# Critical Point

Entry by Andrew Capulli

**Definition: Critical Point** **( for mixtures)**

The Critical Point indicates the Chi value *X* (Chi is an energy of interaction parameter) that separates the situation where mixtures are always stable and the situations where mixtures are unstable or metastable where the resultant mixtures then separate into components (eventually if metastable and instantly is unstable). Take a mixture of two liquids for example A and B: the mixture is stable when the two liquids are completely miscible in one another. The mixture is metastable when the fluids are 'temporarily miscible' but eventually separate into their separate states (actually, the 'separate states' formed when the mixture is metastable are one state of highly concentrated A with a small amount of B and a state of highly concentrated B with a small amount of A). The mixture is unstable when the liquids do not mix at all. The critical point represents the point on the phase diagram (see [C] in the figure below) where the mixture can be stable, metastable, or unstable. Physically, the mixture would never really be at the critical point for any observable amount of time; the mixture will follow one of the 'solutions' (stable, metastable, or unstable) once the critical point is reached.

The "liquid-vapor critical point" (thermodynamic critical point) is often discussed in terms of phase diagrams and a brief discussion of this is found at: http://en.wikipedia.org/wiki/Critical_point_%28thermodynamics%29.

**Example Discussion of the Critical Point**

*For AP225 students, this discussion is based on the diagrams used in the course notes. Here we are looking at the Chi Parameter as a function of temperature T. We can likewise look at it as a function of pressure for example*

The critical point is where the spinodal line and the coexist curve are equal ( as defined in the Jones text used in Soft Matter AP225). The critical point is where *either a stable or unstable solution can occur* (*X*=2 in figures below). At the critical point the free energy of the system is a straight line with no minima; at any *X* less than 2, no contribution is given to the production of the diagram showing *X* versus composition. Given the curves in [A], the diagram *X* versus composition ([B]) can be created. Points on the metastable portions of the curves (the concave down portions) construct the inner parabola seen in [B] while points in the stable regions of the curves (the concave up portions)construct the outer parabola in [B]. To obtain [C] (the phase diagram as we know it!) it is assumed that *X* goes as 1/temperature so [C] is [B] inverted; essentially *X* is increased with temperature decrease and decreased with temperature increase. So, **the critical point represents the 'situation' (concentration and temperature) where mixtures are stable and where they are not stable (ie metastable or completely unstable which will then separate into phases).** As seen in this phase diagram, and what is somewhat intuitive, this system of arbitrary components is miscible at any concentration ratio given a high enough temperature (the critical point or critical temperature!).

*Curves in [A] 'summarized' from the plot below. (AP225 Class Notes)*

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