Gas solubility in dilute solutions: A novel molecular thermodynamic perspective
by
Ariel A. Chialvo & Knoxville, TN 37922-3108, U.S.A.
The Journal of Chemical Physics Accepted for publication April 9, 2018 &
To whom correspondence should be addressed (
[email protected]) - ORCID # 0000-0002-6091-4563
ABSTRACT We present an explicit molecular-based interpretation of the thermodynamic phase equilibrium underlying gas solubility in liquids, through rigorous links between the microstructure of the dilute systems and the relevant macroscopic quantities that characterize their solution thermodynamics. We apply the formal analysis to unravel and highlight the molecular-level nature of the approximations behind the widely used Krichevsky-Kasarnovsky [Journal of the American Chemical Society 57, 2168 (1935)] and Krichevsky-Iliinskaya [Acta Physicochimica 20, 327 (1945)] equations for the modeling of gas solubility. Then, we implement a general molecular-based approach to gas solubility and illustrate it by studying Lennard-Jones binary systems whose microstructure and thermodynamic properties were consistently generated via integral equation calculations. Furthermore, guided by the molecular-based analysis we propose a novel macroscopic modeling approach to gas solubility, emphasize some usually overlook modeling subtleties, and identify novel interdependences among relevant solubility quantities that can be used as either handy modeling constraints or tools for consistency tests.
I.
Introduction Thermodynamics of dilute solutions has been the subject of considerable
theoretical and practical interest
1,2,3,4
closely linked to the molecular theory of fluid
mixtures and consequently to the development of regression approaches for the macroscopic modeling of experimental data associated with chemical engineering bio-chemical
8,9
, and geo-chemical processes
10
5-7
,
. Among this wealth of theoretical
development and experimental data there has been (understandably) room for confusion regarding the meaning of solvation quantities and processes, or as Ben-Naim argued, “between solvation thermodynamics and conventional standard thermodynamics of solution” when proposing a rigorous analysis to identify and define precisely the standard free energy of solution according to the so-called C (mole fraction, molarity, and molality)-process of solvation 11. In the quest for accurate engineering modeling of dilute solutions we typically must confront two interconnected realities, namely, (i) the required highly precise
2
experimental determination of thermodynamic data
12
and (ii) the need for
fundamentally-based and molecularly inspired thermodynamic models
2,13-15
. In this
work we will focus on the second reality; specifically, in pursuit of unambiguous molecular-based description of the phase equilibrium equations for the solubility of sparingly soluble gases in liquids, we provide a microscopic interpretation of and highlight the molecular-level nature behind the approximations underlying the widely used Krichevsky-Kasarnovsky 16 and Krichevsky-Iliinskaya 17 gas solubility correlations. Since the study of gas solubility entails dealing with the isothermal-isobaric composition of binary systems involving a gaseous solute at equilibrium with its dilute counterpart in a condensed solvent phase, we will analyze the conditions of vapor-liquid equilibrium in terms of both the species residual and the resulting system excess properties. The rationale motivating this approach hinges around the advantages these two types of quantities encompass in the interpretation of the link between the (microscopic) intermolecular interactions and the resulting (macroscopic) thermodynamic behavior
14,18
. In fact, as lucidly discussed by Abbott and Nass
19
, it is typically not
practical to deal directly with a solution molar property M (TPx ) in the thermodynamic description
of
a
system,
but
rather
its
deviation
molar
counterpart
ΔM (TPx ) ≡ M real (TPx ) − M model (TPx ) defined as the difference between the actual (real) and a precisely-defined model behavior at identical state conditions (e.g., temperature and pressure) and composition (e.g., mole fractions). To make the microscopic-macroscopic link both explicit and unambiguous, we invoke here two types of molar deviation quantity ΔM (TPx ) namely, the residual M R (TPx ) ≡ M real (TPx ) − M IG (TPx ) and the excess M E (TPx ) ≡ M real (TPx ) − M IS (TPx )
properties. The first of these represents the deviation from the Ideal Gas (i.e., null interactions among species) behavior, while the second is the deviation from an Ideal Solution reference (i.e., described by a precisely-defined model containing the ideal gas plus assimilation contributions due to species distinguishability20).
Thus, M R (TPx )
measures the contribution of the actual interaction forces to the property M (TPx ) while
3
M E (TPx ) assesses the contribution from the differences between species intermolecular interactions (i.e., molecular asymmetry) in solution. While the choice for the model reference might be somewhat arbitrary, the LewisRandall ideal solution 21-22} offers us two practical advantages: it allows us (i) to describe the thermodynamic excess properties as differences of residual properties between the actual solution and each of its pure component counterparts (as well as their temperature derivatives
23
) and consequently (ii) to identify the explicit connections between (the
magnitude and sign of) the molar excess properties and the differences (in nature and strength 18,24) of the intermolecular forces between species in solution. These two features are essential for the versatile and successful modeling of solvation phenomena either over the entire composition range 25 or involving dilute multicomponent systems regardless of their aggregation state and including highly compressible (near critical) media 3,14,26. As we have discussed extensively elsewhere
14,15
, novel separation processes on
the one hand depend on our ability to tune the solvation behavior of species in solution according to required applications including food and pharmaceutical processing capture of anthropogenic gases
7,28
, water reforming and chemical synthesis
29
6,8,27
,
, novel
5
materials preparation and characterization . On the other hand, all of these processes either take place in or involve media with simultaneous solvation of gases, non-polar and ionic species whose behavior can only be interpreted, and consequently described, by macroscopic correlations through a fundamental understanding of the solvation processes in terms of microstructural changes undergone by the fluid environment 14. Within this context our first goal is the explicit molecular-based interpretation of the thermodynamic phase equilibrium equations underlying gas solubility in liquids obtained by drawing unambiguous links between the microstructure the
relevant
macroscopic
quantities
that
characterize
the
30
of the system and
resulting
solution
thermodynamics. As a consequence of this analysis we will highlight the molecular-level nature of the approximations used in the Krichevsky-Kasarnovsky Iliinskaya
17
16
and Krichevsky-
equations for the modeling of gas solubility. Then, our second goal is to
tackle the implementation of a general molecular-based approach to gas solubility for real systems and its illustration by modeling a binary system with interactions described by
4
the Lennard-Jones pair potential, whose microstructure and thermodynamic properties are consistently generated via integral equation (IE) calculations. To pursue these goals we first discuss in section II the fundamental equations underlying the isothermal-isobaric vapor-liquid equilibrium associated with gas solubility in conjunction with the rigorous molecular-based solvation formalism for dilute solutions proposed previously
26
to provide microscopic understanding of the macroscopic
counterparts. In section III we invoke the proposed gas solubility analysis to perform the microscopic interpretation of the Krichevsky-Kasarnovsky and Krichevsky-Iliinskaya equations for the modeling of gas solubility, highlighting the molecular meaning of their approximations. Moreover, to assess the relative contributions of the individual quantities affecting the phase equilibrium we study in section IV a well-characterized model, the Lennard-Jones system, via IE calculations according to Percus-Yevick (PY) approximation for the solution of the atomic Ornstein-Zernike (OZ) equation. Supported by the IE results of the previous section, we propose in section V a novel macroscopic modeling approach to gas solubility guided by the molecular-based analysis developed in this work. Furthermore, in section VI we identify novel interdependences among relevant solubility quantities and discuss their modeling implications. Finally, we close our work with a summary of findings and future outlook.
II.
Thermodynamics of gas solubility A. Phase equilibrium fundamentals Here we focus our attention on the solubility of a pure gas in a pure liquid at a
given temperature T and pressure P and study the thermodynamic quantities that characterize the equilibrium solubility of the gaseous solute in the presence of the vapor– liquid coexisting phases. After invoking the general criterion of physicochemical phase equilibrium, i.e., µiL (TP { x}) = µiV (TP { y}) where µiα (!) denotes the chemical potential of the i − solute species in the α − phase characterized by the thermodynamic conditions
(!) , we can rewrite it following the (φ − γ ) method 21,31 to describe the vapor ( V ) and liquid ( L ) phases respectively, i.e.,
5
fˆiV (TPyi ) = fˆi L (TPxi )
(1)
yiφˆiV (TPyi ) = xiφˆiL (TPxi ) = xi f i o (TP) γ iLR,L (TPxi ) P
The variables yi and xi in Eqn. (1) represent the mole fraction of the i − solute species in the vapor and liquid phase, respectively, φˆiV (TPyi ) and φˆiL (TPxi ) are the corresponding phase equilibrium partial molar fugacity coefficients, while γ iLR,L (TPxi ) denotes LewisRandall activity coefficient at the prevailing state conditions and composition. Typically, the behavior of the i − solute in the vapor phase is accurately characterized by an equation of state (EoS) such as Peng-Robinson liquid phase involves a solution model based on Henry’s law
33
32
while that for the
for the ideal solution
reference, i.e.,
f i HL (TPxi ) = xi Hi,ISj (TP)
(2)
where Hi,ISj (TP) represents Henry’s law constant of the i − solute in the j − solvent at the
(TP )
state conditions, with the superscripts IS and HL denoting the condition of ideal
solution under the reference state given by Henry’s law. Thus, the actual behavior of the
i − solute in the j − solvent becomes described by the deviation from the behavior represented by Eqn. (2) and accounted for the activity coefficient based on the corresponding IS reference as follows;
fˆi L (TPxi ) = γ iHL (TPxi ) xi Hi,ISj (TP)
(3)
where we identify explicitly Henry’s law ideal solution through the superscript HL , though the literature usually involves the notation γ iHL (TPxi ) ≡ γ i∗ (TPxi ) for that purpose. Consequently, from Eqn. (1) we have that lim ⎡⎣ fˆi L (TPxi ) xi ⎤⎦ = lim ⎡⎣ fˆiV (TPyi ) xi ⎤⎦ xi →0 xi →0 T T = Hi,ISj (TP)
6
(4)
Considering that lim P (T ) = Ps (T ) = Ps, j (T ) ≡ Ps and that lim γ iHL (TPs xi ) = 1 then for xi →0
xi →0
tabulation purposes it becomes convenient to define Henry’s law constant at a reference pressure, whose natural choice is that of the j − solvent saturation pressure Ps (T ) at the (subcritical) temperature of the system; in other words,
Hi,ISj (TP) = Hi,ISj (TPs ) exp
(∫
P Ps (T )
⎡υˆ ∞,L (TP) RT ⎤ dP ⎣ i ⎦
)
(5)
where the exponential term is usually known as the Poynting correction and 4
γ iHL (TPxi ) = γ iHL (TPs xi ) exp
(∫
P Ps (T )
{⎡⎣υˆ (TPx ) − υˆ (TP)⎤⎦ L i
i
∞,L i
}
RT dP
)
(6)
Then, from Eqns. (3)-(6) we can rewrite Eqn. (3) as a function of the saturation pressure and the resulting Poynting correction term as follows;
fˆi L (TPxi ) = xiγ iHL (TPxi ) Hi,ISj (TP) = xiγ iHL (TPs xi ) Hi,ISj (TPs ) exp
(
P
∫ Ps (T ) ⎡⎣υˆiL (TPxi ) RT ⎤⎦ dP
where we highlight the cancellation of the term exp
(∫
P
Ps (T )
)
{υˆ (TP ) ∞,L i
(7)
)
RT } dP between
the isothermal pressure dependence of γ iHL (TPxi ) and Hi,ISj (TP) in Eqns. (6) and (3), respectively. At this point we invoke the relation between the composition dependence of the activity coefficients for the i − solute describing the deviations from Henry’s law and Lewis-Randall rule 31, i.e.,
γ iHL (TPxi ) = γ iLR (TPxi ) γ iLR (TPxi = 0 ) = γ iLR (TPxi ) γ iLR, ∞ (TP )
(8)
Now, by recalling the definition of the Lewis-Randall activity coefficient as the ratio between the partial molar fugacity coefficient of the i − solute at the prevailing state conditions and composition and the corresponding pure component counterpart, i.e.,
7
γ iLR (TPxi ) = φˆi (TPxi ) φio (TP ) , we can recast Eqn. (7) after introducing Eqn. (8) in the following alternative form,
fˆi L (TPxi ) = xi ⎡⎣φˆi (TPs xi ) φˆi∞ (TPs )⎤⎦ Hi,ISj (TPs ) exp
(∫
P Ps (T )
)
⎡υˆ L (TPx ) RT ⎤ dP (9) ⎣ i i ⎦
The obvious appeal of Eqn. (9) lies in the ratio φˆi (TPs xi ) φˆi∞ (TPs ) = f ( xi ) where, because we are dealing with dilute solutions, f ( xi ) can be expressed as a second-order isothermal-isobaric composition expansion of φˆi (TPs xi ) around the infinite dilution condition φˆi∞ (TPs )
26,34
, i.e.,
(
)
f ( xi ) = exp ⎡⎣ −kij (TPs ) xi − 0.5xi2 ⎤⎦
(10)
whose consequent fixed-composition isothermal-pressure derivative becomes 26
(
υˆi (TPxi ) = υˆi∞ (TP) − RT ∂kij ∂P
) ( x − 0.5x ) T
i
2 i
(11)
Note that we kept the notation (TPs ) as the variables for φˆi∞ (TPs ) and kij (TPs ) in Eqns. (10)-(11) to highlight their origin, though we should recognize that, because the temperature dependence of the solvent saturation pressure, Ps (T ) , we actually have
φˆi∞ (T ) and kij (T ) along the orthobaric curve.
