equilibrium and stability. phase separation in ethanol blended gasoline 1. three-component system:...
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Equilibrium and Stability
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
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 I need to know what is the equilibrium state of a system?
Moving Toward Equilibrium State
n components
m phases
Assumption 1
• 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
• Heat transfer and/or expansion work with/on surroundings occurs reversibly (why?)
1. Are changes occurring in the system reversible or irreversible?
Notice: Changes will occur in the system, because it IS NOT at chemical/phase equilibrium
Moving Toward Equilibrium State
n components
m phases
Consequence 1
1. When does the inequality applies?
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
dSsurr = dQsurr/Tsurr = -dQ/T
From 2nd law dSuniverse ≥ 0 dSsurr + dS ≥ 0
universe
Consequence 2
dQ ≤ TdS
From 1st law dQ = dU + PdV
Consequence 3
dU + PdV – Tds ≤ 0
dU + PdV –TdS ≤ 0
Minimum Energy Maximum Entropy
Criterion for equilibrium
(dS)U,V ≥ 0
Rigid and Isentropic Isolated
Isothermal and Isobaric Rigid and Isothermal
(dU)S,V ≤ 0
(dG)T,P ≤ 0 (dF)T,V ≤ 0
Criterion for equilibrium
Isothermal and Isobaric
(dG)T,P ≤ 0
dU + PdV – TdS ≤ 0
(dU + d(PV) – d(TS) ≤ 0)T,P
d(kx) = kdx
d(x + y) = dx + dy
(d(U + PV – TS) ≤ 0)T,P
G = U + PV - TS
(d(G) ≤ 0)T,P
1. What state functions are more easily controlled in a chemical process?
Processes occur spontaneously in the direction that G decreases (at constant T and P)
At equilibrium, dG = 0 (at constant T and P)
Analogy with a mechanical systemEquilibrium
Position
Potential Energy
z
U = mg .( z )
x
U = mg .( x2)
0.0
Energy DerivativedU = mg .( x ) dx
At equilibriumdU = 0
Gibbs Free Energy of Mixing
Equilibrium Position
At equilibriumdG = 0
Gibbs Free EnergyG = G( xA)
DGmix = G – SxiGi
1. What is the difference between system I and system II?
Mixing requires: G < SxiGi
A
DGmixA > α (DGmix) α + β(DGmix) β
DGmixA < α (DGmix) α + β(DGmix) β System I
System II
To see video showing temperature-induced phase separation in E10, click here
Clear Liquid(one phase)
Clear Liquid(phase I)
Turbid Liquid(phase II)
SYSTEM: Ethanol-Gasoline-Water
In cold weather (winter) storage tank in car can be colder than storage tank in gas station
Shape of ΔGmix changes with temperature
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
Analytical Approach In terms of gi
0ln1)/(
lnln
1)/(
1
1
221
2
2211
2121
2
dx
d
xdx
RTGd
xxRT
G
xxdx
RTGd
E
E
E
Alternative criteria, at constant T and P, valid for each of the components: 0
0ˆ
1ln
1
1
1
1
11
1
dx
d
dx
fd
xdx
d
See derivation of this criterion posted in the web site
Implications for VLE equilibrium
d(ln u)/dv = (1/u) (du/dv)
Calculus note:
1. What is the equivalent criterium for component 2?
Reminder:
dx1 = -dx2
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
For low pressure VLE, an ideal gas phase, let us demonstrate that:
1
1
21
11
1
)(1
dx
dy
yy
xy
dx
dP
P
What does the above equation implies for the sign of dP/dx1?
Implications for VLE equilibrium
Gibbs-Duhem Equation
What does the above equation implies at the azeotropic point?
Implications for VLE equilibrium
11
1
1 /
/
dxdy
dxdP
dy
dP du/dv = (du/dw)/(dv/dw)
Calculus note:
What does the above equation implies for the sign of dP/dy1?
What does the above equation implies at the azeotropic point?
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 DGmix
A < α (DGmix) α + β(DGmix) β
For VLE equilibrium
sign same thehave )(,,
,0
1111
1
1
xydy
dP
dx
dP
dx
dy