01 Solution thermodynamics

Fugacity, activity and phase equilibrium.

A solvent’s selectivity, capacity, losses and regeneration energy are thermodynamic quantities. This chapter sets out the equations that predict them, from the equilibrium criterion to local-composition models, group contribution, COSMO-RS and cubic equations of state.

Computed for this page NRTL with α = 0.3 and τ12 = τ21 = τ (Eqs. 1.7–1.9). Color: Gibbs energy of mixing. White line: binodal, the coexisting liquid compositions (minima of ΔmixG); dashed: spinodal, the stability limit. The moving tie-line joins two coexisting phases. For a constant interaction energy, τ = Δg/RT, so larger τ means lower temperature.

You will see

  • The equilibrium criterion and the γ–φ formulation
  • Wilson, NRTL, UNIQUAC and UNIFAC
  • COSMO-RS and the Peng–Robinson equation of state
  • How the founder’s published designs used them

Labs in this chapter

1.1 Why thermodynamics decides solvent performance

An extraction solvent is judged by how strongly it takes up one component and rejects another, how much of it dissolves into the product stream, and how much energy it takes to regenerate. Each of these follows from activity coefficients and phase equilibrium. In the 2013 design study the founder co-authored (Karunanithi & Mehrkesh, part of his doctoral work), selectivity, solvent power and solvent loss were all computed from infinite-dilution activity coefficients:

β = γ∞B,Sγ∞A,S,    SP = 1γ∞A,S,    SL = 1γ∞S,B
(1.1)

A: aromatic solute (toluene); B: aliphatic (n-heptane); S: solvent. γ∞i,j: activity coefficient of i infinitely diluted in j.

Published research Historical founder research

For toluene/n-heptane, UNIFAC predicted β∞ = 87.262, SP = 0.67193 and SL = 0.006471 for the designed liquid (1-methylpyridinium tetrafluoroborate in the paper’s notation), against sulfolane’s values of 6.8023, 0.3719 and 0.0065. The SP and SL values were used as constraints (SP > 0.3719, SL < 0.0065, nearly binding); β was the objective. These are model predictions; the liquid was not synthesized in that study. Karunanithi & Mehrkesh, AIChE J. 2013, 59, 4627–4640, Table 16.

1.2 Fugacity and the equilibrium criterion

Fugacity turns the chemical potential into a pressure-like quantity that behaves well in mixtures. Two phases α and β are in equilibrium when, for every component, the fugacities are equal:

μi = μi∘ + RT ln fifi∘
(1.2)
fiα = fiβ    for all i
(1.3)

Vapor fugacities are written with a fugacity coefficient φi = fi/(yiP), and liquid fugacities with an activity coefficient γi = fi/(xi fi∘). Equating them gives the γ–φ form of vapor–liquid equilibrium:

yi φi P = xi γi Pisat φisat exp[ViL (P − Pisat)RT]
(1.4)

The exponential is the Poynting correction. For a sparingly soluble gas, the liquid-side reference is Henry’s constant, fi = xi Hi. The CO2 screen the founder co-authored (Farahipour, Mehrkesh & Karunanithi, Chem. Eng. Sci. 2016) estimated it as H = Psatγ∞, with γ∞ from COSMO-RS and Psat from the Peng–Robinson equation (Methods 02). Above the critical temperature, as for CO2 at 45 °C, no true vapor pressure exists, and the pure-liquid reference state becomes hypothetical; Psat must then be extrapolated.

1.3 Excess Gibbs energy and phase stability

Activity coefficients are partial molar derivatives of one function, the excess Gibbs energy. Any model for GE therefore yields self-consistent activity coefficients for every component:

GERT = Σi xi ln γi,    ln γi = [∂(nGE/RT)∂ni]T,P,nj≠i
(1.5)
Σi xi d ln γi = 0    (constant T, P)
(1.6)

Equation 1.6, the Gibbs–Duhem equation, is a consistency test for measured data. Phase splitting follows from the Gibbs energy of mixing:

ΔmixGRT = Σi xi ln xi + GERT
(1.7)

A binary mixture is unstable wherever this curve is concave in composition. The equilibrium phases are the two points that share a common tangent, which is the lower convex hull of the curve. The NRTL lab below finds them that way.