Therefore, from Eqns. (9)-(11) the
equilibrium partial molar fugacity of the dilute i − solute in the liquid phase becomes fˆi L (TPxi ) = xi Hi,ISj (TPs ) exp ⎡ exp ⎢− ⎣
{∫
P Ps (T )
(∫
(∂k
ij
P Ps (T )
)
⎡υˆ ∞,L (TP) RT ⎤ dP × ⎣ i ⎦
)
}(
∂P dP + kij (TPs ) T
⎤ xi − 0.5x ⎥ ⎦ 2 i
(12)
)
Alternatively, by noting that the {!} − term in the second exponential of Eqn. (12) reduces to
{∫
P Ps (T )
(∂k
ij
)
}
∂P dP + kij (TPs ) = kij (TP) T
8
(13)
we then get the following equivalent expression to Eqn. (12):
fˆi L (TPxi ) = xi Hi,ISj (TPs ) exp
(∫
P Ps (T )
(
⎡υˆ ∞,L (TP) RT ⎤ dP − k (TP) x − 0.5x 2 ⎣ i ⎦ ij i i
))
(14)
The alternative expressions Eqns. (12) and (14) for the liquid-phase partial molar fugacity of the dilute i − solute in vapor-liquid equilibrium are based on the self-consistent secondorder composition dependence of the thermodynamic partial molar quantities derived and discussed extensively elsewhere 26,34. B. Molecular-based interpretation of relevant gas solvation quantities To improve our understanding of the modeling capabilities of the above macroscopic expressions we now invoke Kirkwood-Buff’s fluctuation formalism of mixtures
35
to make the interpretation of the relevant quantities associated with gas
solvation by linking them with the behavior of the evolution of the microstructure of the resulting dilute solutions14,15. The quantities of interest in this context are kij (TP) , its pressure and temperature derivatives, i.e., ⎡⎣∂kij (TP) ∂P⎤⎦ and ⎡⎣∂kij (TP) ∂T ⎤⎦ , as well as υˆi∞ (TP) . T P
We have
already shown that the coefficient kij (TP) in the second-order truncated composition expansion of the partial molar fugacity coefficients of the species in a dilute binary mixture can be identified microscopically by the following expression 3,26
(
)
kij (TP) = Gii∞ + G ojj − 2Gij∞ υ oj
(15)
−1
where υ oj ≡ ρ oj (TP) describes the molar volume of the pure solvent at the prevailing ⊗ state conditions, and Gαβ denotes the Kirkwood-Buff total correlation function integral,
TCFI ,
35
for the αβ − interactions at the composition condition of either pure
component, ⊗ = o , or infinite dilution, ⊗ = ∞ , i.e., ∞
⊗ Gαβ (TPx ) = 4π ∫ hαβ⊗ (TPx,r)r 2 dr
(16)
0
9
The integrand in Eqn. (16) comprises the radial total pair correlation function ⊗ hαβ (TPx,r ) ≡ gαβ⊗ (TPx,r ) − 1 , where the corresponding radial pair distribution function ⊗ gαβ (TPx,r ) characterizes the (short- plus long-range) microstructure of the system, i.e.,
the environment of β − species around α − species at the state conditions and composition (TPx ) . ∞ By definition, the total pair correlation function hαβ (TP,r ) of an infinitely dilute
mixture includes direct pair correlations associated with the perturbation of the
α − solvent structure caused by the presence of the infinitely dilute β − solute as well as indirect pair correlations associated with the propagation of the structural perturbation. This feature, extensively discussed elsewhere 3,14,36,37 allows the segregation between two ⊗ sources of contributions to Gαβ , and consequently to kij (TPs ) ; namely the solvation
process (short-range direct pair correlations), and the propagation through the solvent of the structural perturbation (a distance given by its correlation length) that scales with the solvent’s isothermal compressibility. The short-/long-range split becomes rather handy especially when dealing with highly compressible media (e.g., dilute near critical solutions) because it allows the unambiguous interpretation of the solvation phenomena while avoiding the interference of diverging compressibility-driven quantities. In fact, we have shown that we can split the TCFI ’s of an infinitely dilute binary solution into the direct correlation function integrals DCFI , comprising the short-range interaction contributions, and the indirect correlation function integrals ICFI , measuring the longrange interaction contributions, i.e., G ojj = C ojj + κ oj kTC o2 υ o2 jj j
Gij∞ = Cij∞ + κ oj kTCij∞C ojj υ o2 j
Gii∞ = Cii∞ + κ oj kTCij∞2 υ o2 j
10
(17)
(
υ oj − C ojj where j − species denotes the solvent, κ oj ≡ βυ o2 j
)
is the isothermal
compressibility of the pure solvent, and Cij⊗ represents the integral of the direct correlation function cij⊗ (TPx,r ) , i.e., ∞
Cij⊗ (TPx ) = 4π ∫ cij⊗ (TPx,r)r 2 dr
i, j = 1,2
0
(18)
which is the short-range counterpart to Gij⊗ according to the Ornstein-Zernike equation 38. According to Eqns. (16)-(18) we can now provide a microstructural interpretation for the coefficient kij (TP) in Eqn. (15) through the fundamental link between the finite local solvent density perturbation caused by the presence of the solute ⎯ expressed as an ∞
isochoric-isothermal pressure change (∂P ∂xi )Tυ o
37,39
⎯ and the range of its propagation
j
that scales as the isothermal compressibility of the pure solvent, κ oj . Therefore, by invoking Eqns. (17) we can also split kij (TP) ,
Eqn. (15), into kijsolvation (TP) and
compressibility driven kijκ (TP) contributions, i.e.,
( = (C
) )
( + (κ κ
)( ) ⎡ υ ) ⎢(∂P ∂x ) ⎣
kij (TP) = C ojj + Cii∞ − 2Cij∞ υ oj + κ oj κ oIG υ oj C ojj − Cij∞ j o jj
+ Cii∞ − 2Cij∞ υ oj
o j
oIG j
2
i
2
⎤ T υ oj ⎥ ⎦ ∞
o j
(19)
= kijsolvation (TP) + kijκ (TP) = υ oj kT is the ideal gas solvent compressibility. For all practical purposes, where κ oIG j
we can recast kijsolvation (TP) and kijκ (TP) in terms of measurable macroscopic quantities, e.g., 2
∞ ⎤ ⎡ kijsolvation (TP) = κ oj κ oIG υ oj ⎢(∂P ∂xi )Tυ o ⎥ j ⎣ j ⎦
(
kijκ (TP) = − ∂ ln φˆi ∂ x j
(
)
)
∞ P,T
(20) 2
∞ ⎤ ⎡ − κ oj κ oIG υ oj ⎢(∂P ∂xi )Tυ o ⎥ j ⎣ j ⎦
(
)
11
(21)
∞
where (∂P ∂xi )Tυ o ⎯ the finite isochoric pressure perturbation upon addition of an j
infinitely dilute i − solute into an otherwise pure j − solvent ⎯ can also be unambiguously described and calculate as either an isothermal perturbation of the solute partial molar volume or solvent molecular population around the solute limiting slope ∂ ln φˆi ∂ x j
(
)
∞ P,T
(
= ∂ ln γ iLR ∂ x j
)
∞ P,T
13,37,39
, with the
available from accurate methods 40.
The partial molar volume of the solute species at infinite dilution, υˆi∞ (TP) , is typically assumed pressure independent in the Poynting correction during the evaluation of the solute fugacity, e.g., Eqn. (7)
31,41,42
, a conjecture that can introduce significant
uncertainties on the calculated quantities in the sequential approach 31. In order to assess the accuracy of such approximation we must analyze the molecular origin of the underlying pressure dependence of υˆi∞ (TP) , and for that purpose we invoke its expression from Kirkwood-Buff’s fluctuation formalism in terms of either TCFI ’s or DCFI ’s as follows 3,39
υˆi∞ (TP) = υˆ oj + G ojj − Gij∞
(
υˆi∞ (TP) = υ oj − Cij∞
(22a)
) (1− ρ C ) o j
o jj
(22b)
(
)
υˆi∞ (TP) = υˆ oj + C ojj − Cij∞ + κ oj kTC ojj C ojj − Cij∞ υ o2 j !# #"## $ !### #"#### $ υˆi∞ ( SR)
(22c)
υˆi∞ ( LR)
where we can distinguish in principle two sources of pressure dependence, i.e., one coming from the pure j − solvent and the other from the infinite dilute i − solute interacting with its surrounding j − solvent environment.
In Eqn. (22c) SR and LR
indicate the short- (solvation) and long-range (compressibility-driven) contributions to the quantity 3,36. The other relevant solvation quantity is the finite isochoric pressure perturbation upon addition of an infinite dilute i − solute into an otherwise pure j − solvent 3, i.e.,
12
(∂P ∂x )
∞
i Tυ o j
( kT (C
= ρ o2 kT G ojj − Gij∞ j = ρ o2 j
o jj
− Cij∞
) (1+ ρ G ) ) o j
o jj
(23)
which can also be unambiguously described as an isothermal perturbation of either the solute partial molar volume or solvent molecular population around the solute
13,37,39
. In
fact, Eqn. (23) provides a venue to identify three reference solute-solvent interaction ∞
asymmetries and assess the behavior of (∂P ∂xi )Tυ o involving an infinitely dilute solute j
with decreasing solute-solvent interaction asymmetry — namely, (a) a real solute in a Lewis-Randall ideal solution, (b) a real solute behaving identically to the solvent, and (c) an ideal gas solute in a real solution as discussed in detail in Appendix A. As we will ∞
discuss in the next section, (∂P ∂xi )Tυ o plays a central role in the microscopic j
interpretation of the sources of the pressure dependence of υˆi∞ (TP) .
C. Molecular-based inspired macroscopic correlations The macroscopic thermodynamic counterpart of the molecular interpretation of ∞
υˆi∞ (TP) and (∂P ∂xi )Tυ o can be written in a revealing form as follows 3,39: j
∞ ⎤ ⎡ υˆi∞ (TP) = υ oj ⎢1+ κ oj (∂P ∂xi )Tυ o ⎥ ⎣ j ⎦
(24)
so that after invoking Eqn. (23) and the first term of Eqn. (23c) we obtain,
(∂P ∂x )
∞
i Tυ o j
(
= ρ o2 kT υˆi∞ ( SR) − υ oj j
)
(25) ∞
where the link between υˆi∞ (TP) and (∂P ∂xi )Tυ o can be pursued in terms of the j
difference of partial molar volumes between the i − solute at infinite dilution, υˆi∞ ( SR) , and the pure j − solvent, υˆ oj 3,36.
