Ind. Eng. Chem. Res. 1991, 30, 2006-2011
2006
Relative Measurements of Activity Coefficients at Infinite Dilution by Gas Chromatography Hasan Orbeyt and Stanley I. Sandler* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
In this paper, a relative measurement technique for infinite dilution activity coefficients by gas-liquid chromatography (GLC) is introduced. The method makes use of an internal standard whose infinite dilution activity coefficient in the solvent loaded on the column is known, and allows simultaneous, rapid, and accurate measurements of infinite dilution activity coefficients. The technique is particularly suitable for evaluation of activity coefficients of solutes in volatile solvents; such systems are difficult to investigate by the conventional GLC method due to solvent stripping. Some experimental results obtained by the method proposed here are presented and compared with those obtained earlier by other methods. As part of this work, the governing equations for the GLC technique taking into account the volatility of the solvent were developed.
Introduction In the design of separation processes and the operation of separation units in the chemical and related industries, the understanding of phase behavior of the species involved is essential. In many cases, to obtain complete equilibrium data is prohibitively time-consuming and expensive, and thus models, parameters of which are obtained from a minimal number of experimental measurements, are employed. In this context, the evaluation of infinite dilution activity coefficients is particularly important: normally the greatest deviation from the ideal solution behavior is exhibited in this region. By definition the infinite dilutioq activity coefficient is yi(lim x i 0) = 7: = (lim x i O ) f i / x A ,whereAziis the mole fraction of species i in the liquid mixture, f i is the fugacity of component i in the liquid mixture, and 6is the standard-state fugacity (Tiegs et al., 1986). Infinite dilution activity coefficients can be used to obtain parameters in activity coefficient models which can then be used to model low pressure equilibrium phenomena. Moreover, at least for moderately nonideal systems, infinite dilution activity coefficient data can be used to evaluate binary interaction parameters of equation of state models (Pividal, 1991). There are numerous methods for measuring infinite dilution activity coefficients. Some of the classical ways are ebulliometry, static cell measurements, liquid-liquid chromatography, and gas-liquid chromatography. These and other techniques have recently been reviewed by Tiegs et al. (1986). Here we consider the gas-liquid chromatographic technique (GLC). The idea of obtaining solution properties from GLC measurements was first employed by Littlewood et al. (1955),and initially the technique was used to evaluate the infinite dilution activity coefficient of a volatile solute in a practically nonvolatile solvent which acts as the stationary phase in the column. Later the technique was extended with some modifications to study cases in which the solvent was also somewhat volatile (Pecsar and Martin, 1966; Barker and Hilmi, 1967; Shaffer and Daubert, 1969; Chatterjee et al., 1972; Eon and Guichon, 1973; Barr and Newsham, 1987). This application of the GLC technique is particularly interesting for chemical engineering applications since in most cases of industrial concern all species are to some extent volatile. However, the application of the GLC method to the determination of infinite dilution activity coefficients of
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* Author to whom
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correspondence should be addressed.
Permanent address: Chemical Engineering Department, Middle East Technical University Ankara, Turkey. 0888-5885/91/2630-2006$02.50/0
solutes in volatile solvents presents a number of problems, especially that the amount of solvent in the column changes continuously due to stripping by the carrier gas. Since knowledge of the exact amount of solvent in the column is essential for the accurate determination of the infinite dilution activity coefficient, this necessitates continuous monitoring of the weight of the solvent in the column and some sort of averaging of the initial and final masses of the solvent. In this work, starting from a suggestion made almost 30 years ago by Freeguard and Stock (1962), we propose a relative measurement technique for infinite dilution activity coefficients with gas chromatography that eliminates some of the difficulties of the conventional GLC technique when employed with volatile solvents. This method, which involves the use of an internal standard whose infinite dilution activity coefficient in the solvent is known, greatly enhances the GLC method for rapidly and accurately determining infinite dilution activity coefficients with relatively simple equipment. We present the theory of this new method, report some experimental results obtained, and compare these results with those obtained earlier by other methods. As part of this work, we develop the equations for the GLC technique which include the volatility of the solvent phase. Special cases of o w very general equations have been obtained by previous investigators (Desty et al., 1962; Everett, 1965; Barker and Hilmi, 1967).
