Chapter 14 Part III- Equilibrium and Stability. A system with n components and m phases Initially in a non-equilibrium state (mass transfer and chemical.

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Chapter 14 Part III- Equilibrium and Stability

A system with n components and m phases Initially in a non-equilibrium state (mass transfer and chemical reactions may take place) T and P are uniform System is in thermal and mechanical equilibrium with surroundings What changes may happen to the system? What will be the final state of the system?

Changes to the system translate to: Heat exchange Expansion work By the second law, what happens to the entropy?

dS sys + dS surr > 0 dS sys - dQ/T > 0  dS sys > dQ/T dQ < T dS sys By the first law, dU sys =- PdV sys + T dS sys dU sys + PdV sys < T dS sys, or dU sys + PdV sys -T dS sys < 0

Valid for any closed system The inequality determines the direction of change between non-equilibrium states The equality holds for changes between equilibrium states (reversible)

dU sys + PdV sys -T dS sys < 0 Special cases: – At V and S constant (dU sys ) SV < 0 – At U and V constant (dS sys ) UV > 0

Process at constant T and P dU TP + d(PV) TP –d(TS) TP < 0 d(U+PV-TS) TP < 0 (dG) TP < 0  All irreversible processes at constant T and P tend to decrease the Gibbs free energy

Equilibrium criterion For a closed system at constant T and P, the Gibbs free energy is a minimum Given an expression for G, we find the set of composition values that minimize G

At equilibrium, differential changes may occur The system is not static !!! At constant T and P changes may happen but they do not change G. Therefore: (dG) TP = 0

 G mixing = G –  x i G i If the system is stable, G must decrease, therefore G <  x i G i, G –  x i G i < 0 For curve II, the system has a lower G by splitting into two phases than in a single phase (at compositions between x 1  and x 1  )

Stability criterion for a single phase binary system At constant T and P,  G and its first and second derivatives must be continuous functions of x 1, and the second derivative must everywhere be positive

Alternative stability criterion: Relation to G E Since At constant T and P

Other alternative stability criteria Alternative criteria, at constant T and P, valid for each of the components:

How the stability criteria affect VLE? How is the criterion for component 2?

For an ideal gas mixture, you can show that Then the stability criterion is dy 1 /dx 1 > 0 What does it mean for a y-x diagram?

For the liquid phase, at constant T and P Low pressure VLE, assume ideality of gas phase, you can show What can we say about the sign of dP/dx1?

Therefore, what is the sign of dP/dy1? What happens at an azeotrope?