13
Equations (24)-(25) highlight immediately three factors contributing to the isothermal pressure dependence of υˆi∞ (TP) , where two of them are properties of the pure solvent. We note that, according to the semi-empirical Tait equation 43, κ oj ( P)T exhibits an inversely proportionality with pressure so that
κ oj (TP) = κ oj (TPo ) ⎡⎣1+ nκ oj (TPo ) P ⎤⎦
(26)
1n
υ oj (TP) = υ oj (TPo ) ⎡⎣1+ nκ oj (TPo ) P ⎤⎦
(27)
where P = P − Po , Po is a reference pressure (e.g., Po = Ps, j (T ) ), and 3 ! n !15 . Thus, according to Eqns. (26)-(27), the pure j − solvent quantities contribute significantly to the pressure dependence υˆi∞ (TP) . In addition, after invoking O’Connell et al.’s correlation for simple solutes 13 for the difference of DCFI ’s in the right-hand side of Eqn. (23) as follows C ojj − Cij∞ = a + b⎡⎣exp cρ oj −1⎤⎦
( )
(28)
we can piece together the sought isothermal pressure dependence associate with the solute-solvent interactions, i.e.,
(∂P ∂x )
∞
i Tυ o j
2 n
= kT ρ o2 TPo ) ⎡⎣1+ nκ oj (TPo ) P ⎤⎦ j (
×
1 n ⎤⎞ ⎧ ⎫ ⎛ ⎡ ⎨ a + b ⎜exp ⎢cρ oj (TPo ) ⎡⎣1+ nκ oj (TPo ) P ⎤⎦ ⎥⎟ −1⎬ ⎦⎠ ⎭ ⎝ ⎣ ⎩
(29)
The analysis based on the IE calculations (vide infra section §IV) of the isothermal pressure dependence of the three factors in Eqn. (24), i.e., Eqns. (26)-(28) for the H 2 (i) − Ar( j) system, result in n ≅ 9.056 , a ≅ 5.4606 , b ≅ 0.9831 , and c ≅ 0.78306
indicating that there are two competing contributions to the isothermal-pressure dependence of υˆi∞ (TP) . The first contribution originates in the solute-solvent interaction ∞
asymmetry, i.e., encapsulated in (∂P ∂xi )Tυ o , while the other comes from the pure solvent j
14
and results in an opposite pressure trend (i.e., they scale as either P (
−1 n)
⎡−(1+n) n⎤ ⎦
or P ⎣
) as
clearly depicted in Figure 1, where we can highlight the small negative departure of the actual
Poynting
correction
from
the
approximated
value
when
assuming
υˆi∞ (TP) = υˆi∞ (TPs ) . For a common solvent, the observed compensating behavior will ∞
depend squarely on the magnitude of the (∂P ∂xi )Tυ o , and consequently the accuracy of j
the conjectured incompressible υˆi∞ (TP) in the Poynting correction will be a direct manifestation of the balance of the above two competing contributions. For example, after replacing Eqns. (26)-(27) and (29) (based on Tait’s equation) into Eqn. (24) we obtain
the
resulting
Poynting
integral PC = exp
(∫
P Ps (T )
⎡υˆ ∞,L (TP) RT ⎤⎦ dP ⎣ i Tait
)
for
representative values of υ oj (TPo ) and κ oj (TPo ) whose behavior is illustrated in Figure 2. This plot provides clear evidence of the significant contribution of the Poynting correction to Henry’s law constant as ( P − Ps )T increases, not only from the magnitude of
υˆi∞ (TPs ) but also from the actual isothermal pressure dependence υˆi∞ (TP) . In fact, the resulting PC ( P)T for 39.6 ≤ υˆi∞ (TPs ) cm3 mol ≤ 84.6 exhibits a quadratic rather than exponential, pressure dependence in the Ps (T ) ≤ P ( atm) ≤ 110 interval, i.e., an indication
(
of the non-negligible pressure effect on υˆi∞ (TP) , while PC P = 100,υˆi∞ (TPs ) by
a
ΔPC = exp
factor
(∫
P Ps (T )
1.25.
Under
)
these
conditions
the
υˆi∞,L (TP)Tait dP RT − exp ⎡⎣υˆi∞,L (TPs )Tait ( P − Ps ) RT ⎤⎦ will
)
T
increases departure
amount
to
7 < ΔPC ( % ) < 12 .
III.
Macroscopic modeling of gas solubility as conventionally interpreted and implemented At this point we have a rigorous thermodynamic framework, complemented with
the statistical mechanical interpretation of the relevant solvation quantities, for the rational analysis of existing correlations widely used in the study of gas solubility, i.e.,
15
the Krichevsky-Kasarnovsky, the Krichevsky-Iliinskaya equations, and variations of them21,31,41. In what follows we identify the embedded assumptions associated with these modeling expressions, discuss their microscopic consequences, and highlight some overlooked modeling subtleties. A. Special case 1: Krichevsky-Kasarnovsky equation The composition and pressure dependence of the partial molar fugacity coefficient (or its activity counterpart) becomes frequently blurred by the inaccuracies of the experimental measurements following the condition of high dilution of the system under consideration. Consequently, it is frequently assumed that γ iLR (TPs xi ) ≅ γ iLR,∞ (TPs ) , i.e.,
γ iHL (TPs xi = 0) = 1 with υˆiL (TPs xi ) ≅ υˆi∞,L (TPs ) independent of the system pressure so that Eqn. (7) reduces to
fˆi L (TPxi ) ≅ xi Hi,ISj (TPs ) exp ⎡⎣( P − Ps ) υˆi∞,L (TPs ) RT ⎤⎦
(30)
which is known in the literature as the Krichevsky-Kasarnovsky equation of gas solubility 16. According to the microscopic interpretation of Eqn. (12) derived above, the validity of the Krichevsky-Kasarnovsky equation as a modeling tool hinges around the
(
condition kij (TPs ) = ∂kij ∂P
)
T
= 0 within its second exponential, whose microstructural
manifestation according to Eqn. (15) is represented by the following two conditions,
Δ ij (TPs ) ≡ Gii∞ + G ojj − 2Gij∞ = 0
(31)
(∂Δ
(32)
ij
∂P
)
T
=0
These microscopic conditions indicate that the solubility of a gas in a binary solution would obey the Krichevsky-Kasarnovsky equation if the system behaved as a Lewis-
(
)
Randall ideal solution 15, i.e., Gij∞ = 0.5 Gii∞ + G ojj , not only at the solvent’s saturation pressure Ps (T ) but also at any system pressure along the chosen isotherm.
16
Note that the microscopic expression Eqn. (31) represents even a more restrictive constraint than that in the traditionally invoked condition γ iHL (TPs xi ) ≅ 1 , in the conventional derivation of the Krichevsky-Kasarnovsky equation; that is, Eqn. (31) tells us that not only γ iHL (TPs xi = 0) = 1 but also γ iLR (TPs xi = 0) = γ iLR,∞ (TPs ) = 1 . In other words, we should stress that the Krichevsky-Kasarnovsky equation is a rigorous result only for systems obeying Lewis-Randall ideality (vide infra Appendix A), i.e., those for which
fˆi (TPxi ) = f i o (TP) xi → γ iLR (TPxi ) = 1 ∀xi
(
υˆi (TPxi ) = υ (TPxi ) + x j G jj − Gij
)
(
→ υˆi∞ (TP) = υ oj (TP) + G ojj − Gij∞
)
IS
(33)
Hi,ISj (TPs ) = lim ⎡⎣ fˆi (TPxi ) xi ⎤⎦ = f i o (TPs ) xi →0 Obviously, this assumption is a rather restrictive condition for the gas solubility behavior in real systems, one that constrains the potential use of the Krichevsky-Kasarnovsky equation to systems whose solute exhibits extremely small deviation from Lewis-Randall ideality, i.e., to those whose solute-solvent interaction asymmetry is negligibly small. B. Special case 2: Krichevsky-Iliinskaya equation As highlighted by the derived Eqn. (12), the full and general description of the partial molar fugacity of the i − solute in the condensed phase requires the knowledge of the pressure and composition dependences of the i − solute partial molar fugacity,
φˆi (TPxi ) , or Henry’s law activity coefficient, γ iHL (TPs xi ) , to account for the system pressure and composition dependences in the second exponential term of Eqn. (12). Unfortunately, such information is rarely available hence the regression of solubility measurements
frequently
proceeds
via
conjectured
approximations
such
as
υˆiL (TPs xi ) ≅ υˆi∞,L (TPs ) in the Krichevsky-Kasarnovsky equation as already discussed above. The additional factor in the second exponential terms of Eqn. (12) encompasses the composition dependence of the isobaric-isothermal partial molar fugacity (or its associated Henry’s law activity) coefficient for the i − solute in the condensed phase. In principle, according to the specialized literature including Refs.
41,42,44
, the Krichevsky-
Iliinskaya equation accounts for the non-ideality originated in the solute-solvent
17
interactions (vide infra, Appendix B) via a simple two-suffix Margules (or Porter),
(
)
ln γ iHL = A (TP) x 2j −1 , i.e., Eqn. (12) simplifies to the familiar expression, fˆi L (TPxi ) = xi Hi,ISj (TPs ) exp ⎡⎣( P − Ps ) υˆi∞,L (TPs ) RT + A (TPs ) x 2j −1 ⎤⎦
(
)
(34)
known in the literature as the Krichevsky-Iliinskaya equation for the ij − binary system 17
. If we express the Margules/Porter equation in terms of the i − solute mole fraction,
(
)
(
)
i.e., A (TP ) x 2j − 1 = −2A (TP ) xi − 0.5xi2 where A ≡ ln γ iLR,∞ (vide infra Appendix B), then the direct comparison between Eqns. (34) and (12) allows us to identify 2A (TPs ) = kij (TPs ) and provide a meaningful molecular-based interpretation for the
Margules/Porter coefficient A (TPs ) in this type of dilute system as follows: A (TPs ) ≡ 0.5⎡⎣ Gii∞ + G ojj − 2Gij∞ υ oj ⎤⎦ TPs
(
)
(35)
= 0.5kij (TPs )
Note that the Krichevsky-Iliinskaya equation as conventionally written, Eqn. (34), invokes also a pressure and composition independent partial molar volume for the
i − solute, i.e., υˆiL (TPs xi ) ≅ υˆi∞,L (TPs ) . Considering the fundamental pressure dependence for Henry’s law activity coefficients 4, or their partial molar fugacity counterparts, i.e.,
⎡∂ln γ HL (TPx ) ∂P⎤ = ⎡υˆ L (TPx ) − υˆ ∞L (TP)⎤ RT ⎣ i i ⎦Txi ⎣ i i i ⎦
(36)
then the presumed υˆiL (TPs xi ) ≅ υˆi∞,L (TPs ) condition means that the Margules/Porter parameter must also be pressure independent, ∂A (TP ) ∂P = 0 , for the KrichevskyIliinskaya Eqn. (34) to be self-consistent according to our derived Eqn. (12). In other
(
)
words, Eqn. (35) should actually read A (T ) ≡ 0.5 ⎡⎣ Gii∞ + G ojj − 2Gij∞ υ oj ⎤⎦ and from a T microscopic viewpoint, its pressure independence would signify an imposed artificial compensation between the pressure effects on the system microstructure as described by
(G
∞ ii
)
+ G ojj − 2Gij∞ and the system’s saturation density. However, we will argue that Eqn.
18
(34) does not characterize properly the actual Krichevsky-Iliinskaya gas solubility equation (vide infra §VI).
IV.
Gas solubility behavior of model binary solutions Because we recognize the challenges behind obtaining accurate gas solubility
measurements
45
, and to avoid any ambiguity originated in experimental data
uncertainties, here we provide a set of thought experiments to generate accurate gas solubility data for a simple model system to illustrate Eqns. (12)-(14). Our choice of model system is a binary mixture interacting via Lennard-Jones pair potentials, whose microstructure and thermodynamic properties are consistently determined via integral equation (IE) calculations. In these thought experiments
46
we have full control of the
species interaction strength and hence a complete knowledge of the links between the solute-solvent
molecular
interaction
asymmetry,
the
resulting
microstructural
manifestation, and ultimately the corresponding thermodynamic properties of interest to gas solvation. The rationale behind the illustration and the choice of the model is manifold, namely: (a) the model’s well-characterized orthobaric phase envelope, (b) the suitable solution of the atomic Ornstein-Zernike integral equations Yevick (PY) approximation
47
38
according to the Percus-
, which provides the required complete microstructural
information consistent with the thermodynamic state and the infinite dilution conditions, and (c) the accurate access, resulting from the internal consistency, to the magnitude and sign of the individual quantities affecting the phase equilibrium that affords the direct assessment of the impact of the missing terms in Krichevsky-Kasarnovsky, KrichevskyIliinskaya and, for that matter, any other equations for the correlation of gas solubility. For that purpose we invoked the approach of McGuigan and Monson
48
to
determine the pair correlation functions, their integrals, and the thermodynamic properties of infinitely dilute binary Lennard-Jones systems, a methodology already used successfully for this purpose
3
. The illustration involves the Lennard-Jones
parameterization of Lotfi and Fischer
49
as a crude, yet surprisingly acceptable
representation of the second virial coefficient and Henry’s law constant for the H 2 (i) − Ar( j) system according to the available perturbation theory and molecular
19
dynamics simulation results
50,51
.
For the model solvent argon we use the critical
conditions of the model as predicted by the PY-IE following the method used elsewhere 52
, resulting in ε jj k = 114.2K and σ jj = 3.367 Å , while for the solute hydrogen we
adopted the parameters from Lotfi and Fischer
49
, i.e., ε ii k = 32.1K and σ ii = 2.892 Å .
Finally, for the unlike-pair interaction size and energy parameters we determined their deviations 53
from
the
conventional
Lorentz-Berthelot
combining
rules
η = σ ij σ ijLorentz ≈ 1.07584 and ξ = ε ij ε ijBerthelot ≈ 0.96398 required to describe reasonably
and simultaneously Bij (T )
54
and Hi, j (TPs ) 55 around T = 100K .
In order to determine the isothermal-pressure dependence of the solubility of the
i − gas solute in the j − condensed solvent we apply the alluded IE-PY approach to calculate the macroscopic quantities in Eqn. (12) or (14), from the resulting system microstructure, i.e., Hi,ISj (TPs ) , υˆi∞,L (TP) , kij (TP) , and fˆiV (TPyi ) (see Appendix C for details) as a venue to solve the phase equilibrium condition fˆiV (TPyi ) = fˆi L (TPxi ) required in the macroscopic (φ − γ ) modeling of gas solubility discussed in the next section.
V.
