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Suppose we have a system at constant . We minimize .

This means we make a line and the curve *A*(*N*). We want to
minimize the distance between the line and curve we make
a *tangent*. If the curve is *above* tangent, the point is
locally stable, otherwise--unstable.

- One phase--one minimum

- Binodal--two equal minima

- Metastable point and stable point--two
*unequal*minima (we changed and do not have a bitangent anymore!)

- Unstable point and stable point--minimum and maximum (even
greater change in )

Spinodal divides metastable points from stable points!

In -*T* space we have a *coexistence curve*:

In *N*-*T* space we have a *coexistence region*

Why the difference? Because is intensive, and *N* is extensive
variable!

Is there an ending point for coexistence curve?

Two possibilities:

- 1.
- Free energies of two phases
*A*and_{1}*A*are_{2}*different*curves--ending point! Liquid-solid phase transition.

- 2.
*A*and_{1}*A*are parts of_{2}*same*curve--ending point is possible! it is called*critical point.*Liquid-gas phase transition.

© 1997

Thu Oct 2 21:02:12 EDT 1997