Theory of Elution Processes The elution of a substance from a chromatographic column is assumed to occur with a peak velocity that is proportional to local carrier gas velocity. The proportionality constant 5 is the retention ratio, which is taken to be equal to the fraction of the total number of solute molecules that are in the gas phase at equilibrium in a differential volume element in the column. That is
t=-
nf nf
+ n\
and the local distribution coefficient K1is defined as where n , is the number of moles of solute, u1and d are the volumes of the liquid and gas phases per unit length of the column, and superscripts 1 and g represent liquid and gas phases respectively. The ratio u ’ / u g is a characteristic of column packing [neglecting small possible dependencies on local pressure and concentration of solute (Cruickshank et al., 196611. 0 1991 American
Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991 2007 Combining (1)and (2) we have [ = 1/(1 + kpl/ug)
(3)
If the mean gas velocity averaged over the column cross section is u, then the mean speed of elution peak, w ,is w = [U = u / ( l + K ~ u ' / u=~U) U ~ / ( U +~ Kid) (4) where the term uug is the local volumetric flow rate of the carrier gas in the column, F(1). Since w = dl/dt, where dl is a differential length of column, one may write ( ~ + g Kid) dl = F(1) d t (5) where F(1) dt is the volume of the carrier gas that passes out of the column while the peak migrates the distance dl in the time dt. Due to the uncertainties in defining ug, it has been suggested that instead of F(1) dt, the "volume of the carrier gas which passes the elution peak during its progress from 1 to 1 + dl" (Cruickshank et al., 1966) be used. This quantity is
dVN = F(I) dt - U B dl = Kid dl
(6) Even though the distribution coefficient Kl is a function of local pressure and a pressure gradient exists in the column, all descriptions of the GLC method developed to date assume that its value can be calculated at an average column pressure and can be kept constant during the integration of (6). Thus (7)
where V, is the net retention volume of the carrier gas, v' is the amount of solvent on the column, and KL is the distribution coefficient evaluated at the average column pressure over the column length L. Note that in (7) the gas volume is expressed at column conditions and is related to retention volume measured at the column exit
to the evaluation of infinite dilution activity coefficients from GLC measurements considering the volatility of the solvent phase and clarify the assumptions involved. Assuming that equilibrium is established between the gas and the liquid phases during elution, we have the following expression for the solute (l), neglecting the very small pressure dependence of saturated liquid molar volume of the pure solute, E?, x l T 1 ( ~ p , x l$) ~ ( T , P " exp[vi(P)~I PB,)/W 11 = Y A P (10) In (lo), x1 is the mole fraction of solute in the liquid phase, y1 is the activity coefficient of the solute at the column temperature T and average column pressure P, $7 is the fugacity coefficient of pure solute 1 at the column temperature and corresponding saturation pressure, yl is the mole fraction of the solute in the gas phase, and d1 is the fugacity coefficient of the solute in the gas phase at T and P. Assuming that because of the moderate pressures involved the virial equation of state truncated after the second virial coefficient can account for gas-phase nonidealities, the fugacity coefficients appearing in (10) are In $1 = Bllq/(RT) In
$1=
(P/RT){2Cy,B1j - Bm,j 1
(11)
(12)
where Bll is the second virial coefficient of the solute at T and Bmix
=
CCYiYjBij i l
(13)
Equation 12 differs for the cases of volatile and nonvolatile solvents. If the solvent is nonvolatile, there are two components in the gas phase, solute and carrier gas, and at the limit of infinite dilution the gas phase is pure carrier gas. On the other hand, with a volatile solvent there are three components in the gas phase (solute, solvent, and Here Fois the volumetric flow rate of the carrier gas at the carrier gas) and, even in the limit of infiiite dilution of the column exit, t is the time for the solute peak to pass solute, the mole fraction of the solvent in the gas phase through the column, tmfis the time for an inert gas to pass may not be very small. To account for the presence of through the column, Pais ambient pressure, T and Taare solvent in the gas, one must be able to measure its mole the column and ambient temperatures, respectively, Pkzo fraction in the gas mixture. is the saturation pressure of water vapor (not to be conConsidering the more general case of a volatile solvent, fused with the saturation pressure of the solvent, which one may rewrite (12) as happens to be water in the experiments reported here) at ambient temperature, and is James-Martin factor which (14) In $1 = (P/RT){2blBll + YZBIZ + Y3B13) - Bmix) accounts for the pressure gradient over the column where the subscript 2 and 3 indicate carrier gas and solvent, respectively. Next, we define an average partition coefficient KL as This corrected net retention volume VN is one of the experimental parameters that must be measured in order to determine the absolute value of the infinite dilution activity coefficient of a species in a solvent by the gas chromatographic method.