Modeling of isothermal gas solubility The self-consistent microstructural information generated in the previous section,
and the associated macroscopic solvation quantities along the chosen solubility isotherm, become the accurate sources to find unambiguous answers to the relevant gas solubility modeling issues, including: (i) how the assumed γ iHL (TPs xi ) ≅ γ iHL,∞ (TPs ) = 1 and its consequent behavior described by Eqn. (33) compare against the actual behavior of the model system, (ii) how large the impact of the assumed pressure independence of
υˆi∞,L (TP) ! υˆi∞,L (TPs ) might be in the calculation of the Poynting correction, and (iii) how accurate the approximation υˆiL (TPs xi ) ≅ υˆi∞,L (TPs ) might be to describe the pressure dependence of gas solubility.
20
Toward that end, except for issue (iii) that will be addressed in section §VI for reasons that will become clear below, we invoke the derived molecular-based expressions, Eqns. (12)-(14), for the description of isothermal gas solubility in binary aqueous systems, where the modeling proceeds according to a sequential approach. The raw data from the actual solubility experiments must comprise the isothermal pressure dependence of the vapor- and liquid-phase compositions, yi and xi , the second virial coefficient Bij (T ) for the ij − pair interactions, and a value for υˆi∞,L (TPs ) from an independent experimental source. After defining ℑα xi − 0.5xi2 = ln ⎡⎣ fˆiV (TPyi ) xi ⎤⎦ − Piυˆi∞,L (TPs ) RT based on Eqn. (12), its plot allows us
(
)
to extract simultaneously ln Hi,ISj (TPs ) and kij (TPs ) as follows:
(
)
lim ℑα xi − 0.5xi2 = ln Hi,ISj (TPs ) xi →0
(−)lim ⎡⎣∂ℑα xi − 0.5xi2 xi →0
(
(37a)
) ∂( x − 0.5x )⎤⎦ = k (TP ) i
2 i
ij
s
(37b)
In Figure 3 we plotted the isothermal-composition dependence of the modeling function
(
)
ℑα xi − 0.5xi2 for the H 2 (i) − Ar( j) system as described by the IE calculations discussed in §IV. This function exhibits an advantageous quasi-linear behavior that allows the reliable extraction of the two relevant solvation quantities associated with the limiting conditions, Eqns. (37a-b), i.e., ln Hi,ISj (TPs ) ≅ 0.502 and kij (TPs ) ≅ 10.02 . These results provide an answer to issue (i) above, i.e., for a rather conservative low i − solute concentration we have, after B2, that 0.6 ≤ γ iHL (TPs xi ≤ 0.05) ≤ 1.0 ; therefore, the assumption γ iHL (TPs xi ) ≅ γ iHL,∞ (TPs ) = 1 becomes less plausible as the system departs from the Lewis-Randall ideality, even for the small interaction asymmetries encountered in real systems (e.g., vide infra CH 4 − H 2O ). Then, from the pressure dependence of either Eqn. (12) or Eqn. (14) we also have that
21
⎡∂ln fˆ V x i i ⎣
(
∂P⎤⎦ = ⎡⎣υˆi∞,L (TPs ) RT − ∂kij ∂P T
)
(
{∫ ⎡∂ln fˆ V x i i ⎣
(
P Ps (T )
(∂k
)
∂P dP + kij
ij
T
) ( x − 0.5x )⎤⎦ − (TP )}⎡⎣∂( x − 0.5x ) 2 i
i
T
s
2 i
i
∂P⎤⎦ T
∂P⎤⎦ = ⎡⎣υˆi∞,L (TPs ) RT − ∂kij ∂P xi − 0.5xi2 ⎤⎦ − T T 2 kij (TP) ⎡⎣∂ xi − 0.5xi ∂P⎤⎦ T
)
)(
(
(
)
(38a)
(38b)
)
In addition, after considering the limiting condition lim P (T ) = Ps (T ) ≡ Ps it follows that, xi →0
⎡∂ x − 0.5x 2 ∂P⎤ = (1− x ) ⎡∂( P − P ) ∂x ⎤ ⎣ i i s i ⎦T ⎣ i ⎦T
(39)
lim ⎡⎣∂( P − Ps ) ∂xi ⎤⎦ = Hi,ISj (TPs ) T xi →0
(40)
lim ⎡⎣∂ xi − 0.5xi2 ∂P⎤⎦ = 1 Hi,ISj (TPs ) T
(41)
(
xi →0 P→Ps
)
(
)
Consequently, Eqns. (38a-b) reduce to the equivalent forms lim ⎡⎣∂ℑ β (TPxi ) ∂P⎤⎦ = υˆi∞,L (TPs ) RT − kij (TPs ) lim ⎡⎣∂ xi − 0.5xi2 ∂P⎤⎦ T xi →0 xi →0 T
(
= υˆ
∞,L i
(TP ) s
IS i, j
RT − kij (TPs ) H
)
(42)
(TP ) s
with ℑ β (TPxi ) = ln ⎡⎣ fˆiV (TPyi ) xi ⎤⎦ as illustrated in Figure 4 and
lim ⎡⎣∂ℑα ( Pi ) ∂Pi ⎤⎦ = − kij (TPs ) Hi,ISj (TPs ) T xi →0
(43)
from which we can determine kij (TPs ) after substituting Hi,ISj (TPs ) from Eqns. (38a-b). In order to address issue (ii) we analyze the behavior of the Poynting correction as
ΔPC = exp
(∫
P Ps (T )
)
υˆi∞,L (TP) dP RT − exp ⎡⎣υˆi∞,L (TPs ) ( P − Ps ) RT ⎤⎦ with υˆi∞,L (TP) generated
by the IE calculations along the T = 100K isotherm and P = 40atm , i.e., slightly below the limit of stability for this mixture. On the right ordinate of Figure 1 we display the two exponential terms of ΔPC (TP) where we clearly observe a very small over-prediction
22
PC ⎡⎣TP,υˆi∞,L (TPs )⎤⎦ relative to PC ⎡⎣TP,υˆi∞,L (TP)⎤⎦ , i.e., ΔPC (TP) PC ⎡⎣TP,υˆi∞,L (TPs )⎤⎦ ! 2% . Obviously, the departure ΔPC (TP) becomes monotonically larger with increasing values of υˆi∞,L (TP) and system pressure.
VI.
Discussion and outlook We frequently find in textbooks and specialized literature the Krichevsky-
Iliinskaya equation written as either (i) ln ⎡⎣ fˆi L (TPxi ) xi ⎤⎦ = ln Hi,ISj (TPs ) + ( P − Ps ) υˆi∞,L RT + A x 2j −1
(
)
(44)
under the clarification that “…equation (8.3.9) assumes that the partial molar volume of the solute is independent of pressure and composition over the pressure and composition ranges under consideration” 41 or (ii) ln ⎡⎣ fˆiV (TPyi ) xi Hi,ISj (TPs )⎤⎦ − ( P − Ps ) υˆi∞,L (TPs ) RT = A (TPs ) x 2j −1
(
)
(45)
with fˆiV (TPyi ) = yi PφˆiV (TPyi ) , where the author explicitly approximates the actual solute partial molar volume as υˆiL (TPxi ) ≅ υˆi∞,L (TPs ) whose state conditions are (TPs ) ≡ TPσ , j
(
)
in our notation 4,56. Here we argue that, as long as the Henry’s law activity coefficient is described by
(
)
the two-suffix Margules expression, ln γ iHL = A 1− x 2j , such an approximation is both unnecessary and misleading; in fact, by a simple look at the rigorous Eqns. (11)-(14), it becomes clear that the previous Eqns. (44)-(45) might be considered alternative interpretations of the original Krichevsky-Iliinskaya equation in that υˆi∞,L (TPs ) comes from the first term of the composition dependence of the partial molar volume of the i − solute, υˆiL (TPxi ) = υˆi∞,L (TP) − RT ∂kij ∂P
(
) ( x − 0.5x ) — Eqn. (11) — T
i
2 i
where
A = 0.5kij (vide infra Appendix B). This argument is not only of academic interest but also of practical relevance in that (a) it identifies the microscopic nature and characterizes the actual source of solution non-ideality through the parameter A , and (b) it highlights
23
the fact that the state conditions (TP) in the last term of the Krichevsky-Iliinskaya
(
)
(
)
equation, A (TP) x 2j −1 ≡ ⎡⎣ln γ iLR,∞ (TP)⎤⎦ x 2j −1 , are the result of the pressure integration given by Eqn. (13), not a result of the alleged approximation indicated in in Eqns. (44)(45) above. This implicit pressure dependence conspires against the conventional linear regression ℑα ( zi ) = ln ⎡⎣ fˆiV (TPyi ) xi Hi,ISj (TPs )⎤⎦ − υˆi∞,L (TPs ) ( P − Ps ) RT ≡ 2 A ( zi ) , where T zi = xi − 0.5xi2 , to extract the Margules/Porter parameter A (TP) from its composition
slope 31; because the system pressure is not constant and xi = ℑ ( P)T , the slope A (TP) is not constant along the plotted isotherm. The Krichevsky-Kasarnovsky equation has been frequently used to extract
υˆi∞,L (TP) from solubility measurements at elevated pressures, and the conventional wisdom suggests that under those state conditions the gas solubility might be already large enough to invalidate the underlying assumption γ iHL (TPs xi ) ≅ 1 , rendering unreliable the υˆi∞,L (TP) values 4,41. Our molecular-based analysis, encapsulated in Eqns. (31)-(33), unravels an overlooked subtlety on the above scenario; namely, that the KrichevskyKasarnovsky equation becomes a better descriptor of gas solubility with isothermal decreasing solute-solvent molecular asymmetry, which in turn makes the i − solute more soluble in the j − solvent environment. In other words, the conjectured γ iHL (TPs xi ) ≅ 1 is not the consequence of the xi → 0 limiting condition, but rather of
γ iHL (TPxi ) ≡ γ iLR (TPxi ) γ iLR,∞ (TP) → 1 with γ iLR,∞ (TP) → 1 as Δ∞ij → 0 limiting conditions, i.e., when the strength of the solute-solvent interactions approaches that of the average between solvent-solvent and solute-solute interactions regardless of the system composition (vide infra, Appendix A). Consequently, the regressed value of υˆi∞,L (TP) will approach that of the solute in the corresponding Lewis-Randall ideal solution, i.e.,
(
υˆi∞,IS (TP) = υ oj − lim Δ∞ij + Gij∞ − Gii∞ ∞ Δ ij →0
)
A4
o i
= υ (TP)
24
An instructive example of these overlooked modeling subtleties can be extracted from the recent work of Hou et al. 57 who illustrated the failure of the KrichevskyKasarnovsky equation for the solubility of CO2 in the CO2 − H 2O system along the
T ≈ 423K isotherm by highlighting in their Figure 14 that ⎡⎣∂ln fˆi L xi
(
)
∂Pi ⎤⎦
T ,xi →0
0 and consequently ∞
(υˆ
∞ i
)
υ oj = 1+ κ oj ( ∂P ∂xi )Tυ > 0 . In fact, in the context of the observation by Hou et al. 57, ∞
this behavior provides a hint into the likely effect of the missing term in the KrichevskyKasarnovsky equation, later added to the Krichevsky-Iliinskaya equation, e.g., vide supra Eqn. (42),
⎡∂ln fˆ L x i i ⎣
(
)
∂Pi ⎤⎦
Txi →0
= υˆi∞,L (TPs ) RT − 2 A (TPs ) Hi,ISj (TPs ) ≶ 0
(46)
with Pi = P − Ps . Clearly, Eqn. (46) can be satisfied by υˆi∞,L (TPs ) > 0 since
A (TPs ) ≡ 0.5ρ oj Δ∞ij ≶ 0 (vide infra Appendix B), i.e., a necessary condition for A (TPs ) to be able to describe positive/negative deviations from the Lewis-Randall ideal solution reference (e.g., see Figure 8 and related discussion in Ref. 25). A key feature underlying the sequential approach discussed in section §V is its exclusive reliance on the i − solute phase equilibrium condition alone, i.e., no need for the corresponding j − solvent counterpart. This scenario precludes the automatic compliance of thermodynamic constraints including Gibbs-Duhem and Maxwell relations, and consequently, it may lead to thermodynamic inconsistencies (see e.g., §5 of Ref. 15). In contrast, the simultaneous approach provides an alternative avenue for the determination of the gas solubility parameters by applying readily available non-linear optimization methodologies 60. The success of these optimized regressions will be greatly enhanced by invoking the i − solute and corresponding j − solvent phase-equilibrium conditions,
25
vide infra Appendix D, with the addition of suitable constraints to ensure thermodynamic as well as solvation parameter consistency and avoid potential unphysical modeling results as we discuss below. Beyond the required compliance with Gibbs-Duhem and Maxwell relations for the second-order composition expansion underlying the phase-equilibrium conditions 26, we recognize additional relations from the definition of Hi,ISj (TP) , Eqn. (4), where we identify two alternative expressions linking Hi,ISj (TP) to an infinite dilution coefficient with well-defined microstructural nature, i.e.,
Hi,ISj (TP) = P lim ⎡⎣φˆiL (TPxi )⎤⎦ xi →0
= Pφ
o,L i
T
(TP)γ
LR,∞ i
(47)
(TP)
where Pφio,L (TP) = f i o,L (TP) denotes the pure component fugacity of the i − solute, while
γ iLR,∞ (TP) = lim ⎡⎣φˆiL (TPxi ) φio,L (TP)⎤⎦ , at the prevailing state conditions. Thus, by xi →0
T
invoking Eqn. (35) in conjunction with B2 we have
Hi,ISj (TPs ) = f i o (TPs ) exp ⎡⎣0.5kij (TPs )⎤⎦
(48)
kij (TPs ) = 2ln ⎡⎣Hi,ISj (TPs ) f i o (TPs )⎤⎦
(49)
where once again Ps is the saturation pressure of the j − solvent at the prevailing temperature. Consequently, we can now provide additional insights into Eqn. (46), by rewriting the ratio 2 A (TPs ) Hi,ISj (TPs ) in two alternative ways as follows:
2 A (TPs ) Hi,ISj (TPs ) = kij (TPs ) exp ⎡⎣−0.5kij (TPs )⎤⎦ f i o (TPs ) = 2ln ⎡⎣Hi,ISj (TPs ) f i o (TPs )⎤⎦ Hi,ISj (TPs )
(50)
Equation (50) reveals a (so far) hidden molecularly-based link between the coefficient A (TPs ) and Henry’s law constant Hi,ISj (TPs ) , and consequently highlights the fact that the Krichevsky-Iliinskaya slope, identified by Eqn. (46), comprises three
26
macroscopic quantities but actually only two variables. More importantly, Eqns. (48-49) provide a novel avenue for a few relevant observations including (i) the reformulation of the gas solubility, Eqn. (14), in a pair of equivalent forms as follows: ln ⎡⎣ fˆi L (TPxi ) xi ⎤⎦ = x 2j ln ⎡⎣Hi,ISj (TPs ) f i o (TPs )⎤⎦ + ln f i o (TPs ) + 2 j ij
o
≅ 0.5x k (TPs ) + ln f i (TPs ) + υˆ
∞,L i
o
P Ps (T )
(TP ) ( P − P )
ln ⎡⎣ fˆi L (TPxi ) xi ⎤⎦ = 0.5x 2j kij (TPs ) + ln f i o (TPs ) + 2 j ij
∫
s
s
∫
≅ 0.5x k (TPs ) + ln f i (TPs ) + υˆ
P Ps (T )
∞,L i
⎡υˆ ∞,L (TP) RT ⎤ dP ⎣ i ⎦ RT
⎡υˆ ∞,L (TP) RT ⎤ dP ⎣ i ⎦
