1. Equilibrium and Stability

2. Phase Separation in Ethanol Blended Gasoline
1. Three-component system: Ethanol, water, and gasoline
2. Up to three phases depending on concentration XetOH, XH2O, Xgas
3. Phase separation can be triggered by drop in temperature
4. Different levels of engine failure depending on phase fed

3. An Arbitrary Thermodynamic System
n components
m phases
Surroundings
System
Closed System:
dn = 0
1. What happens if the system is in equilibrium?
2. What happens if the system is not in equilibrium?
3. Why do I need to know what is the equilibrium state of a system?

4. Moving Toward Equilibrium State
Assumption 1
n components
m phases
T and P are uniform throughout the system
System is in thermal and mechanical equilibrium
= 0, and = 0
Assumption 2
System is in thermal and mechanical equilibrium with surroundings
Notice: Changes will occur in the system, because it IS NOT at chemical/phase equilibrium
Heat transfer and/or expansion work with/on surroundings occurs reversibly (why?)
1. Are changes occurring in the system reversible or irreversible?

5. Moving Toward Equilibrium State
Consequence 1
n components
m phases
dSsurr = dQsurr/Tsurr = -dQ/T
From 2nd law  dSuniverse ≥ 0 dSsurr + dS ≥ 0
Consequence 2
universe
dQ ≤ TdS
Notice: Since U, S, and V (and T and P) are state functions. Consequence 3 is true for ANY closed-system of uniform T and P
From 1st law  dQ = dU + PdV
Consequence 3
dU + PdV – Tds ≤ 0
1. When does the inequality apply?

6. dU + PdV –TdS ≤ 0
Minimum Energy
Maximum Entropy

7. Criterion for equilibrium
Rigid and Isentropic
Isolated
(dU)S,V ≤ 0
(dS)U,V ≥ 0
Isothermal and Isobaric
Rigid and Isothermal
(dG)T,P ≤ 0
(dF)T,V ≤ 0

8. Criterion for equilibrium
Isothermal and Isobaric
(dG)T,P ≤ 0
dU + PdV – TdS ≤ 0
d(kx) = kdx
d(x + y) = dx + dy
(dU + d(PV) – d(TS) ≤ 0)T,P
1. What state functions are more easily controlled in a chemical process?
(d(U + PV – TS) ≤ 0)T,P
G = U + PV - TS
Processes occur spontaneously in the direction that G decreases (at constant T and P)
(d(G) ≤ 0)T,P
At equilibrium, dG = 0 (at constant T and P)

9. Analogy with a mechanical system
Gibbs Free Energy of Mixing
Equilibrium Position z
Equilibrium Position
0.0 x
Potential Energy
U = mg .( z )
U = mg .( x2)
Energy Derivative
Gibbs Free Energy
dU = mg .( x ) dx
G = G( xA)
At equilibrium
At equilibrium
dG = 0
dU = 0

10. DGmixA < α (DGmix) α + β(DGmix) β
DGmix = G – SxiGi A
Mixing requires: G < SxiGi
1. What is the difference between system I and system II?
DGmixA < α (DGmix) α + β(DGmix) β
System I
DGmixA > α (DGmix) α + β(DGmix) β
System II

11. SYSTEM: Ethanol-Gasoline-Water
Clear Liquid
(one phase)
Clear Liquid
(phase I)
In cold weather (winter) storage tank in car can be colder than storage tank in gas station
Turbid Liquid
(phase II)
Shape of ΔGmix changes with temperature
To see video showing temperature-induced phase separation in E10, click here

12. Analytical approach  Stability in terms of GE
Not only does (ΔGmix)T,P have to be negative, but also:
(d2ΔGmix/dx12 > 0)T,P (why?)
Since T is constant, we can divide both sides by RT
(d2(ΔGmix/RT)/dx12 > 0)T,P
For a binary system  ΔGmix/RT = x1lnx1 + x2lnx2 + GE/RT
Constant T and P

13. Analytical Approach  In terms of gi
See derivation of this criterion posted in the web site
Alternative criteria, at constant T and P, valid for each of
the components:

14. Implications for VLE equilibrium
Calculus note:
d(ln u)/dv = (1/u) (du/dv)
1. What is the equivalent criterion for component 2?
Reminder:
dx1 = -dx2

15. Implications for VLE equilibrium
Since T is constant, we can divide both sides by RT
Constant T and P
Let us demonstrate that for an ideal gas:
What is the connection to variations in compositions?
dy1/dx1 > 0  dy1 > 0; dx1 > 0 or dy1 < 0; dx1 < 0

16. Implications for VLE equilibrium
Gibbs-Duhem Equation
For low pressure VLE, an ideal gas phase, let us demonstrate that:
What does the above equation imply for the sign of dP/dx1?
What does the above equation imply at the azeotropic point?

17. Implications for VLE equilibrium
Calculus note:
du/dv = (du/dw)/(dv/dw)
What does the above equation imply for the sign of dP/dy1?
What does the above equation imply at the azeotropic point?

18. Summary In an isolated system entropy is maximized at equilibrium
In an T, P controlled system Gibbs free energy is minimized at equilibrium
Two components will be mixed into a single phase if ΔGmix < 0, and DGmixA < α (DGmix) α + β(DGmix) β
For VLE equilibrium