1.4 Local-composition models: Wilson, NRTL, UNIQUAC

Local-composition models assume the composition around a molecule differs from the bulk because interaction energies differ. Wilson’s equation (1964) handles strongly non-ideal miscible mixtures. It cannot represent liquid–liquid immiscibility, which is why NRTL and UNIQUAC are used for extraction systems.

NRTL (Renon and Prausnitz, 1968), binary form:

GERT = x1x2[τ21G21x1 + x2G21 + τ12G12x2 + x1G12],    Gij = exp(−α τij)
(1.8)
ln γ1 = x22[τ21(G21x1 + x2G21)2 + τ12G12(x2 + x1G12)2]
(1.9)

τij: dimensionless interaction parameters (often written Δgij/RT); α: non-randomness parameter. ln γ2 follows by exchanging indices 1 and 2.

UNIQUAC (Abrams and Prausnitz, 1975) splits ln γ into a combinatorial term, from molecular size and shape, and a residual term, from interaction energies:

ln γi = ln γiC + ln γiR
(1.10)
ln γiC = ln Φixi + z2 qi ln θiΦi + li − Φixi Σj xjlj
(1.11)
ln γiR = qi[1 − ln(Σj θjτji) − Σj θjτijΣk θkτkj]
(1.12)

Φi = rixi/Σ rjxj (volume fraction); θi = qixi/Σ qjxj (area fraction); li = (z/2)(ri − qi) − (ri − 1); z = 10; τij = exp(−Δuij/RT).

Lab 1.1 Methodology

NRTL phase-split explorer

Set the parameters; the lab computes γ1, γ2 and ΔmixG/RT from Eqs. 1.7–1.9 and finds coexisting liquid phases from the lower convex hull. The parameters are inputs, not fitted to a specific mixture.

1.5 Group contribution: UNIFAC

UNIFAC (Fredenslund, Jones and Prausnitz, 1975) keeps the UNIQUAC structure but builds each molecule from functional groups, so it can predict mixtures never measured. Molecular parameters are sums over groups, ri = Σk νk(i)Rk and qi = Σk νk(i)Qk, and the residual term becomes a sum of group activity coefficients:

ln γiR = Σk νk(i)[ln Γk − ln Γk(i)]
(1.13)
ln Γk = Qk[1 − ln(Σm ΘmΨmk) − Σm ΘmΨkmΣn ΘnΨnm],    Ψmn = exp(−amnT)
(1.14)

Γk: group activity coefficient in the mixture; Γk(i): in pure i; Θm: group area fraction; amn: group-interaction parameter fitted to data.

Ionic liquids strain this framework because the groups carry charge. The 2013 formulation followed the approach of Roughton and co-workers (2012): cation core, anion and alkyl groups are treated as separate groups, with interactions between the ionic groups assumed zero because the ions stay strongly paired. The amn values are empirical parameters fitted to equilibrium data. Learned latent descriptors (Chapter 3) take that idea further.

1.6 COSMO-RS: thermodynamics from surface charge

COSMO-RS (Klamt, 1995) starts from a quantum-chemical calculation of each molecule in a virtual conductor. The screening charge density σ on the molecular surface is summarized as a histogram, the σ-profile p(σ). Surface segments interact pairwise through an electrostatic misfit energy and a hydrogen-bond term, and the chemical potential of a segment in solvent S satisfies a self-consistent equation:

μS(σ) = −RTaeff ln ∫ pS(σ′) exp{aeffRT[μS(σ′) − emisfit(σ, σ′) − ehb(σ, σ′)]} dσ′
(1.15)
emisfit(σ, σ′) = α′2(σ + σ′)2
(1.16)

The chemical potential of a solute then follows by integrating μS(σ) over the solute’s own σ-profile, plus a combinatorial term. No group-interaction parameters are needed (a small set of global and element-specific parameters, periodically re-fitted by the developer), so COSMO-RS can screen ions that group-contribution tables do not cover. The electrostatic-potential surfaces in Methods 01 show the kind of charge distribution this starts from, though COSMO-RS uses conductor-screened charges, not gas-phase potentials.

