Ch 14: Phase Equilibria

I. Condensed Phase ↔ VaporII. TR: Eqns Vi. and Vj.

III. More CyclesIV. Surface Tension

I. Condensed Ph. ↔ Vapor

• At constant T and p, the Extremum Principle states that equilibrium is associated with ΔG = 0

μc = μv Recall Example 7.5

• If gas is ideal, μv = kT ℓn (p/pint0) Eqn 11.50

Use Lattice Model for Condensed Phase

• Treat liquids and solids the same (ie ignore long range forces in solids)

• ΔStrans = 0 (condensed phase atoms held “in place”)

• ΔF = ΔU = f(trans only)• Let attractive interaction energy = wAA

which is negative and independent of T.

Use Lattice Model for Condensed Phase

• Assume N atoms each with z nearest neighbors (n.n.), ΔF = ΔU = 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. (ΔUremove), closing the hole (ΔUclose) and opening the hole(ΔUopen = - ΔUclose).

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 = Δs/Δv = Δh/T Δv Eqn Vi (TR)• Applies to s ↔ ℓ; v ↔ ℓ; s ↔ v, s1 ↔ s2, etc

Clausius-Clapeyron Eqn

• Applies to s ↔ g and ℓ ↔ g.• Assume ideal gas, Δv = vg, Δh ≠ f(p,T)• Then Clapeyron Eqn becomes CC Eqn• d ℓn p = Δh/RT2 dT • ℓn (p2/p1) = [- Δh/R][1/T2 – 1/T1] Eqn 14.23

Clausius-Clapeyron Eqn

• ℓn (p2/p1) = [- Δh/R][1/T2 – 1/T1] Eqn 14.23• Measure p vs T to find Δh/R = -slope or Δh

for sublimation and vaporization. Fig 14.8, Table 14.1

• Δhvap = - z wAA/2 Eqn 14.24• Prob 3, 7, 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

IV. Surface Tension (γ)

• Surface = interface between two phases (e.g. liquid and vapor).

• Surface tension = free energy cost to increase surface area = γ

• 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

Surface Tension (γ)

• U = [wAA/2] [Nz-n] Eqn 14.25

• γ = (∂F/∂A)T,V,N = (∂U/∂N)T,V,N = - [wAA/2a] – γ increases as wAA increases (becomes more

negative)– γ increases as a decreases (molecular area

decreases)– γ has units of dyn/cm = force/length = erg/cm2

Table 14.2

Surface Tension (γ)

• U = [wAA/2] [Nz-n] Eqn 14.25

• γ = (∂F/∂A)T,V,N = (∂U/∂N)T,V,N = - [wAA/2a]

• Eqn 14.24 + 14.28 γ = Δhvap /za Fig 14.12

• Prob 2,4