# Page:Popular Science Monthly Volume 74.djvu/487

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JOSIAH WILLARD GIBBS

and great capacity for the rigors of formal logic, and the rapid movement of his mind as he clears away the underbrush and covers the vast area of his new territory is wearying and even confusing to the reader. The intention of the present resume is frankly journalistic, aiming only to emphasize such points in the Gibbsian theory as have been thrown into strongest relief by their relation to recent science. These are:

The General Equation of Thermodynamics.—In applying the laws of dynamics to thermal phenomena, Clausius had shown that if we differentiate with respect to the volume of a body, we obtain its pressure with reversed sign; if we differentiate with respect to its entropy we obtain its temperature on the thermodynamic scale; the energy of the body can then be expressed as a function of its volume and entropy, the differential coefficients with respect to the latter being the pressure (with negative sign) and the temperature. Gibbs has extended these principles to the formulation of a fundamental equation of thermodynamics, in which the new departure is taken of introducing the masses of the chemical components of a system as variables, the differential coefficients in this case being certain new conceptions which he terms the "potentials" of the substances considered. From this equation most of the principles and formulae of thermodynamics can be deduced. It lies at the basis of the new aggregate of sciences called "energetics"[1] as well as of mathematical chemistry, in which all spontaneous changes of substance or state are regarded as more or less direct consequences of the second law. The equations of Clausius and Gibbs, although exceedingly general and difficult of application to chemistry, are exact, representing the physical facts.[2]

The Chemical Potentials.—In the fundamental equation of Gibbs we distinguish two classes of variables, of which the volume, entropy and masses of the component substances are looked upon as magnitudes or capacities, while the temperature, the pressure and the potentials are to be thought of as qualitative, being non-measurable, nonadditive physical intensities of the system considered. Thus the pres-

If ${\displaystyle \epsilon ,t,\eta ,p}$ and ${\displaystyle v}$ represent the energy, temperature, entropy, pressure and volume of a homogeneous substance respectively, the equation of Clausius may be written ${\displaystyle d\epsilon =td\eta -pdf}$. It is applicable to all one-component systems, such as steam in a boiler. The equation of Gibbs, which is applicable to any chemical system whatever, is written

${\displaystyle d\epsilon =td\eta -pdv+\mu _{1}dm_{1}+\mu _{2}dm_{2}\ldots +\mu _{n}dm_{n},}$

where ${\displaystyle \mu _{1},\mu _{2}\ldots }$ denote the chemical potentials, and ${\displaystyle m_{1},m_{2}}$ the masses of the chemical components of the system.

1. Thoughout this paper, "energetics," thermodynamics and physical chemistry are regarded as practically identical in scope, in the original sense in which Gibbs referred to all material systems as "actually thermodynamic," or Ostwald to "das glänzendste Gebiet der heutigen Physik und Chemie, die reine Thermodynamik, oder da dieser Name viel zu eng ist, die reine Energetik."
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