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Phase Equilibria: Condensed Phases, Vapor, and Surface Tension Analysis

This chapter explores the principles of phase equilibria, focusing on the transition between condensed phases and vapor. Key concepts include the Extremum Principle, the Lattice Model for condensed phases, and the critical relationships governing chemical potential (μ) and Gibbs free energy (ΔG). The Clapeyron and Clausius-Clapeyron equations are extensively discussed, outlining phase equilibrium conditions at constant temperature and pressure. Additionally, the chapter addresses concepts of surface tension and its dependence on molecular interactions and area, providing insights into thermodynamic cycles relevant in refrigeration and heat suppression.

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Phase Equilibria: Condensed Phases, Vapor, and Surface Tension Analysis

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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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