Subsections

• Tangents and Minimization
• Phase Diagrams
• Critical Point

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.

• One phase--one minimum
  
  

• Binodal--two equal minima
  
  

• Metastable point and stable point--two unequal minima (we changed $\mu_0$ and do not have a bitangent anymore!)
  
  

• Unstable point and stable point--minimum and maximum (even greater change in $\mu_0$)
  
  Spinodal divides metastable points from stable points!

Phase Diagrams

In $\mu$-T space we have a coexistence curve:

In N-T space we have a coexistence region

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.

2.

A₁ and A₂ are parts of same curve--ending point is possible! it is called critical point. Liquid-gas phase transition.