CONDENSATION AND CRITICAL PHENOMENA. 271 



The theory assumes an equation of condition for a 

 mixture analogous to the equation for single substances. 

 Van der Waals uses the same equation for mixtures as he 



deduced for simple substances, zu's. [p H — J (z' — b) = RT 



(i) but a and b now depend on the composition of the 

 mixture x. Van der Waals finds 



a = a^ [\ — xf + 2 ^ij .r ( I — .r) + a^ x^ 

 and a similar equation for b ; a^ a^ b^ and b^ are the constants 

 for the two components ; a^^ and b^^ depend on the mutual 

 actions. Equation (i) is an equation between/* v T and x. 



Now we may be perfectly assured that this equation is 

 not correct. We saw how even for sino-le substances the 

 same equation fails utterly. But the general results of the 

 theory do not depend on the special form of the equation. 

 We only have to assume, as with single substances, the 

 existence of some equation between p v x and T as de- 

 scribinof the behaviour of the mixture under different 

 circumstances. This equation should yield an isothermal 

 with the double-wave shape at lower temperatures and no 

 unstable parts at higher temperatures. 



It has already been said that Maxw^ell-Clausius' criterion 

 is not adequate to find the border-curve because of the 

 different composition of the co-existing phases. This 

 criterion was equivalent to the condition that the same 

 number of molecules move from the liquid to the gas in the 

 opposite direction. But as there are two substances the latter 

 condition must now be applied to both. In thermody- 

 namical language this double condition may be combined 

 with the condition of the equality of pressure in the two 

 phases into one condition. Van der Waals uses the con- 

 dition in this form : in a given volume the free energy -^ 

 (i// = J - r)7, £ = energy, j? = entropy) is a minimum. One 



of the properties of j// is ^ = — /. 



In the case of one substance ^ is a function of v (at 

 constant temperature). Below the critical temperature the 

 ■^ X V curve has a double tangent, the slope of which de- 

 termines the vapour pressure, the two points of contact re- 



