150 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



the condition is not satisfied, if equilibrium subsists, it will be at least 

 practically unstable. 



Hence, the phase at any point of a fluid mass, which is in stable 

 equilibrium under the influence of gravity (whether this force is due 

 to external bodies or to the mass itself), and which has throughout the 

 same independently variable components, is completely determined by 

 the phase at any other point and the difference of the values of the 

 gravitational potential for the two points. 



Fundamental Equations of Ideal Gases and Gas-Mixtures. 



For a constant quantity of a perfect or ideal gas, the product of the 

 volume and pressure is proportional to the temperature, and the 

 variations of energy are proportional to the variations of temperature. 

 For a unit of such a gas we may write 



pv = at, 

 de = c dt, 

 a and c denoting constants. By integration, we obtain the equation 



in which E also denotes a constant. If by these equations we elimi- 

 nate t and p from (11) we obtain 



e E 7 a e E 7 



de - drt --- dv, 



C V G 



4 



de dv 



or c - ^,=aw a . 



e E v 



The integral of this equation may be written in the form 



clog - = q alogv H, 



where H denotes a fourth constant. We may regard E as denoting the 

 energy of a unit of the gas for t = ; H its entropy for t = 1 and v = 1 ; 

 a its pressure in the latter state, or its volume for t = 1 and p = 1 ; 

 c its specific heat at constant volume. We may extend the application 

 of the equation to any quantity of the gas, without altering the values 



e r\ v 



1U, 11 Wt5 HUUStltUUtJ 



This will give 



of the constants, if we substitute , , for e, n, v, respectively. 



m m m 



, eEm r\ . , m / KK \ 



clog = - ZT+alog . (25o) 



cm m * v 



This is a fundamental equation (see pages 85-89) for an ideal gas of 

 invariable composition. It will be observed that if we do not have 

 to consider the properties of the matter which forms the gas as 



