1. General Phase Equilibrium
Lecture 4
General Phase Equilibrium
Minimum Energy Principle
Degrees of freedom
Gibbs-Duhem equation
Equilibrium in multiphase systems
Gibbs phase rule
Problems

2. Minimum Energy Principle
Maximum entropy principle:
For isolated system the entropy is maximum in equilibrium
Minimum energy principle:
For a system kept at constant entropy and volume the energy is minimum in equilibrium
The two principles are equivalent

3. Materials Stability Minimum Energy
Taylor expansion of U (S,V) around equilibrium a c b

4. Materials Stability -2 Equivalent to Equivalent to Equivalent to
We have,

5. Materials Stability - 3

6. Materials Stability - 4
Thus,
Thermal stability
Mechanical stability

7. State function and equilibrium
NVE dS0
NVT dF0
NPT dG0
VT d(pV) 0

8. Degrees of freedom
Definition: Number of independently variable intensive properties of the system
Example: For a single phase system number of intensive variables describing the system C+2, where C is number of components. For example with N components there is n chemical potentials + pressure + temperature. However, there is one relationship, so-called Gibbs-Duhem equation that relates C=2 components with each other making only C+1 independent components. For example a single component gas has only 1+1 degrees of freedom (e.g., p,V)

9. Gibb-Duhem equation
From definition of G
But we also know that
Thus

10. Multiphase multicomponent system
The total energy of the system is the sum of the energy of each phase
i labels components, j labels phases

11. Equilibrium in multiphase multicomponent system
0th law of thermodynamics - each phase has the same T
dF = 0 at const T and leads to hydrostatic equilibrium
dG = 0 at const P and T gives distributive equilibrium - chemical potential of a given component is the same in every phase

12. Chemical equilibrium In equilibrium at const P and T
Example: 2H2(gas)+O2(gas) 2H2O(liquid)
With dT=dP=0
In general for reaction AA+ BB CC+ DD

13. Gibbs Phase Rule At each phase the Gibbs-Duhem equation holds
Therefore # of degrees of freedom, DOF = C+2-P, where P is number of phases

14. Problem 5.2 (modified) CO (gas) + H2O (gas) H2(gas) + CO2 (gas)
1. Why in general equilibrium composition depends on pressure
2. Is this the case for the reactions:
and
CO (gas) + H2O (gas) H2(gas) + CO2 (gas)
2H2(gas)+O2(gas) 2H2O(liquid)