24 GRAPHICAL METHODS IN THE 



and in part vapor. The properties of such a mixture are very simply 

 and clearly exhibited in the volume-entropy diagram. 



Let the temperature and the pressure of the mixture, which are 

 independent of the proportions of vapor, solid and liquid, be denoted 



by if and p'. Also let V, L and S (fig. 9) 

 be points of the diagram which indicate 

 v the volume and entropy of the body in 

 three perfectly defined states, viz : that of 

 a vapor of temperature if and pressure p\ 

 that of a liquid of the same temperature 

 and pressure, and that of a solid of the 

 same temperature and pressure. And let 

 v V) i\ v , V L , rj L , v s , ri s denote the volume and 





Fi 9 entropy of these states. The position of 



the point which represents the body, when 

 part is vapor, part liquid, and part solid, these parts being as /*, i/, 

 and 1 fji i/, is determined by the equations 



V = fJLV v + W L + (1 - JUL - V )V a , 



where v and rj are the volume and entropy of the mixture. The 

 truth of the first equation is evident. The second may be written 



f-Hf 



or multiplying by if, 



The first member of this equation denotes the heat necessary to 'bring 

 the body from the state S to the state of the mixture in question 

 under the constant temperature if, while the terms of the second 

 member denote separately the heat necessary to vaporize the part ju, 

 and to liquefy the part v of the body. 



The values of v and r\ are such as would give the center of gravity 

 of masses //, v and 1 /z v placed at the points V, L and S.* Hence 

 the part of the diagram which represents a mixture of vapor, liquid 

 and solid, is the triangle VLS. The pressure and temperature are 

 constant for this triangle, i.e., an isopiestic and also an isothermal 

 here expand to cover a space. The isodynamics are straight and equi- 



distant for equal differences of energy. For -7- = p' and -^~ = t', 

 both of which are constant throughout the triangle. 



* These points will not be in the same straight line unless 



t' (nv - rjs) : t'tiL - ifor) : : i>r - v s : V L - v s , 



a condition very unlikely to be fulfilled by any substance. The first and second terms 

 of this proportion denote the heat of vaporization (from the solid state) and that of 

 liquefaction. 



