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 do I need to know what is the equilibrium state of a system?

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?

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?

dU + PdV –TdS ≤ 0 Minimum Energy Maximum Entropy

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

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)

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

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

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

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 See derivation of this criterion posted in the web site Alternative criteria, at constant T and P, valid for each of the components:

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

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

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?

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?

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