(TP ) ( P − P ) s
(51a)
s
(51b)
RT
(ii) the estimation of the challenging υˆi∞,L (TPs ) from the independent knowledge of Henry’s law constant Hi,ISj (TPs ) or kij (TPs ) , (iii) the a priori assessment of the conjectured γ iHL (TPxi ) ≅ 1 condition in either Krichevsky-Iliinskaya or KrichevskyKasarnovsky equation, (iv) the consistent evaluation of the corresponding Henry’s law activity coefficient for the dilute i − solute in a j − solvent, i.e.,
(
ln γ iHL (TPs xi ) = −kij (TPs ) xi − 0.5xi2
) (
= 2ln ⎡⎣ f i o (TPs ) Hi,ISj (TPs )⎤⎦ xi − 0.5xi2
)
(52)
and consequently (v) the detection of potential inconsistencies among solvation parameters in the solubility data regression, (vi) as well as the formal link between the microstructural behavior of the dilute solution under vapor-liquid equilibrium and additional relevant solvation quantities including the i − solute distribution coefficient
K i∞ (TPs ) as well as the corresponding Ostwald coefficient L∞i, j (TPs ) to be discussed below. Regarding the alluded a priori assessment of conjectured behaviors, in items (iiiiv), we note that Dhima et al.’s gas solubility study61 reported the regression of the quantity γ iHL (TPs xi ) Hi,ISj (TPs ) rather than the usual Hi,ISj (TPs ) in their Eqn. (5), and properly indicated that the plot of ln ⎡⎣ fˆi L (TPxi ) xi ⎤⎦ as a function of ( P − Ps ) will exhibit
27
a linear behavior with slope υˆi∞,L (TPs ) RT with an ordinate at the origin equal to
γ iHL (TPs xi ) Hi,ISj (TPs ) as long as both γ iHL and υˆi∞,L behaved as independently of pressure and composition. The constancy of the product γ iHL (TPs xi ) Hi,ISj (TPs ) , however, becomes a direct consequence of Eqns. (48) and (52), i.e.,
γ iHL (TPs xi ) Hi,rIS (TPs ) = f i o (TPs ) exp ⎡⎣0.5kij (TPs ) x 2j ⎤⎦
(53)
≅ f i o (TPs ) exp ⎡⎣0.5kij (TPs )⎤⎦ since the j − solvent mole fraction x j ! 1 , regardless of any condition of pressurecomposition independence for γ iHL and υˆi∞,L .
The interdependence of the solvation parameters embodied in Eqns. (48-49) provides a quick test of consistency for regressed quantities as suggested in item (v). In fact, an instructive illustration on the alluded internal consistency regards the study of CH 4 − H 2O binary system. On the one hand Bender et al. 62 reported in their Figure 9
the temperature dependence of the Margules parameter ACH −H O (T ) according to its 4
2
description from two equations of state (EoS). On the other hand, Dhima et al. 61 analyzed this aqueous system and determined the corresponding Henry’s law constant, IS i.e., HCH
4 ,H 2O
(T = 344K ) = 5930 MPa .
At the water saturation conditions
o fCH (TPs ) ≅ Ps (T = 344K ) ≅ 0.0306 MPa then according to Eqn. (49) we have 4
that ACH −H O (TPs ) ≅ 12.2 . The comparison between this value and the graphical 4
2
estimations from Bender et al.’s Figure 9, ACH −H O (TPs ) ≈ 6.4 and PR 4
2
ACH −H O (TPs ) RK ≈ 10.6 according to the Peng-Robinson and Redlick-Kwong EoS 4
2
∞ calculations indicates an underestimation of the actual activity coefficient γ CH by a factor 4
of 66 and 5, respectively, and provides convincing evidence for the lack of internal consistency of the thermodynamic results in Ref. 62, which has been previously implied 63
.
28
As mentioned in item (vi), Eqns. (48-49) allow the formal link between the descriptors of the microstructure of the phases in vapor-liquid equilibrium and the
i − solute distribution coefficient K i∞ (TPs )
64
as derived in appendix E, results from
invoking the phase equilibrium condition Eqn. (1) and the alternative expression for Henry’s law constant, Eqn. (47), i.e., K i∞ (TPs ) = ⎡⎣ f i o (TPs ) Ps (T )⎤⎦exp 0.5kij (TPs ) − β Ps ⎡⎣δ jj (T ) − Bii (T )⎤⎦
{
}
E3a
where we can also express f i o (TPs ) in terms of the microstructure of the pure i − solute at the prevailing state conditions as depicted by Eqns. E4a-b in appendix E. Likewise, the corresponding Ostwald coefficient L∞i, j (TPs ) becomes 65
L
∞ i, j
(TP ) =
( )
s
Ps (T ) ρio,L (TPs ) f i o (TPs ) ρio,V (TPs )
(
)
{
}
exp β Ps ⎡⎣δij (T ) − Bii (T )⎤⎦ − 0.5kij (TPs )
E5b
where Bij T = lim Gij TPxi and δij (T ) = lim Δ∞ij (TP) . ⊗
ρ→0
ρ→0
VII. Summary In this work we have developed an explicit molecularly-based interpretation of the thermodynamic phase equilibrium underlying gas solubility through the identification of fundamental links between the microstructure of the infinite dilute reference systems and the relevant macroscopic quantities that characterize the resulting dilute solutions and phases in equilibrium. The formalism hinges around the thermodynamic self-consistent second-order truncated composition expansion of the partial molar quantities of the species regardless of its aggregation state 15,26, which (a) provides the tools to identify and assess at a molecular-level the nature of the approximations behind the commonly used equations for the modeling of gas solubility, (b) highlights the relevant subtleties behind the conjectured approximations associated with the macroscopic modeling and, (c) delivers a general fundamentally- and molecularly-based platform for gas solubility modeling.
29
We have illustrated the derived relationships by generating proxy data from IE calculations of a model system and unraveled novel interdependences between the solvation parameters relevant to gas solubility in dilute systems. This analysis results in, among other things, the reformulation of the gas solubility problem leading to the a priori assessment of the validity of modeling assumptions, the detection of potential inconsistencies among gas solubility regressed quantities, and the formal links between Ostwald coefficient, the solute distribution coefficient, and the microstructural behavior of the system at vapor-liquid phase equilibrium. ACKNOWLEDGEMENTS The author expresses his gratitude to Prof. Peter A. Monson for providing years ago the original version of his IE-PY code used in the reported calculations, and to Dr. Sebastian Chialvo for the critical reading of the manuscript. REFERENCES 1
2 3 4 5 6 7
8 9
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31
28
29
30 31 32 33 34
35 36 37 38 39 40
41
42 43 44
Meullemiestre, M. Turk, S. Perino, A. S. Fabiano-Tixier, and M. Abert-Vian, Innovative Food Science & Emerging Technologies 41, 357 (2017). L. Chahen, T. Huard, L. Cuccia, V. Cuzuel, J. Dugay, V. Pichon, J. Vial, C. Gouedard, L. Bonnard, N. Cellier, and P. L. Carrette, International Journal of Greenhouse Gas Control 51, 305 (2016); M. Lucquiaud and J. Gibbins, Chemical Engineering Research & Design 89 (9), 1553 (2011). L. J. Guo and H. Jin, International Journal of Hydrogen Energy 38 (29), 12953 (2013); Z. Knez, E. Markocic, M. K. Hrncic, M. Ravber, and M. Skerget, Journal of Supercritical Fluids 96, 46 (2015). In this context the term microstructure actually denotes volume integrals over pair correlation functions defining the Kirkwood-Buff integrals H. C. Van Ness and M. M. Abbott, Classical Thermodynamics of Nonelectrolyte Solutions. (McGraw Hill, New York, 1982). D. Peng and D. B. Robinson, Industrial & Engineering Chemistry Fundamentals 15 (1), 59 (1976). W. Henry, Philosophical Transactions of the Royal Society of London 93, 29 (1803). A. A. Chialvo, S. Chialvo, and J. M. Simonson, in Gas-Expanded Liquids and near-Critical Media: Green Chemistry and Engineering, edited by K. W. Hutchenson, A. M. Scurto, and B. Subramaniam (2009), Vol. 1006, pp. 66. J. G. Kirkwood and F. P. Buff, Journal of Chemical Physics 19, 774 (1951). A. A. Chialvo, P. T. Cummings, J. M. Simonson, and R. E. Mesmer, Journal of Chemical Physics 110, 1075 (1999). A. A. Chialvo, P. G. Kusalik, P. T. Cummings, J. M. Simonson, and R. E. Mesmer, Journal of Physics-Condensed Matter 12 (15), 3585 (2000). J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, 2nd. ed. (Academic Press, New York, 1986). A. A. Chialvo and P. T. Cummings, Molecular Physics 84, 41 (1995). D. A. Jonah, Chemical Engineering Science 41 (9), 2261 (1986); A. C. Chialvo, Canadian Journal of Chemistry-Revue Canadienne De Chimie 70 (6), 1645 (1992). J. M. Prausnitz, R. N. Lichtenthaler, and E. G. de Azevedo, Molecular Thermodynamics of Fluid Phase Equilibria, 2nd. ed. (Prentice-Hall, Englewood Cliffs, 1986). E. Wilhelm, Journal of Thermal Analysis and Calorimetry 108 (2), 547 (2012). J. R. Macdonald, Reviews of Modern Physics 38 (4), 669 (1966). J. M. Prausnitz, in Advances in Chemical Engineering, edited by T. B. Drew, G. R. Cokelet, J. W. Hoopes, and T. Vermeulen (Academic Press, 1968), Vol. 7, pp. 139; R. E. Gibbs and H. C. van Ness, Industrial & Engineering Chemistry Fundamentals 10 (2), 312 (1971); J. J. Carroll and A. E. Mather, Journal of Solution Chemistry 21 (7), 607 (1992); E. Wilhelm, in Developments and Applications in Solubility (The Royal Society of Chemistry, 2007), pp. 3; I. Tosun, in The Thermodynamics of Phase and Reaction Equilibria (Elsevier, Amsterdam, 2013), pp. 447; E. Wilhelm and R. Battino, in Volume Properties: Liquids, Solutions and Vapours, edited by T. L. Emmerich 32
45 46 47 48 49 50 51
52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67
Wilhelm (The Royal Society of Chemistry, 2015), pp. 273; F. Y. Jou, A. E. Mather, and K. A. G. Schmidt, Journal of Chemical and Engineering Data 62 (9), 2761 (2017). E. Wilhelm, Crc Critical Reviews in Analytical Chemistry 16 (2), 129 (1985). A. A. Chialvo and L. Vlcek, Pure and Applied Chemistry 88 (3), 163 (2016). J. K. Percus and G. J. Yevick, Physical Review 110, 1 (1958). D. B. McGuigan and P. A. Monson, Fluid Phase Equilibria 57, 227 (1990). A. Lotfi and J. Fischer, Molecular Physics 66, 199 (1989). J. Fischer, M. Bohn, and M. Gebauer, Fluid Phase Equilibria 17 (1), 131 (1984); A. Lotfi, Ruhr-Universität, 1993. We should not expect the Lennard-Jones model to be an accurate representation of this binary mixture at low temperature as a result of likely significant quantum effects from the infinitely dilute molecular hydrogen [Q. U. Wang, J. K. Johnson, and J. Q. Broughton, Molecular Physics 89 (4), 1105 (1996)] A. A. Chialvo and J. Horita, Journal of Chemical Physics 119 (8), 4458 (2003). J. Fischer, D. Möller, A. A. Chialvo, and J. M. Haile, Fluid Phase Equilibria 48, 161 (1989); A. A. Chialvo, Molecular Physics 73, 127 (1991). J. H. Dymond and E. B. Smith, The Virial Coefficients of Pure Gases and Mixtures: A Critical Compilation. (Clarendon Press, 1980). M. Orentlicher and J. M. Prausnitz, Chemical Engineering Science 19 (10), 775 (1964). E. Wilhelm, in Experimental Thermodynamics, edited by R. D. Weir and T. W. De Loos (Elsevier, 2005), Vol. 7, pp. 137. S. X. Hou, G. C. Maitland, and J. P. M. Trusler, Journal of Supercritical Fluids 73, 87 (2013). P. G. Debenedetti and R. S. Mohamed, Journal of Chemical Physics 90, 4528 (1989). J. M. H. Levelt Sengers, Journal of Supercritical Fluids 4, 215 (1991). W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN; The Art of Scientific Computing, 2nd ed. (Cambridge University Press, New York, 1993). A. Dhima, J.-C. de Hemptinne, and J. Jose, Industrial & Engineering Chemistry Research 38 (8), 3144 (1999). E. Bender, U. Klein, W. P. Schmitt, and J. M. Prausnitz, Fluid Phase Equilibria 15 (3), 241 (1984). R. D. Deshmukh and A. E. Mather, Fluid Phase Equilibria 35 (1-3), 313 (1987). E. Wilhelm, Thermochimica Acta 162 (1), 43 (1990). E. Wilhelm, Pure and Applied Chemistry 57 (2), 303 (1985). A. Ben-Naim, Molecular Theory of Solutions. (Oxford University Press, Oxford, 2006). This expression can also be derived from the virial theorem as done elsewhere [A. A. Chialvo, P. T. Cummings, and Y. V. Kalyuzhnyi, Aiche Journal 44 (3), 667 (1998)]
33
68 69 70 71 72 73
This is a direct consequence of Widom's potential distribution theorem for a non-interacting particle [Journal of Physical Chemistry 86 (6), 869 (1982)] C. A. Cerdeirina and B. Widom, Journal of Physical Chemistry B 120 (51), 13144 (2016). J. Alvarez and R. Fernandez-Prini, Fluid Phase Equilibria 66, 309 (1991). J. L. Alvarez and R. F. Prini, Journal of Solution Chemistry 37 (10), 1379 (2008). R. Fernandez-Prini, J. L. Alvarez, and A. H. Harvey, Journal of Physical and Chemical Reference Data 32 (2), 903 (2003). An alternative compeling argument is given in section 3.22 of Ben-Naim’s "Solvation Thermodynamics", Ref 11.