Published research Historical founder research

COSMO-RS is predictive but not exact. In the CO2 screen (published 2016), predicted Henry’s constants were systematically high, and a linear correction fitted on 50 measurements cut the average error on 18 held-out systems from 36.6% to 12.7%. Methods 03 re-analyzes that correction.

1.7 Cubic equations of state: Peng–Robinson

Equations of state give fugacity coefficients for gases and, for cubic forms, for both phases at once. The Peng–Robinson equation (1976) is a standard choice for CO2 and hydrocarbons:

P = RTV − b − a(T)V(V + b) + b(V − b)
(1.17)
a(T) = 0.45724 R2Tc2Pc[1 + κ(1 − √Tr)]2,   b = 0.07780 RTcPc,   κ = 0.37464 + 1.54226ω − 0.26992ω2
(1.18)

In terms of the compressibility factor Z, with A = aP/(RT)2 and B = bP/(RT), the equation is a cubic whose largest root is the vapor and smallest the liquid:

Z3 − (1 − B)Z2 + (A − 3B2 − 2B)Z − (AB − B2 − B3) = 0
(1.19)
ln φ = Z − 1 − ln(Z − B) − A2√2 B ln Z + (1 + √2)BZ + (1 − √2)B
(1.20)

The saturation pressure is the pressure at which ln φL = ln φV, found by successive substitution P ← P·φL/φV. The lab does exactly this for CO2 and compares the result with NIST reference data.

Lab 1.2 Computed for this page

Peng–Robinson CO2, checked against NIST

Constants from NIST (Span–Wagner equation of state): Tc = 304.1282 K, Pc = 7.3773 MPa, ω = 0.22394. Points are NIST saturation pressures; the line is computed here from Eqs. 1.17–1.20.

Saturation pressure of CO2: NIST vs Peng–Robinson
NIST Chemistry WebBook, SRD 69 (Span & Wagner, J. Phys. Chem. Ref. Data 1996, 25, 1509–1596). At 288.15 K: NIST 50.87 bar; Peng–Robinson (this lab) 50.96 bar; the CO2 screening paper used 50.83 bar.

References

  1. Renon, H.; Prausnitz, J. M. Local compositions in thermodynamic excess functions for liquid mixtures. AIChE J. 1968, 14, 135–144. doi:10.1002/aic.690140124
  2. Wilson, G. M. Vapor–liquid equilibrium. XI. A new expression for the excess free energy of mixing. J. Am. Chem. Soc. 1964, 86, 127–130. doi:10.1021/ja01056a002
  3. Abrams, D. S.; Prausnitz, J. M. Statistical thermodynamics of liquid mixtures: a new expression for the excess Gibbs energy of partly or completely miscible systems. AIChE J. 1975, 21, 116–128. doi:10.1002/aic.690210115
  4. Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. Group-contribution estimation of activity coefficients in nonideal liquid mixtures. AIChE J. 1975, 21, 1086–1099. doi:10.1002/aic.690210607
  5. Klamt, A. Conductor-like screening model for real solvents: a new approach to the quantitative calculation of solvation phenomena. J. Phys. Chem. 1995, 99, 2224–2235. doi:10.1021/j100007a062
  6. Peng, D.-Y.; Robinson, D. B. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, 59–64. doi:10.1021/i160057a011
  7. NIST Chemistry WebBook, SRD 69: carbon dioxide, thermophysical properties of fluid systems (Span–Wagner equation of state). webbook.nist.gov
  8. Karunanithi, A. T.; Mehrkesh, A. Computer-aided design of tailor-made ionic liquids. AIChE J. 2013, 59, 4627–4640. doi:10.1002/aic.14228
  9. Farahipour, R.; Mehrkesh, A.; Karunanithi, A. T. A systematic screening methodology towards exploration of ionic liquids for CO2 capture processes. Chem. Eng. Sci. 2016, 145, 126–132. doi:10.1016/j.ces.2015.12.015
  10. Roughton, B. C.; Christian, B.; White, J.; Camarda, K. V.; Gani, R. Simultaneous design of ionic liquid entrainers and energy efficient azeotropic separation processes. Comput. Chem. Eng. 2012, 42, 248–262. doi:10.1016/j.compchemeng.2012.02.021

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