as a length average. The JamesMartin average, tjJ.v,is a time average, and is thus different.) However, the apparent plate height contribution is, from Equation 7, CIDJhf. I n view of the fact (11) that t j = 2v,/(l P ) , we have the following difference between apparent and average plate height
+
where f 3 is given by fa
3(P
+ 1)(P2 - 1)/4(P3 - 1)
(26)
a term always less than unity when P > 1. This leads t o the rather astonishing conclusion that the measured plate height is less than its lengthaverage value in so far as the liquid contribution is concerned-Le., the apparent number of plates generated by the column is greater than the number that one would find in adding together all the individual plates. In the column, in effect, fact as P +
produces 33% more plates than indicated by the average plate height. If this factor, alone, were involved, a column with a large C I contribution would perform better with a large pressure drop than with none a t all. However the advantage of this effect is offset by the disadvantage (of pressure gradients) mentioned by Keulemans. I n regard to the C l term, the two effects are apparently equal and opposite, and thus there is no inherent advantage or disadvantage in operating a column with a large pressure drop. LITERATURE CITED
(1) Ambrose, D., Ambrose, B. A., “Gas Chromatography,” p. 113, van Nostrand, Princeton, 1962. ( 2 ) Dal Xogare, S., Juvet, R. S., Jr.,
“Gas-Liquid Chromatography,’’ Chap. IV, Interscience, New York, 1962. ( 3 ) DeFord, D. D., Loyd, R. J., Ayers, B. O., AI~AL.CHEM.35, 426 (1963). (4) Desty, D. H., Goldup, A., S,yanton, W. T., “Gas Chromatography, Chap.
VIII, M. Brenner, el al., ed., Academic Press, New York, 1962. (5) Giddings, J. C., ANAL. CHEM. 32, 1707 (1960). (6) Zbid., 34, 314 (1962). ( 7 ) Ibid., 35, 1338 (1963). (8) Giddings, J. C., Seager, S. L., Stucki, L. R., Stewart, G. H., Zbid., 32, 867 (1960). ( 9 ) Keulemys, A. I. M., “Gas Chroma tography, 2nd ed., pp. 143, 149, Reinhold, New York, 1959.
(IO) Littlew:od, A. B., “Gas Chroma tography, p. 191, Academic Press, New York, 1962. ( 1 1 ) Littlewood, A. B., “Gas Chromatography, 1958,” p. 35, D. H. Desty, ed., Butterworths, London, 1958. (12) PurEell, J. H., “Gas Chromatography, Chap. 8, Wiley, New York, 1962. (13) Stewart, G. H.,
Seager, S. L., Giddings, J. C., ANAL.CHEM.31, 1738
(1959).
RECEIVEDfor review August 26, 1963. Accepted December 16, 1963. This work was supported by the Atomic Energy Commission under Contract AT-( 11-1)748.
Equilibrium Considerations at Finite Solute Concentrations in Gas Liquid Chromatography ROBERT S. HENLY, ARTHUR ROSE, and ROBERT F. SWEENY Applied Science Laboratories, Inc., State College, Pa., and The Pennsylvania State University, University Park, Pa.
b A general expression relating the partition coefficient Ki o f any solute, i, in a multicomponent liquid solution to concentration has been derived in this work. This expression, which is
readily reduces to that for a binary (one-solute) system. The only limiting assumption is that the gas phase follows the perfect gas law. Analysis of the equations presented here shows that for both ideal and nonideal multicomponent liquid systems there is interaction among the solutes at finite solute concentrations. This is exemplified b y theoretical chromatograms calculated for several hypothetical ternary (two-solute) systems. For a binary (one-solute) system it is shown that the solute partition coefficient increases with increasing solute concentration for ideal systems, nonideal systems with positive deviations, and some nonideal systems with negative deviations. For other nonideal systems with negative deviations, the partition coefficient first decreases and then increases as the solute concentration increases. This i s most likely to occur with systems which exhibit extreme negative deviations.
744
ANALYTICAL CHEMISTRY
A
OF THE VARIOUS published theories of gas liquid chromatography predict symmetrical solute bands or peaks if the partition coefficients are independent of concentration (linear partition isotherms). Asymmetrical peaks have long been predicted qualitatively for gas liquid chromatography if the partition coefficients vary with concentration (curved partition isotherms) (11, 12, 16). Recently this latter case has been predicted quantitatively for a single solute by means of computer calculations (5,f5).For these calculations, the partition coefficient K was assumed t o be a linear function of solute concentration, and arbitrary constants were used. It has been shown quantitatively that nonlinear adsorption isotherms result in asymmetrical peaks in liquid solid and gas solid chromatography (6, ‘7, 9). Glueckauf has considered theoretically two solute systems in liquid solid chromatography (6, 7 ) . The manner in which K varies with concentration in gas liquid chromatography has usually been qualitatively related t o solution nonideali t y alone. It has been commonly stated in the literature that K is independent of concentration for a n ideal solution, and that K increases with increasing solute concentration for nonideal solutions
LL
with positive deviations and decreases with increasing solute concentration for nonideal solutions with negative deviations. These conclusions were arrived at by relating K , t o concentration only through the solute activity coefficient, yi, in Equation 1 which is limited to infinite dilution of the solute in the solvent (11, 12, 1 4 .