34
APPENDIX A: Effect of solute-solvent interaction asymmetries on relevant gas solvation quantities ∞
From the microscopic definition of υˆi∞ (TP) and its link to (∂P ∂xi )Tυ o , Eqns. j
(22a)-(23) and the molecular identification of the limiting composition slope kij (TP) , Eqn. (15), we can analyze the impact, and derive the expressions, of the strength of the solute-solvent interaction for a few precisely-defined molecular asymmetry; namely (a) a real solute in a Lewis-Randall ideal solution, (b) a real solute behaving identically to the solvent, and (c) an ideal gas solute in a real solution The first scenario defines the Lewis-Randall ideal solution for a binary system, a reference
for
which
the
excess
chemical
potential
of
an
i − species
µ iE (TPxi ) = kT ln γ iLR (TPxi ) ≡ 0 25,66 that translates into Δ ijIS (TPxi ) = 0 , i.e., Δ ijIS (TPxi ) ≡ GiiIS + G jjIS − 2GijIS = 0 ∀TPxi
A1
so that, for the i − solute at infinite dilution, the microscopic manifestation of the LewisRandall ideal solution reads as follows,
(
)
Gii∞,IS + G o,IS − 2Gij∞,IS = 0 → Gij∞,IS = 0.5 Gii∞.IS + G o,IS ≠0 jj jj
A2
and consequently, from Eqn. (15) kijIS (TP) = 0 . Thus from A2 Eqn. (23) becomes
(∂P ∂x )
∞,IS
i Tυ o j
(
= −kT ρ
o2 j
IS
) (1+ ρ G ) (G − G ) (1+ ρ G ) ≠ 0
= kT ρ o2 G ojj − Gij∞ j ∞ ii
∞ ij
o j
IS
o jj
o j
A3
o jj
Likewise, from Eqn. (22), we obtain
( − (G
υˆi∞,IS (TP) = υ oj + G ojj − Gij∞ = υ oj
∞ ii
− Gij∞
) )
IS
IS
A4
= υio (TP)
35
The second scenario is a more restrictive condition for the Lewis-Randall ideal solution, one in which all TCFI ’s are identically the same, G ojj = Gii∞ = Gij∞ , resulting in IS ,G ojj =Gij∞ =Gii∞
kij
∞,IS ,G ojj =Gij∞ =Gii∞
(TP) = 0 , υˆi
∞,IS ,G ojj =Gij∞ =Gii∞
= υˆ oj = ρ o(−1) , and therefore, (∂P ∂xi )Tυ o j
= 0,
j
i.e., this scenario describes an infinite dilute solution where the i − species and the
j − species are only distinguishable by their label, yet, the strength of like- and unlike pair interactions are all the same as that in the pure j − solvent 36. Then, the third scenario involves a non-interacting, or ideal gas, infinite dilute particle immersed in an otherwise real pure solvent, whose microstructural signature is a uniform distribution of the solute-solvent interactions described by Gij∞ ( PT ) = 0 . Thus, after invoking Eqn. (23) we find that67,
(∂P ∂x )
∞,IG _ i
i Tυ o j
( ) − (κ )
= kT ρ o2 G ojj 1+ ρ oj G ojj j
(
= κ
o,IG j
−1
o j
)
−1
A5
where IG _ i superscript denotes the ideal gas behavior of the i − solute in solution,
κ o,IG = 1 kT ρ oj is the ideal gas isothermal compressibility of the j − solvent at the j = 1+ ρ oj G ojj . In addition, from Eqns. (22) and (23) we prevailing conditions, and κ oj κ o,IG j arrive at the expression for the partial molar volume of the ideal gas solute at infinite dilution in the real solvent, i.e.,
(
)
υˆi∞,IG _ i (TP) = υ oj κ oj κ o,IG = kT κ oj j
A6
The expression A6 highlights the fact that υˆi∞,IG _ i (TP) > 0 , a behavior that characterizes an infinite dilute solute as either volatile 59 or weakly repulsive (i.e., at the border between repulsive and weakly attractive) with Gij∞ = 0 .
58
species given the fact that υˆi∞,IG > 0 simultaneously
Moreover, according to Eqn. (15) the coefficient of the composition
36
(
)
expansion for this case becomes kijIG _ i (TP) = G ojj υ oj = κ oj κ o,IG −1 , a negative quantity j since κ oj < κ o,IG that translates into kijIG _ i (TP) < 0 . j From the definition of Henry’s law constant, Hi, j (TPs ) = Ps exp ⎡⎣βµ iR,∞ (TPs )⎤⎦
A7
= kT ρ oj (TPs ) exp ⎡⎣βµ iR,∞ T ρ oj ⎤⎦
( )
where we have invoked the link between µ iR,∞ (TP) and µ iR,∞ (T ρ )
19
, and after the fact
that for an ideal gas i − solute µ iR,∞,IG _ i T ρ oj = 0 68, we obtain Hi,IGj _ i (TPs ) = kT ρ oj (TPs ) .
( )
As an illustration of the effect of the solute-solvent molecular asymmetry on gas solubility we plot in Figure 5 the isothermal liquid-phase equilibrium composition x1 resulting from the phase equilibrium condition fˆ1V (TPy1 ) = fˆ1L (TPx1 ) . For that purpose
fˆ1L (TPx1 ) is represented by the prescriptions around Henry’s law discussed above involving the H 2 − Ar system along the T = 100K isotherm according to the IE calculations of section §IV. This figure highlights the balance between the Poynting and the non-ideality corrections to Henry’s law solubility, a feature that is typically overlooked and lost during the regression of solubility quantities, whose simultaneous inaccurate representations for υˆi∞ (TP) and γ iHL (TPxi ) lead to a reasonable fit due to compensation of those corrections. This is precisely one of the scenarios where we need to have available a tool for the detection of and test for internal thermodynamic inconsistencies. APPENDIX B: Microscopic interpretation of the Margules/Porter parameter A(T , P) As we have already identified it, according to Eqn. (12), The A(T , P) parameter in the Krichevsky-Iliinskaya equation can be rigorously expressed as follows,
37
(
)
A (TP ) ≡ 0.5 ⎡⎣ Gii∞ + G ojj − 2Gij∞ υ oj ⎤⎦ TP
(35)
= 0.5 ρ oj Δ ij∞ where the Margules/Porter equation reads 21, g E (TPxi ) RT = A (TP) xi x j
B1
= xi ln γ iHL (TPxi ) + x j ln γ LR TPxi ) j (
with ln γ iHL (TPxi ) = ln γ iLR (TPxi ) − ln γ iLR,∞ (TP) and consequently, A = ln γ iLR,∞ (TP) , i.e.,
(
ln γ iHL (TPxi ) = −2 A (TP) xi − 0.5xi2
)
B2
ln γ LR TPxi ) = A (TP) xi2 j (
We have also indicated that ρ oj Δ ij∞ can be determined as the limiting slope of the limiting composition dependence of ln γ i (TPxi )
15,25
. Alternatively, we can express
A (TP ) in terms of other thermodynamic properties of the infinite dilute solute and the corresponding pure solvents as,
A (TP ) = − 0.5 ρ oj ⎡⎣ 2Bi⊗ (TP ) + υ oj − 2υˆ ∞j + kTκ oj ⎤⎦
B3
where Bi⊗ (TP ) is the osmotic second virial coefficient 69, so that, from Eqn. (16), ∞
Bi⊗ (TP ) = − 0.5Gii∞ = −2π ∫ hii∞ (TP)r 2 dr
B4
0
(
and the pure solvent isothermal compressibility κ oj = υ oj + G ojj
)
kT , with ρ oj = 1 υ oj . In
addition, after invoking Eqn. (22a), from B3-B4 we recover Eqn. (35), i.e.,
(
)
A (TP ) = − 0.5 ρ oj ⎡⎣ −Gii∞ + 2 υ oj − υˆi∞ + G ojj ⎤⎦ = − 0.5 ρ oj ⎡⎣ −Gii∞ + 2Gij∞ − G ojj ⎤⎦ = 0.5 ρ oj Δ ij∞
38
(35)
Note that A3 has been previously derived by Alvarez et al. 70, denoted b2 (TP ) in their Eqn. (7), as the coefficient for the linear composition dependence of Henry’s law activity coefficient for the i − solute, their Eqn. (6), i.e.,
ln γ iHL = b2 (TP) xi
B5
However, we must highlight, as we have discussed extensively elsewhere 14,15,26, that the linear composition dependence in A5 as well as in Eqn. (11) in Ref. 71 and Eqn. A1 of Ref. 72 are thermodynamically inconsistent within the framework of composition perturbation expansion approaches, i.e., their consistency requires a quadratic composition term as that given above by A1-A2 to satisfy simultaneously Gibbs-Duhem and Maxwell relations 73. Moreover, we should also note that b2 (TP ) does not account just for the solute-solute interactions (or correlations at infinite dilution) as frequently claimed in the literature 71,72, but also for the solvent-solvent, and most importantly, the solute-solvent interactions. This is a crucial observation in that even when we are in the presence of a non-interaction ideal gas solute there is a contribution from the solute-
(
)
solvent interactions, i.e., b2 (TP) = kijIG _ i (TP) = κ oj κ o,IG −1 resulting from the solutej solvent interaction asymmetry as discussed in Appendix A. APPENDIX C: Integral equation calculations of the gas solubility quantities The solution of the Ornstein-Zernike integral equation provides the three TCFI ’s ⎯ Gii∞ , G ojj , Gij∞ ⎯ and corresponding DCFI ’s ⎯ Cii∞ , C ojj , Cij∞ ⎯ for the infinite dilution system at the relevant state conditions, i.e., from the saturation condition to the pressure of interest along the chosen isotherm. For each state condition we determine the following quantities,
