Introduction to phase equilibrium
Chapter 10 (but also revision from Chapter 6) Equilibrium Absence of change Absence of a driving force for change
Example of driving forces Imbalance of mechanical forces => work (energy transfer) Temperature differences => heat transfer Differences in chemical potential => mass transfer Energies Internal energy, U Enthalpy H = U + PV
Gibbs free energy G = H TS Helmholtzfree energy A = U - TS Phase Diagram Pure Component
f e d c b a Describeprocess from (a) to (f) as volume is compressed at constant T. P-T for pure component P-V diagrams pure component Equilibrium condition for coexistence of two phases (pure component)
Review Section 6.4 At a phase transition, molar or specific values of extensive thermodynamic properties change abruptly. The exception is the molar Gibbs free energy, G, that for a pure species does not change at a phase transition Equilibrium condition for coexistence of two phases(pure component, closed system)
d(nG) = (nV) dP (nS) dT Pure liquid in equilibrium with its vapor, if a differential amount of liquid evaporates at constant T and P, then d(nG) = 0 n = constant => ndG =0 => dG =0 Gl = Gv Equality of the molar or specific Gibbs free energies (chemical potentials) of each phase Chemical potential in a mixture:
Single-phase, open system: Chemical potential of component i in the mixture Phase equilibrium: 2-phase mixture (n components)
Two phases, a and b andn components: Equilibrium conditions: mia = mib (for I = 1, 2, 3,.n) Ta = Tb Pa = Pb For a pure component ma = mb
For a pure component, fugacity is a function of T and P For a mixture of n components
mia = mibfor all i =1, 2, 3,n in a mixture: Fugacity is a function of composition, T and P Lets recall Raoults law for a binary
We need models for the fugacity in the vapor phase and in the liquid phase Raoults law Model the vapor phase as a mixture of ideal gases:
Model the liquid phase as an ideal solution VLE according to Raoults law: Today Applications of Raoults law
Qualitative description of phase diagrams Acetonitrile (1)/nitromethane (2)
Antoine equations for saturation pressures: Calculate P vs. x1 and P vs. y1 at 75 oC Diagram is at constant T
Bubble line 66.72 Dew line 0.75 Calculate the P-x-y diagram
Knowing T and x1, calculate P and y1 Bubble pressure calculations Knowing P and y, get T and x
Dew point calculation Diagram is at constant T
59.74 0.43 In this diagram, the pressure
Is constant 78oC 0.51 0.67 Calculate a T-x-y diagram
(1) (2) get the two saturation temperatures Then select a temperature from the range between T1sat and T2sat At the selected T, summing (1) and (2) solve for x1 Given P and x, get T and y Iterate to find T, then calculate y
(II) (III) Estimate P1sat/P2sat using a guess T Then calculate P2sat from (III) And then get T from (I) Calculate P1sat/P2satfrom (II) Then calculate P2sat And then get T from (I) Until convergence of T In this diagram, the pressure
is constant Dew points Bubble points 78oC 76.4 0.51 0.75 Knowing P and y, get T and x
Start from point c last slide (70 kPa and y1= 0.6 Iterate to find T, and then calculate x
(II) (III) Calculate P1sat/P2satfrom (II) Then calculate P1sat from (III) And then get T from (I) Until convergence of T Estimate P1sat/P2sat using a guess T Then calculate P1sat from (III) And then get T from (I) 79.6 0.44 F=2-p+N For a binary F=4-p For one phase: P, T, x (or y) Subcooled-liquid above the upper surface Superheated-vapor below the under surface L is a bubble point W is a dew point LV is a tie-line Line of critical points Each interior loop represents the PT
behavior of a mixture of fixed composition In a pure component, the bubble and dew lines coincide What happens at points A and B? Critical point of a mixture is the point where the nose of a loop is tangent to the envelope curve Tc and Pc are functions of composition, and do not necessarily coincide with the highest T and P At the left of C, reduction
of P leads to vaporization At F, reduction in P leads to condensation and then vaporization (retrograde condensation) Important in the operation of deep natural-gas wells At constant pressure, retrograde vaporization may occur Fraction of the overall system that is liquid Minimum and maximum of the more volatile species obtainable by distillation at this pressure (these are mixture CPs) azeotrope This is a mixture of very dissimilar components The P-x curve in (a) lies below
Raoults law; in this case there are stronger intermolecular attractions between unlike than between like molecular pairs This behavior may result in a minimum point as in (b), where x1=y1 Is called an azeotrope The P-x curve in (c) lies above Raoults law; in this case there are weaker than between like molecular pairs; it could end as L-L immiscibility This behavior may result in a maximum point as in (d), where x1=y1, it is also an azeotrope Usually distillation is carried
out at constant P Minimum-P azeotrope is a maximum-T (maximum boiling) Point (case b) Maximum-P azeotrope is a minimum-T (minimum boiling) Point(case d) Ki = yi/xi Ki = Pisat/P Read Examples 10.4, 10.5, 10.6 Limitations of Raoults law
When a component critical temperature is < T, the saturation pressure is not defined. Example: air + liquid water; what is in the vapor phase? And in the liquid? Calculate the mole fraction of air in water at 25oC and 1 atm Tc air