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Subsections


Geometric Interpretation and Phase Diagrams

Tangents and Minimization

Suppose we have a system at constant $\mu_0$. We minimize $A-\mu_0
N$.

This means we make a line $A-\mu_0
N$ and the curve A(N). We want to minimize the distance between the line and curve $\Rightarrow$ we make a tangent. If the curve is above tangent, the point is locally stable, otherwise--unstable.

Phase Diagrams

In $\mu$-T space we have a coexistence curve:
\begin{figure}
\psfrag{mu}{$\mu$}
 \psfrag{T}{$T$}
 \psfrag{Phase I}{Phase I}
 \...
 ...\psfrag{Coexistence Curve}{Coexistence Curve}
 \includegraphics{muT}\end{figure}

In N-T space we have a coexistence region
\begin{figure}
\psfrag{N}{$N$}
 \psfrag{T}{$T$}
 \psfrag{Phase I}{Phase I}
 \psf...
 ...able}{Metastable}
 \psfrag{Unstable}{Unstable}
 \includegraphics{NT}\end{figure}

Why the difference? Because $\mu$ is intensive, and N is extensive variable!

Critical Point

Is there an ending point for coexistence curve?

Two possibilities:

1.
Free energies of two phases A1 and A2 are different curves--ending point! Liquid-solid phase transition.
\begin{figure}
\psfrag{A}{$A$}
 \psfrag{N}{$N$}
 \includegraphics{2curves}
 \end{figure}

2.
A1 and A2 are parts of same curve--ending point is possible! it is called critical point. Liquid-gas phase transition.
\begin{figure}
\psfrag{A}{$A$}
 \psfrag{N}{$N$}
 \includegraphics{1curve}
 \end{figure}

next up previous
Next: Gibbs Phase Rule Up: Phase Equilibria Previous: Free Energy Minima and

© 1997 Boris Veytsman and Michael Kotelyanskii
Thu Oct 2 21:02:12 EDT 1997