υˆi∞,L (TP) ← Eqn. (22a − b)
C1
kij (TP) ← Eqn. (15)
C2
39
Hi, j T , Pj,s = f i o T , Pj,s γ iLR,∞ T , Pj,s
(
)
(
)
(
= P φˆ T , Pj,s ∞ s i
(
)
C3
)
where the activity coefficient γ iLR,∞ T , Pj,s is calculated as follows (see Appendix B for
(
)
details), A (TP) = 0.5ρ oj Δ∞ij
C4
= ln γ iLR,∞ (TP)
while the fugacity coefficient f i o T , Pj,s = Pj,s exp ⎡⎣βµ ior T , Pj,s ⎤⎦ results from the density integral,
(
)
(
)
⎡ ρ( P ) ⎤ µ ior T , Pj,s = −β −1 ⎢ ∫ 0 j ,s Ciio d ρ + ln z T , Pj,s ⎥ ⎣ ⎦
(
)
(
µ ior T , Pj,s = −β −1
(
)
{∫ (
ρ Pj ,s
0
)
C5a
)⎡ o o ⎤ ⎣Gii 1+ ρGii ⎦ d ρ + ln z T , Pj,s
(
(
)
(
)}
C5b
)
where the compressibility z T , Pj,s can be written as (e.g., see Appendix C of Ref. 3)
( )
ρ Pj ,s
ρCiio d ρ
(
)
0
(
)
( )⎡ o o ⎤ ⎣ρGii 1+ ρGii ⎦ d ρ 0
z T , Pj,s = −ρ −1 ∫ z T , Pj,s = −ρ −1 ∫
ρ Pj ,s
(
C6a
)
C6b
for the supercritical state conditions T * = kT ε ii and P* = Pj,sσ ii3 ε ii . Finally, the fugacity
fˆiV (TPyi ) is determined in terms of the truncated virial equation of state, the consistent low-density counterpart of the second-order truncated composition expansion of the partial molar quantities for liquid mixtures used throughout this analysis (see §IV-B of Ref. 26 for details), i.e.,
fˆiV (TPyi ) = yiφˆiV (TPyi ) P = P − Pj,s exp ⎡⎣β P Bii − y 2j δij ⎤⎦
(
)
(
)
40
C7
where δij (T ) = lim Δ∞ij (TP) = Bii (T ) + B jj (T ) − 2Bij (T ) , whose individual contributions can ρ→0
be determined from the available tabulation, i.e., Bij* kT ε ij = 3Bij 2πσ ij3 , for the
(
)
Lennard-Jones fluid 54. APPENDIX D: Simultaneous solution of the gas-solubility phase equilibrium While we have focused almost exclusively on the i − solute phase equilibrium equations for the current analysis, those for the j − solvent can also help in that task. In fact, the counterparts of Eqn. (1) for the solvent are, fˆjV (TPyi ) = fˆjL (TPxi )
D1
y jφˆVj (TPyi ) = x jφˆ jL (TPxi )
where we kept the i − solute mole fraction as the descriptor of the phase compositions. By invoking the second-order composition expansion 26 to both phases, the partial molar fugacity of the j − solvent in each phase becomes, fˆjV (TPyi ) = P (1− yi ) exp β ⎡⎣ B jj P − Pj,s − Pδij yi2 ⎤⎦
}
D2
fˆjL (TPxi ) = Pj,s (1− xi ) exp ⎡⎣βυ oj P − Pj,s + 0.5kij xi2 ⎤⎦
D3
{
(
)
(
)
(
)
where υ oj is the molar volume of the pure j − solvent and γ LR TPxi ) = exp 0.5kij xi2 as j ( derived in Appendix B. Thus, after solving the phase equilibrium condition, D1, we define the plotting function ℑγ xi2 = ln ⎡⎣ fˆjV (TPyi ) Pj,s (1− xi )⎤⎦ − βυ oj P − Pj,s
( )
(
)
whose
limiting composition slope becomes,
lim ⎡⎣∂ℑγ xi2 xi →0
( )
∂xi2 ⎤⎦ = 0.5kij TPj,s
(
T
)
D4
Note that if kij (TP) exhibits weak pressure dependence, the limiting condition D4 can be replaced by ⎡⎣∂ℑγ xi2
( )
∂xi2 ⎤⎦ ≅ 0.5kij TPj,s . In Figure 6 we illustrate the well-behaved T
(
)
41
( )
nature of ℑγ xi2 and the limiting condition for the H 2 (i) − Ar( j) system from the IE calculations reported in section §IV. However, we note that despite its linear appearance
ℑγ xi2 displays a very small curvature such that lim ⎡⎣∂ℑγ xi2 xi →0
( )
( )
( ) ( ) (TP ) = 5.04 .
average xi2 − slope of ℑγ xi2 is Δℑγ xi2 given by D4, i.e., 0.5kij
∂xi2 ⎤⎦ ≅ 5.01 while the T
Δxi2 ≅ 4.86 in comparison with the true value
j,s
APPENDIX E: Derivation of the novel expressions for K i∞ (TPs ) , L∞i, j (TPs ) , and their limiting behaviors Starting the phase equilibrium condition Eqn. (1) and the alternative expression for Henry’s law constant, Eqn. (47), we have that,
K i∞ (TPs ) = lim ( yi xi ) xi ,yi →0 P→Ps
= φˆi∞,L (TPs ) φˆi∞,V (TPs )
E1
= Hi,ISj (TPs ) Psφˆi∞,V (TPs ) Moreover, for the vapor phase away from the j − solvent critical conditions, we have that
φˆi∞,V (TPs ) = exp ⎡⎣β Ps Bii − δij ⎤⎦ 26, i.e.,
(
)
{ = exp {0.5β P lim ⎡⎣2G
} (TP ) − G (TP )⎤⎦}
φˆi∞,V (TPs ) = exp −0.5β Ps lim ⎡⎣Gii∞ (TPs ) − Δ∞ij (TPs )⎤⎦ ρ→0
s ρ→0
∞ ij
s
o jj
E2
s
and, after invoking Eqns. (48) and C5-C7, Eqns. E1-E2 lead to the sought novel interpretation for K i∞ (TPs ) ,
{
} E3a
K i∞ (TPs ) = ⎡⎣ f i o (TPs ) Ps (T )⎤⎦exp 0.5kij (TPs ) − β Ps lim ⎡⎣Gij∞ (TPs ) − 0.5G ojj (TPs )⎤⎦ ρ→0
= ⎡⎣ f i o (TPs ) Ps (T )⎤⎦exp 0.5kij (TPs ) − β Ps ⎡⎣ B jj (T ) − 2Bij (T )⎤⎦
{
}
or in the equivalent form, after invoking δij (T ) = Bii (T ) + B jj (T ) − 2Bij (T ) , as follows:
42
K i∞ (TPs ) = ⎡⎣ f i o (TPs ) Ps (T )⎤⎦exp 0.5kij (TPs ) − β Ps ⎡⎣δ jj (T ) − Bii (T )⎤⎦
{
}
E3b
where kij (TPs ) = ρ oj Δ∞ij is defined by Eqn. (15) and f i o (TPs ) can be expressed by its molecular counterparts in terms of either the DCFI , Ciio (TPs ) , or the TCFI , Giio (TPs ) , for the pure i − solute at the prevailing saturation conditions, i.e.,
f i o TPj,s = Pj,s exp ⎡⎣βµ ior TPj,s ⎤⎦ ⎡ ρ( Pj ,s ) o ⎤ = Pj,s exp ⎢− ∫ Cii d ρ − ln z TPj,s ⎥ 0 ⎣ ⎦ ρ P ⎛ ( j ,s ) o ⎞ ⎡ ρ( Pj ,s ) o ⎤ = κ io,IG TPj,s ⎜ ∫ ρ Cii d ρ ⎟exp ⎢− ∫ Cii d ρ ⎥ ⎣ 0 ⎦ ⎝ 0 ⎠
(
)
(
)
(
(
f i o TPj,s = κ io,IG TPj,s
(
)
(
){ ∫
)
E4a
)
⎡ ρ( Pj ,s ) ⎡ o ⎤ ( )⎡ o ρ Gii 1+ ρGiio ⎤⎦ d ρ exp ⎢− ∫ Gii 1+ ρGiio ⎤⎦ d ρ ⎥ E4b ⎣ ⎣ 0 ⎣ 0 ⎦ ρ Pj ,s
(
)
}
(
)
where κ io,IG TPj,s is the ideal gas isothermal compressibility of the pure i − solute at the
(
)
j − solvent saturation conditions. Consequently, the corresponding Ostwald coefficient L∞i, j (TPs ) becomes 65,
L
∞ i, j
(TP ) = s
Ps (T ) ρio,L (TPs ) f i o (TPs ) ρio,V (TPs )
{
}
exp β Ps ⎡⎣ B jj (T ) − 2Bij (T )⎤⎦ − 0.5kij (TPs )
E5a
or in the equivalent form,
L∞i, j (TPs ) =
Ps (T ) ρio,L (TPs ) f i o (TPs ) ρio,V (TPs )
{
}
exp β Ps ⎡⎣δij (T ) − Bii (T )⎤⎦ − 0.5kij (TPs )
E5b
Note also that if the i − solute behaves as an ideal gas particle then, from Eqns. E1, C5b, and C6b, we have that the vapor-liquid distribution coefficient of an ideal gas
i − solute in a real j − solvent reads,
43
K i∞,IG _ i (TPs ) = z o,V TPs ) z o,L TPs ) j ( j ( =
ρ o,L Pj,s j ρ o,V j
( )∫ (P ) ∫ j,s
ρ oj ,V Pj ,s
( )⎡ o o ⎣ρGii 1+ ρGii ρ oj ,L ( Pj ,s ) ⎡ρG o 1+ ρG o ii ⎣ ii 0
( (
0
)⎤⎦ d ρ )⎤⎦ d ρ
E6
while the corresponding Ostwald coefficient becomes, _i L∞,IG (TPs ) = z o,Lj (TPs ) ρ o,Lj Pj,s i, j
( )
z o,V TPs ) ρ o,V Pj,s j ( j
( )
E7
=1
Here we note that the behavior of the vapor-liquid distribution coefficient K i∞ (TPs ) , Eqns. E1 and E6, obeys the following limiting condition,
lim K i∞ (TPs ) = lim K i∞,IG _ i (TPs ) = 1
T →Tc , j Ps , j →Pc , j
T →Tc , j Ps , j →Pc , j
E8
i.e., regardless of the type of i − solute. Likewise, from Eqn. E5,
lim L∞i (TPs ) = 1
E9
T →Tc , j Ps , j →Pc , j
Finally, for the i − solute in the ideal solution defined by the Lewis-Randall reference we have that γ iLR (TPxi ) = φˆi (TPxi ) φio (TP) = 1 therefore, from E1 we conclude that, K i∞,IS (TPs ) = φio,L (TPs ) φio,V (TPs )
E10
= f i o,L (TPs ) f i o,V (TPs )
where f i o,L (TPs ) is given by E4b, while f i o,V (TPs ) can be obtained by invoking the density limiting behavior Giio (TP) → −2Bii (T ) in E4b. Obviously, as expecteded
lim K i∞,IS (TPs ) = 1
E11
T →Tc , j Ps , j →Pc , j
44
FIGURE LEGENDS ∞
Figure 1: Isothermal-pressure dependence of the thermodynamic quantities (∂P ∂xi )Tυ o , j
υˆi∞ , υ oj , and κ oj (left axis), as well as e
β
P
∫ Ps υˆi∞ dP
and e
βυˆi∞ ( P−Ps )
(right color-
coordinated axes) for the H 2 (i) − Ar( j) system as described by the integral equation calculations involving the Lennard-Jones model along the T = 100K isotherm. Figure 2: Isothermal-pressure dependence of the Poynting correction PC ⎡⎣TP,υˆi∞,L (TP) ⎤⎦ Tait with representative values of υ oj (TPo ) and κ oj (TPo ) for infinitely dilute LennardJones systems as described by integral equation calculations along the T = 100K isotherm.
