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Ch 14: Phase Equilibria

Ch 14: Phase Equilibria. I. Condensed Phase ↔ Vapor II. TR: Eqns Vi. and Vj. III. More Cycles IV. Surface Tension. I. Condensed Ph. ↔ Vapor. At constant T and p, the Extremum Principle states that equilibrium is associated with Δ G = 0  μ c = μ v Recall Example 7.5

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Ch 14: Phase Equilibria

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  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 ΔG = 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) • 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.

  4. 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 = pint0exp (z wAA/2kT) • Creating cavities or holes in a cond. ph. (ΔUremove), closing the hole (ΔUclose) and opening the hole(ΔUopen = - ΔUclose).

  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 = Δs/Δv = Δh/T Δv 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, 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

  7. 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

  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 (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

  10. 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

  11. 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

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