Solution Thermodynamics and Gas Chromatography In order to complete the discussion of the gas chromatographic method, we need to relate the distribution coefficient KL to thermodynamic properties. The solution thermodynamics of gas chromatography has been discussed many times, especially by Desty et al. (19621, Everett (1965), Cruickshank et al. (19661, Windsor and Young (1967), Pewr and Martin (1966),Barker and Hilmi (19671, Laub and Pecsok (1978), and Conder and Young (1979). Here we reexamine the solution thermodynamics that leads
where nl,R,and n3 are the mole numbers of solute, carrier gas and solvent, respectively. Combining (10) and (15) yields
In 4: =In[
+ In P- In e - u " , P - f l ) / ( R T )
(n,+ n3I1Vp4 K,V'(n, n2 n3)g
+ +
3+
In P - l n e +
2008 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991
Now using VN/(n, + n2 + n3) = R T / P = VN gives
1
+ Bmixl + In P -
In yl(T,P,xl)= In
In
+ Bmixand KLV'
+ ( P / R T)12(Y1Bll + Y$12 + Y3B13) - BmixI BllPB,/(RT 1 - [$(P - q ) / ( R T11 (17)
Equation 17 is a general relation between the retention volume VN and the activity coefficient y1 at T , P, and xl. This equation can be simplified by assuming that both solute and solvent concentrations in the gas phase are negligibly small, Le., nl, yl, and y3 0, in which case the equation proposed by Desty et al. (1962) is obtained, or we can consider that a finite amount of solvent is present in the gas phase, in which case an expression similar to that proposed by Barker and Hilmi (1967) is obtained. Note that the activity coefficient obtained from (17) is at the column temperature T and average column pressure P. It is usually more advantageous to express the infinite dilution activity coefficients at the zero-pressure limit. This can easily be obtained from
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In yl(T,P,xl)= In y,(T,O,x,) + g ( D l e x / R T1
+
= In yl(T,O,xl) D,P/(RT) - ulP/(RT
(18)
where Dlex and Dl are excess partial molar and molar volumes of the solute, respectively. In this case, approximating gl = gi, we have In yl(T,O,xl) = In
1
( R T / P+ Bmix) + In P -
or rearranging and retaining only the first terms of the series, we have
Bl1G/(RT1 + g",/(RT)
- D,P/(RT
1 (21)
This equation is similar to that proposed by Barker and Hilmi (1967). To evaluate the infinite dilution activity coefficient of the solute (1)in solvent (3) from the equation above, each of the number of moles of solvent in the column (n;),net retention volume of the solute (VN), and the mole fraction of the solvent in the gas phase (y3) must be measured. In addition, a knowledge of the pure component saturation pressure and liquid molar volume is necessary. The determination of the mole fraction of solvent in the gas may be difficult, so even though not it may not be completely justified, it has further been assumed that this quantity is small enough to be neglected. Thus, as y3 0 and y2 1, (21) becomes
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or
(23)
This equation was first obtained by Desty et al. (1962). They further suggested that since the gas solution is very dilute in solute, D r = gi, and they concluded from their measurements that when helium is used as carrier gas there may not be any need to correct for solute-carrier gas interactions; i.e., one can set B12 = 0. Depending upon the degree of simplification one is willing to accept, one of (20)-(23) can be used for determination of absolute values of infinite dilution activity coefficients from GLC measurements.
Relative Measurements of Activity Coefficients by Gas Chromatography Following the work of others and considering that at low pressures the contribution from gas-phase nonidealitieswill be rather small, we start our analysis from (23). However we note that when two solutes, A and B, are run through the gas chromatograph simultaneously, provided that they are separated completely by the column, the following relation can be written for the ratio of their infinite dilution activity coefficients: = [ ( V N , B G ) / ( V N , A E ) l exp{(2BA,- 2B,2 + - oj;")P/(RT)I expl[(BBB-&)G- (BAA& ) z l / ( R T11 (24) The relative simplicity of this equation stems from the fact that since both species are subject to the same amount of the solvent, n;, the number of moles of solvent in the liquid phase drops out of the expression. This is an important advantage since n\ is difficult to measure. Furthermore, since the volumetric flow rate and inlet outlet pressures are identical for both species (25) v N , B / v N , A = ( t - tref)B/(t - tref)A = a B A and (24) becomes
?;/Ti
=
[(G/
