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Ind. Eng. Chem. Res. 2008, 47, 5758–5760
CORRESPONDENCE A New Reference State for the Calculation of Activity Coefficients: Comment on the Paper “A Molecular Theory of the Activity Coefficient and Their Reference Fugacities at the Supercritical State” Jaime Wisniak*,† and Hugo Segura‡ Department of Chemical Engineering, Ben-Gurion UniVersity of the NegeV, Beer-SheVa 84105, Israel, and Departamento de Ingenierı´a Quı´mica, UniVersidad de Concepcio´n, Concepcio´n 4070405, Chile Martz and Lee1 have recently proposed a rigorous new definition of the reference state for the fugacity of the pure component which, being based on true equilibrium states predicted from statistical mechanics, avoids the indefiniteness problems that occur when calculating the activity coefficient of a supercritical component in a fluid mixture. As we will review below, the problem posted by Martz and Lee is much general than the particular situation they have analyzed, because it also affects determination of the activity coefficient of subcritical vapor-liquid equilibrium (VLE) experiments, which is a key property for assessing the consistency of the data.2 The general definition of the activity coefficient γ of component i in the π-phase is3 γπi )
ˆf π i
where ˆfπi is the effective fugacity of component i in the π-phase and fiπ,0 the fugacity of the reference state. Assuming, as usual, the Lewis and Randall (LR) approach,4 the reference state for calculating activity coefficients is the pure component in the same state of aggregation at the pressure (P) and temperature (T) of the mixture. In addition, at pressures close to atmospheric pressure, the liquid phase is assumed to be incompressible, whereas the behavior of the pure gas or its mixtures may be accurately described by the virial equation of state truncated at the second virial coefficient (B): B V
(2)
where z and V correspond to the compressibility factor and volume of the vapor phase, respectively. According to the approach previously mentioned, the reference fugacities of pure vapors and liquids are given by the following relations:3 f
V,LR )P i
[
( )
Bii P exp RT
) P0i exp (Bii - ViL) f L,LR i
) γV,LR i
(1)
xi fiπ,0
z)1+
are strictly Valid and accurate when applied to globally stable phases at the P, T condition of the mixture. However, as shown in Figure 1, if we constraint our attention to any equilibrium state of the simplest VLE diagram of a binary mixtures (tieline AB in Figure 1a), we conclude that the volatile pure component does not yield global stability in the liquid phase, nor is the pure nonvolatile component globally stable in the vapor phase. In fact, according to the P-T projection of the pure components shown in Figure 1b, the components of a binary mixture become stable liquids aboVe curve CD (the vapor pressure of the volatile component), although they are stable vapors below line EF (the vapor pressure of the nonvolatile component). From eqs 1–4, we can deduce the following relationships for the LR-based activity coefficients of vapor and liquid phases:
PViL P0i + RT RT
(3)
]
(4)
where Pi0 and ViL are the vapor pressure and the molar volume of the pure liquid, respectively, at the temperature and pressure of the solution, and Bii corresponds to the second virial coefficient of component i at temperature T. Equations 3 and 4 * To whom correspondence should be addressed. Fax: (972)-8-6472916. E-mail address:
[email protected]. † Ben-Gurion University of the Negev. ‡ Universidad de Concepcio´n.
) γL,LR i
ˆf V i yi fV,LR i
ˆf L i
) L,LR
xif i
ˆf L i xiP0i
P ) exp (Bji - Bii) RT
[
[
exp - (Bii - ViL)
]
PViL P0i RT RT
(5)
]
(6)
where xi and yi are the equilibrium concentrations of the liquid j i is the partial second virial and vapor phase, respectively, and B j i ) 2∑yjBji - ∑j∑iyjyiBji). Using the condition for coefficient (B vapor liquid equilibrium (VLE) in a multicomponent mixture,3 fˆ Vi ) ˆf Li
(7)
the activity coefficient of the liquid phase can be determined from the VLE data, according to the following standard equation:3 ln
γL,LR ) ln i
( ) yiP
xiP0i
(Bii - VLi )(P - P0i ) + ln γV,LR + i RT
(8)
Because a gas mixture at subcritical conditions has a tendency to behave like a regular solution,5 for a binary mixture, eq 5 simplifies to the well-known relation3
( )
Pδij [i * j;j ) 1, 2;i ) 1, 2] (9) RT where δij ) 2Bij - Bii - Bjj. Equation 8 indicates that the deviation from ideality of a solution in the liquid phase is due to the strange combination of three factors. The first one defines the equilibrium condition, and the second one defines the difference between the deviation from gas ideality of pure component i (Bii) and the value of its molar liquid volume (ViL). The third factor defines the nonideality of the vapor phase of the mixture. Equation 8 has the aestetic advantage that, when lnγV,LR ) yj2 i
10.1021/ie8006566 CCC: $40.75 2008 American Chemical Society Published on Web 07/03/2008
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
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Figure 1. Phase equilibrium diagram and interpretation of the phase stability of pure components on a pressure-temperature (P-T) projection: (0) mixture phase equilibrium point and (O) pure-component phase equilibrium point.
all the deviations from ideality are neglected, it predicts that the activity coefficients of the liquid phase are equal to 1 (that is, the solution is ideal). However, as we mentioned previously, for the ordinary VLE of mixtures under isothermal conditions, at least one of the pure phases required for calculating the liquid phase activity coefficient of each component (liquid or vapor) does not exist as a globally stable system under the conditions of the mixture. The indefiniteness problem mentioned by Martz and Lee1 also appears when calculating the value of the activity coefficient for VLE under isobaric conditions. In every situation of this nature, and as occurs for the case of isothermal systems, the volatile component cannot exist as a pure liquid under the equilibrium conditions; it will always be in the superheated liquid state (not necessarily supercritical). Similarly, the nonvolatile component will be in the subcooled vapor state. This circumstance originates from the arbitrary selection of the standard state for fugacities indicated previously. Most papers that report VLE data under isothermal or isobaric conditions overlook the problem and “calculate” the value of ViL under the assumption that the liquid is incompressible, so that the actual value of ViL can be approximated by the value of ViL in the saturated state. Hence, they use experimental values for this property or they estimate it using the Rackett relation6 or one of its many modifications. In addition, for the case of the vapor phase, the fugacity relations obtained from the virial equation of state (EOS) model are extrapolated for predicting the properties of subcooled vapors. The practical consequence is that the values of the activity coefficients γiL are incorrectly reported, from a rigorous point of view. We can ask the following question: how bad are they? Our experience with the determination of VLE data at approximately atmospheric pressure for a very large number of binary systems indicates that the second term in eq 8 contributes no more than 5% of the total value of γLi for the case of a nonassociating vapor phase, and this much only in the diluted range of the component. For larger concentrations, the contribution decreases to
