HETEROGENEOUS EQUILIBRIUM 279 



K20-Si02, both triangles become zero simultaneously, and the 

 equation becomes indeterminate; the curve is at its end point 

 at the quadruple point in the binary system. 



It is evident that when the phases have the composition 

 indicated above, no maximum is possible in the p-t curve of 

 the univariant equilibrium. However, if the vapor phase, in- 

 stead of being pure H2O, contained a small amount of Si02, the 

 curve would have a horizontal tangent before the phases, L, 

 K2O -8102 41120, and K20-Si02 fell on a straight line, as can 

 readily be seen from the equation of the curve. 



29. Equilibrium, KiO-^SiOi + K^O-J^SiOi-H^O + Solution 

 + Vapor. In the discussion of binary systems, it was seen 

 that when a volatile component is considered, the maximum 

 temperature is not at the composition of a compound, as in 

 condensed systems, but is displaced in the direction of the 

 more volatile component. A similar condition is found in the 

 general case; an example in a ternary system is found along 

 the curve, K2O -28102 + K2O - 48102 - H2O + L + V (curve 46), 

 which goes from Q4 to Qsa. The equation of this curve is 



dp Am Ani 



dt , , . , Aiiv , , . A.\\.o . 



{v^ - v^) + -r- {v' - v'-) - -r~ (^ " " ) 

 A\2i Am 



in which the accents (') and (") refer to the phases, K2O- 28102 

 and K2O- 48102 -H2O, respectively. The condition for a 

 temperature maximum is that the denominator of this expres- 

 sion shall approach zero as a limit; dp/dt becomes infinite. 

 Since the volume difference between vapor and liquid is far 

 greater than that between solid and liquid, the denominator 

 will approach zero as a limit only when the coefficients of the 

 last two volume differences become very large, hence when 

 the area of the triangle, K2O - 2Si02-K20 • 48102 • H2O-L, 

 becomes very small. This point will be reached slightly before 

 the liquid phase lies on the line, K2O - 28i02-K20 - 48102 • H2O, 

 hence the point of maximum temperature has been displaced 

 sUghtly in the direction of the volatile component. 