(
Figure 3: Isothermal-composition dependence of the modeling function ℑα xi − 0.5xi2
)
for the H 2 (i) − Ar( j) system as described by the integral equation calculations involving the Lennard-Jones model along the T = 100K isotherm. Figure 4: Isothermal-pressure dependence of the modeling function ℑ β ( P) = ln ⎡⎣ fˆ1V (TPy1 ) x1 ⎤⎦ and its limiting slope (left red) and the corresponding
(
)
behavior of kij (TP) = Gii∞ + G ojj − 2Gij∞ υ oj (right blue) for the H 2 (i) − Ar( j) system as described by the integral equation calculations involving the LennardJones model along the T = 100K isotherm. Figure 5: Isothermal liquid-phase equilibrium composition x1 resulting from the phase equilibrium condition fˆ1V (TPy1 ) = fˆ1L (TPx1 ) when fˆ1L (TPx1 ) is represented by three distinct prescriptions around Henry’s law for the H 2 (i) − Ar( j) system as described by the integral equation calculations involving the Lennard-Jones model along the T = 100K isotherm. Figure 6: Isothermal-pressure dependence of the modeling function P ℑγ = ln ⎡⎣ fˆjV (TPyi ) Ps x j ⎤⎦ − ∫ ⎡⎣υ oj (TP) RT ⎤⎦ dP and its limiting slope D4 for Ps (T ) the H 2 (i) − Ar( j) system as described by the integral equation calculations involving the Lennard-Jones model along the T = 100K isotherm.
45
xe(integral) c17=c10/(1.6530*exp(c6/0.8756)) c16=c10/(1.6530*exp(c7/0.8756))
c15=c10/1.6530 fraction, y 1
P ⎡ ⎤ exp ⎢β ∫ υˆi∞ dP⎥ or exp ⎡⎣βυˆi∞ ( P − Ps )⎤⎦ ⎣ Ps ⎦
0.1
0.08 0.06 0.04 0.02
0 0.1
reduced quantity
0.08
0.1
0.06
46
0.02 0.04 0.06 0.08
1
1.05 1.3
0.04
0
κ
o j
1.1
υ oj
1.15 1.2
1.25
υˆi∞ i Tυ o j ∞
1.35
0.02
reduced pressure, P σ 3jj ε jj -0.02 0 0.5 1 1.5 2 2.5 3
(∂P ∂x )
3.5
xe(integral) 3 ressure, Pσ 22 ε 22 c17=c10/(1.6530*exp(c6/0.8756)) c16=c10/(1.6530*exp(c7/0.8756)) c15=c10/1.6530 e fraction, x1
3 Pσ 22 ε 22
0 0.1
Figure 1
0.1
0.08
84.6 78.6
υ cm mol
72.6
)
66.6 57.6 51.6 45.6 39.6
2
40
1.8
20
1.6 1.4
80
100
1
1.2
60
1.4
40
1.6
1.8
2.2
20
2
2.4
0
κ oj = 8.3×10−4 atm
0
T = 100K − Po = 4.4atm
1.2 1
120
pressure difference(atm), P − Po
80
(
3
60
o j
2.2
2.6
Poynting correction, PC
2.4
47
pressure (atm)
2.6
100
120
Figure 2
ℑ = ln ⎡ fˆ V TPy1 ) x1 ⎦⎤ − 1 ( ⎣ y2=p2/p α
1
∫
P Ps (T )
⎡ ∞,L ⎤ ⎣υˆ (TP) RT ⎦ dP
1 xe(integral) c17=c10/(1.6530*exp(c6/0.8756)) c16=c10/(1.6530*exp(c7/0.8756)) c15=c10/1.6530
0.1
0.8
0.08
y2=p2/p
y2=p2/p
i
(
( x − 0.5x )
xi →0
2 i
lim ℑα xi − 0.5xi2 ≅ 0.502
Eqn. (14) with kij=0
Eqn.(14) (30) Eqn. (14)
0.06 0.04
Eqn. (14)phase Henry 's law Eqn. (14) HL liquid behavior Eqn. (14) with kij=0
)
Eqn. (14)with withwith kij=0k = 0 Eqn.(14) Eqn. (14) kij=0 Eqn. (30) 12 Eqn. (30) Eqn.(30) Eqn. (30) HL liquid phase behavior HLliquid liquidphase phasebehavior behavior HL
)
0.6
(
)
0.6 0.6 3 0.4 reduced pressure, P σ ε 22 22 0.06
0.08
∞,L ⎡ ˆ υ ∫ Ps (T ) ⎣ i (TP) RT ⎤⎦ dP
(
lim ℑα xi − 0.5xi2 ≅ 0.502
0.02 0.4 0.04 0.4 0.4 0.2
P
)
(−)lim ⎡⎣∂ℑα ∂ xi − 0.5xi2 ⎤⎦ ≅ 10.021 xi →0
0
(−)lim ⎡⎣∂ℑα ∂ xi − 0.5xi2 ⎤⎦ ≅ 10.021 xi →0
0.6
Eqn. (14)
1 0.811
0.8 0.8 0.6 0.8
(
0
y2=p2/p y2=p2/p
1
0.4 0.2
1
0.8
0.6
ℑα = ln ⎡⎣ fˆiV (TPyi ) xi ⎤⎦ −
)
y2=p2/p
0.1
0.08
0.06 0.4
0.2
0
xe(integral) c17=c10/(1.6530*exp(c6/0.8756)) c16=c10/(1.6530*exp(c7/0.8756))
0.06
xi − 0.5xi2
xi →0
c15=c10/1.6530 vapor − phase mole fraction, y 1
0.08
(
0.04 0.02 0 -0.2
0.02
0.04
0.1
0.08 0.08 0.08 0.06
0.02
0.06 0.06 0.06 0.04
0 0.1
0.2 0.2 0.2 0 0 0 0 0
0.1 0.1 0.1 0.08 1
48
0.04
0.02
0 0.1
xe(integral) 3 y2=p2/p reduced pressure, P σ ε c17=c10/(1.6530*exp(c6/0.8756)) 22 22 c16=c10/(1.6530*exp(c7/0.8756)) c15=c10/1.6530 liquid − phase mole fraction, x1
0.1
3 Pσ 22 ε 22
0.08
0 0.1
0.02 0.04 0.06 0.08 0.1
, y1
(integral) 7=c10/(1.6530*exp(c6/0.8756)) 6=c10/(1.6530*exp(c7/0.8756)) 5=c10/1.6530
0.04 0.04 0.04 0.02
Figure 3
0.06
0.08
0.02 0.02 0.02 0 0.1 00 0
ℑ β (TPxi ) = ln ⎣⎡ fˆiV (TPyi ) xi ⎦⎤ + G ojj − 2Gij∞ υ oj
)
ln ⎣⎡ fˆiV (TPyi ) xi ⎦⎤
lim ⎣⎡∂ℑ β ∂P⎦⎤ ≅ −3.0 T
0.2
0.3
0.4
0.5
≅ −3.35
lim ⎡⎣∂ℑα ∂P⎤⎦ = υˆi∞,L (TPs ) RT − kij xi →0 P→Ps
ℑ = ln ⎡⎣ fˆiV (TPyi ) xi ⎤⎦
lim ⎡⎣∂ℑ β ∂P⎤⎦ ≅ −3.0 T
xi →0 P→Ps
(
o jj
0.1
)
0
0
0.02
kij (TP) = G + G − 2Gij∞ υ oj ∞ ii
0.3
reduced solute pressure, Pi σ 3jj ε jj
⎤ ⎦
i
reduced solute pressure, Pi σ 3jj ε jj
ℑ β (TPxi ) = l
0.04
xi →0 P→Ps
0.06 3
0.08
lim ⎡⎣∂ℑ β ∂P⎤⎦ ≅ −3.0 T
0.2
xi →0 P→Ps
lim ⎣⎡∂ℑ β ∂P⎦⎤ ≅ −3.0 T
x1→0 P→Ps
∞ ii
ℑ β ( P) = ln ⎡⎣ fˆiV (TPyi ) xi ⎤⎦
reduced solute pressure, P1 σ 322 ε22
ℑ β = ln ⎣ f i (TPyi ) xi ⎦
0.5
) 0.1
T
(G
lim ⎡⎣∂ℑ β ∂P⎤⎦ ≅ −3.0
0.4
lim ⎣⎡∂ℑ β ∂P⎦⎤ ≅ −3.0 T lim ⎡⎣∂ℑ ∂P⎤⎦ ≅ −3.0 T
0
reduced solute pressure, Pi σ 3jj ε jj reduced solute pressure, P1 σ 322 ε22
49
ℑ = ln ⎡⎣ fˆ1V (TPy1 ) x1 ⎤⎦
∂P⎦⎤ ≅ −3.0 T
ℑ β (TPxi ) = kij (TP) = Gii∞ + G ojj − 2Gij∞ υ oj
Figure 4
(
xi →0 P→P
⎤ ⎦T ≅ −3.0
xi →0 P→Ps
x1→0 P→Ps
0.02
Eqn.(14) (30) Eqn. (14) Eqn. (14)phase Henry 's law Eqn. (14) HL liquid behavior Eqn. (14) with kij=0
Eqn. (14) with kij=0
0
Eqn. (14)
Eqn.Eqn.(14),k (14)with withkij=0 kij=0ij = 0 Eqn. (14) Eqn. (30) Eqn. (30)phase behavior Eqn.(30) Eqn. (30) HL liquid
-0.2
0.2 0.4
0.6 0.6 0.6 0.4
1 0.811
1
0.8 0.8 0.6 0.8
0.04
0.06
0.08
0.6 0.8
0.2 0.2 0.2 0
0.06 0.06 0.06 0.04 0.04 0.04 0.04 0.02
1
0 0 0 0
0.02
y2=p2/p
0.04
0.06
0.08
0 0 0
0.02 0.02 0.02
0.04 0.04 0.04
0.06 0.06 0.06
0.08 0.08 0.08
c15=c10/1.6530 vapor − phase mole fraction, y 1
50
0.08 0.08 0.08 0.06
0.2 0.2 0.2 0
0.02
y2=p2/p y2=p2/p y2=p2/p
HLliquid liquidphase phasebehavior behavior HL
0
xe(integral)
0.4 0.4 0.4 0.2
0.1 0.1 0.1 0.08
0.4 0.4 0.2
3 y2=p2/p reduced pressure, Pσ 22 ε 22 c17=c10/(1.6530*exp(c6/0.8756)) c16=c10/(1.6530*exp(c7/0.8756)) c15=c10/1.6530 liquid − phase mole fraction, x1
HLliquid liquidphase phasebehavior behavior HL
reduced pressure, Pσ 3jj ε jj
0.02
0.1
xe(integral) c17=c10/(1.6530*exp(c6/0.8756)) c16=c10/(1.6530*exp(c7/0.8756))
0.1
0.1
Eqn.Eqn.(14),k (14)with withkij=0 kij=0ij = 0 Eqn. (14) Eqn. (30) Eqn. (30)phase behavior Eqn.(30) Eqn. (30) HL liquid
c15=c10/1.6530 on, y 1
1 y2=p2/p
0.08
0.1 0.1 0.1 0.08
0.08 0.08 0.08 0.06
0.06
0.6 0.6 0.6 0.4
Eqn.(14) (30) Eqn. (14) Eqn. (14)phase Henry 's law Eqn. (14) HL liquid behavior Eqn. (14) with kij=0
0.1
0.08
0.8
0.8 0.8 0.6 0.8
Eqn. (14) with kij=0
0.04 0.04 0.04 0.02
0.6
Eqn. (14)
1 0.811
0.04
0.06
0.08
0.06
0.04
0.02
0 0.1
y2=p2/p y2=p2/p y2=p2/p
1
0.1
vapor − phase mole fraction, yi
0.06 reduced pressure, Pσ 3jj ε jj 0.06 0.06 vapor − phase liquid − phase mole fraction, x 0.04 i 0.4
0.4
0.08
y2=p2/p
xe(integral) c17=c10/(1.6530*exp(c6/0.8756)) c16=c10/(1.6530*exp(c7/0.8756))
0.06
0 0.1
liquid − phase mole fraction, xi
/p
0.04
0.02 0.04 0.06 0.08 0.1
xe(integral) c17=c10/(1.6530*exp(c6/0.8756)) c16=c10/(1.6530*exp(c7/0.8756))
c15=c10/1.6530 se mole fraction, y 1
2
3 duced pressure, Pσ 22 ε 22
Figure 5
0.02 0.02 0.02 0 0.1 00 0 0.1 0.1 0.1
x12
P
lim ℑγ x12 ≅ 5.0
( )
x1→0
⎡ o,L ⎤ ⎣υ2 (TP) RT ⎦ dP Ps (T )
∫
ℑγ = ln ⎣⎡ fˆ2V (TPy1 ) Ps x2 ⎦⎤ −
lim ⎡⎣∂ℑγ xi2
xi →0
51 2 i
T
xi2
lim ⎡⎣∂ℑγ xi2
( ) ∂( x )⎤⎦ xi →0
∫
2 i
T
≅ 5.01
⎡υ o (TP) RT ⎤ dP ⎦ Ps (T ) ⎣ j
P
( ) ∂( x )⎤⎦
ℑγ = ln ⎡⎣ fˆjV (TPyi ) Ps x j ⎤⎦ −
Figure 6
≅ 5.01
xi2
