112 BUTLER ART. D 



i.e., the system can undergo two independent variations, in 

 accordance with (96). Now if the proportions of the two com- 

 ponents are the same in the two phases, i.e., if 



m.\ mi" 

 nh' nii" 



the coefficient of dju in (105) is zero, so that 



{v'm" - v"mi')dv = Wni2" - ■n"m')dt, (106) 



i.e., the same relation between dp and dt holds, as for a single 

 component. For example, in the equilibrium between ammo- 

 nium chloride and its vapor, the latter may contain ammonia 

 and hydrogen chloride, formed by dissociation. These two 

 substances may be regarded as the independently variable 

 components of the system, but if no excess of either of them is 

 added the ratios of their amounts are the same in both phases. 

 Then (106) holds, so that the system behaves as if it had a 

 single component. 



When there are n independent components in the two phases, 

 then in the absence of any restriction on their proportions the 

 number of degrees of freedom is ^^ = n -f 2 — 2 = n. But 

 when the quantities of all components are proportional in the 

 two phases, the equality of the n — 1 ratios of m/, rth', . . . m„' 

 with the n — 1 ratios of m/', mz', . . . mn" gives n — 1 additional 

 conditions, so that the number of degrees of freedom is reduced 

 to one and there is a relation similar to (106) between the 

 variations of temperature and pressure. 



Again, in a system of two components in two phases, at 

 constant temperature, (105) becomes 



dp mi' m^" — mi" W 



T~ = ~' T, — -77 r • (107) 



dm V m2 — V m^ 



If the proportions of the two components are the same in the 



two phases, the numerator of the fraction on the right is zero, so 



that 



dp 



dm 



