Ch 14: Phase Equilibria

1
Ch 14 Phase Equilibria

• I. Condensed Phase ? Vapor
• II. TR Eqns Vi. and Vj.
• III. More Cycles
• IV. Surface Tension

2
I. Condensed Ph. ? Vapor

• At constant T and p, the Extremum Principle
  states that equilibrium is associated with ?G 0
  
• ? µc µv Recall Example 7.5
• If gas is ideal, µv kT ln (p/pint0) Eqn 11.50

3
Use Lattice Model for Condensed Phase

• Treat liquids and solids the same (ie ignore long
  range forces in solids)
• ?Strans 0 (condensed phase atoms held in
  place)
• ?F ?U f(trans only)
• Let attractive interaction energy wAA which is
  negative and independent of T.

4
Use Lattice Model for Condensed Phase

• Assume N atoms each with z nearest neighbors
  (n.n.), ??F ?U Nz wAA/2 Eqn 14.6
• µc (?F/?N)T,V z wAA/2 Eqn 10.41
• µc µv ? p pint0 exp (z wAA/2kT)
• Creating cavities or holes in a cond. ph.
  (?Uremove), closing the hole (?Uclose) and
  opening the hole(?Uopen - ?Uclose).

5
II. Phase Equilibrium Eqns (TR)

• Clapeyron Eqn general phase equil eqn
• At constant T and p, dG -TdS V dp is the
  indicator for equilibrium.
• Since µ partial molar G, µ can be used.
• dµ -sdT vdp
• Consider liquid ? vapor or µl µv
• dp/dT ?s/?v ?h/T ?v Eqn Vi (TR)
• Applies to s ? l v ? l s ? v, s1 ? s2, etc

6
Clausius-Clapeyron Eqn

• Applies to s ? g and l ? g.
• Assume ideal gas, ?v vg, ?h ? f(p,T)
• Then Clapeyron Eqn becomes CC Eqn
• d ln p ?h/RT2 dT
• ln (p2/p1) - ?h/R1/T2 1/T1 Eqn 14.23

7
Clausius-Clapeyron Eqn

• ln (p2/p1) - ?h/R1/T2 1/T1 Eqn 14.23
• Measure p vs T to find ?h/R -slope or ?h for
  sublimation and vaporization. Fig 14.8, Table
  14.1
• ?hvap - z wAA/2 Eqn 14.24
• Prob 3, 7, 8

8
III. Refrigerators and Heat Pumps

• Working fluid operates in a cycle
• Take heat from cold reservoir (qc at Tc,
  refrigerator or outside) and dumps it into high
  temperature (qh at Th, room or house) sink.
• Note cycle in Fig 14.9 showing H vs p
• Determine coefficient of performance c
  gain/work

9
IV. Surface Tension (?)

• Surface interface between two phases (e.g.
  liquid and vapor).
• Surface tension free energy cost to increase
  surface area ?
• Consider lattice model again with N molecules
  total including n on surface with (z-1) n.n. and
  (N-n) in bulk with z n.n.
• Total surface area A na

10
Surface Tension (?)

• U wAA/2 Nz-n Eqn 14.25
• ? (?F/?A)T,V,N (?U/?N)T,V,N - wAA/2a
• ? increases as wAA increases (becomes more
  negative)
• ? increases as a decreases (molecular area
  decreases)
• ? has units of dyn/cm force/length
  erg/cm2 Table 14.2

11
Surface Tension (?)

• U wAA/2 Nz-n Eqn 14.25
• ? (?F/?A)T,V,N (?U/?N)T,V,N - wAA/2a
• Eqn 14.24 14.28 ? ? ?hvap /za Fig 14.12
• Prob 2,4