates of the various points of the surface are made equal to the volume, entropy, and energy of the body in its various effects. It may be interesting to examine the properties of such a surface, which

1. For the demonstration of this equation, and in regard to the units used in the measurement of quantities, the reader is referred to page 2.