Excess thermodynamic functions in the region of a critical solution temperature

Excess Thermodynamic Functions in the Region of a Critical Solution Temperature. 

It is of interest to examine the behaviour of the excess thermodynamic functions near a critical solution temperature. We start from the equation 

In the case of an upper consolute temperature for which d%jdxl (18.84) is negative, this relation can be satisfied if even is zero. 

On the other hand for a lower consolute temperature for which d%jdxl is positive (18.85) we must have 

If this is an upper critical point, will be positive because of (18.84) and kg may or may not be equal to zero, although it must always satisfy 

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