Non-analyticity. The Critical Exponents

If Gm(T, p, x), the appropriate thermodynamic function for a binary mixture, is analytic, the deductions about the behaviour of the various thermodynamic properties are entirely analogous to those for the one-component fluid. The same conclusions arise from any general Taylor series expansion in which all the coefficients are non-zero except those two required to define the critical point [equations (6a, b)]. In particular, the coexistence curve (T vs. x at constant p) should be parabolic, the critical isotherm vs. x at constant T and p) should be cubic, and the molar heat capacity C, ,m should be everywhere finite. 

However, just as is the case for many other kinds of critical phenomena (e.g. the one-component fluid, magnetism, order-disorder transitions in solids, etc.) such predictions do not agree either with the results of careful experimental measurements or with simple theoretical models that can be treated nearly exactly. The coexistence curve is more nearly cubic than parabolic, the critical isotherm is of distinctly higher order than cubic, and the heat capacity Cp,x,m diverges at the critical point. 

At constant pressure, the difference in the mole fractions (x — x ) of the conjugate phases varies as (r° — T) as the critical solution temperature is approached. 

Along a line of constant temperature T = and p = Q.e. the critical isotherm-isobar), A - (and so also px — pi or p% — pi taken separately) approaches the critical composition as (x — x )  

Along the coexistence curve below the critical temperaturef relations similar to equations (12) apply. and its counterparts should diverge as — T v  

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