Equation 1 can be obtained by taking the limit of the general equation derived in this work as the concentrations approach zero. Contrary t o the conclusions based on Equation 1, one of the authors (10) predicted some time ago, on a very qualitative basis, that K should increase with increasing solute concentration for an ideal solution, and that K might not necessarily decrease with increasing solute concentration for a nonideal solution with negative deviations. Subsequently, experimental binary isotherms, which were all curved and convex to the gas phase concentration axis, were independently presented by Freeguard and Stock (4) for three nonideal systems with negative deviations and one nearly ideal system, showing that K increased with increasing solute concentration. All four
systems were actual chromatographic systems. I n a more detailed paper on this experimental work (S), Freeguard and Stock state that their experimental isotherms could be \ erified by calculations, but they gave no details on this. Amaya and Sasak (1) recently presented approximate equations for the solute partition coefficients for binary liquid systems (one solute in a solvent) and for ternary liquid systems (two solutes in a solvent) in terms of liquid phase mole fractions and interaction energies among the vitrious components. Here it was assumed that the gas phase followed the perfect gas law and that all the components including the solvent have the same molar volume. These workers pointed out that the latter assumption was not valid in gas chromatography. Analyzing their equations only for the case of high dilution of the solutes with the solvent, they predicted that the retention times of two solutes in a mixture might differ slightly from the retention times or each solute alone because of interaction between the solutes. This was shown experimentally to be true with small samples, but the differences in retenticn times were very small. The work presented here clarifies and adds to the predictions made by Henly (10) and gives a theoretical basis to the experimental results presented by Freeguard and Stock (4). The expression for a multicomponent liquid system accounts for interactions among solutes and shows t h a t one solute affects another a t finite solute concentrations, both for nonideal and ideal solutions.
Combining Equations 2 through 7 results in Equation 8, the general expression for K ;
For a binary liquid system-i.e., only one solute i-the Z sign disappears in Equation 8, resulting in Equation 9.
Ki
=
RT [ E Pi yi AIS
+ CLi]
(9)
I n order to analyze Equation 9 for a nonideal binary liquid system with negative deviations, a n expression for the derivative dKi/dCLi is required. Since Ki = f (CLi,Y;), Equation 10 can be written for dKJdCLi
By the definition of units
If Equation 11 is differentiated with respect to CLi and solved for dxi/dCLi
By making the proper differentiations of Equation 9
ANALYSIS OF EQUATIONS
Binary Liquid Systems. Although the binary liquid system (one solute) is encountered in gas liquid chromatography, i t is not nearly as common as multicomponent liquid systems. However, most previous theoretical work has been limited to the binary system because of its relative simplicity. Therefore, Equations 9 and 17 for the binary system will be analyzed first. For nonideal binary solutions with positive deviations, yt is greater than 1.0 and usually decreases with increasing solute concentration. Therefore T~ and CLzreinforce each other in Equation 9, and K , increases with increasing solute concentration. This results in a curved partition isotherm which is convex to the gas phase concentration axis. For ideal solutions, K , is a linear ~ function of CL2, increasing as C L increases. Thus a curved isotherm convex to the gas phase concentration axis will also be obtained for an ideal system. For nonideal solutions with negative deviations, y bis less than 1.0 and usually increases with increasing solute concentration. In this case the effect of 7,on K , opposes the effect of CL, in Equation 9, and it is difficult to predict qualitatively the manner in which K , varies with concentration from Equation 9 alone. Hon ever, Equation 17 for dK,/dCL, gives more information on this. If K , is independent of concentration, dK,/dCL, = 0. I n order for this to be true, Equation 17 shows that d In T%/ dz, must follow the relationship
DERIVATION OF EQUATIONS
Consider a gas liquid system at equilibrium at a temperature T . The system consists of a n inert gas which is insoluble in the liqLid phase, a nonvolatile liquid solvert, and n volatile solutes which are diistributed between the gas and liquid phases. The partition coefficient, K,, for any solute i (i = 1, 2, 3 . . . . . . . n) is defined by Equation 2 below.
(14)
Substitution of Equations 12, 13, and 14 into Equation 10 gives
-dKi -
dCLi
Rl'
-p s i [I - (1 + PS
By adding 1 to each side of Equation 11
If the gas phase is assumed to follow the perfect gas law, then the following equations can be writ ten: cci = y t p m PMG
yt =
=
P/RT
r,x,P,"/P
(3)
1
+ llPS PS
CL,
1 1-
= 2%
Substitution of Equation Equation 15 results in
(16) 16 into
Yi =
(4) (5)
I n order for K , to decrease with increasing CLz,dK,/dCL, < 0, and d In y,/dz% must be positive and greater than 1/(1 - zt). If d In Y , / d x , is less than 1/(1 - z,), even though it may be positive, K , hill increase with increasing solute concentration. It can be seen that Equation 17 results in the same conclusions as Equation 9 for ideal solutions and nonideal solutions with positive deviations, since d In Y,/dxL is zero for the former and usually negative for the latter. Integration of Equation 18 gives 7,as a function of 2,.
(17)
Equation 17 is expressed in terms of In 7,and xt for ease of discussion since data from vapor liquid equilibrium studies are usually presented in these terms.
Yi"
I -xi
Equation 19 describes a family of similar T,, x, curves, one for each value of T~ at infinite dilution. One of these curves is shown on semilog coordinates in Figure 1. At x1 = 0, the value of 7, is its value a t infinite dilution, in this case 0.3, and the value of d In Y,/dx, is VOL. 36, NO. 4, APRIL 1964
745
1.0. Both y, and d In y,/dx, continually increase with increasing solute concentration t o infinity as x, approaches 1.0. This relationship for q / b as a function of x, does not fit vapor liquid equilibrium theory nor known experimental data. Therefore a K , independent of concentration would not be expected for any type of binary liquid system. For nonideal binary systems with negative deviations, d In yc/dxt is usually positive and decreases with increasing x,,approaching a value of zero as x, approaches 1.0. The rate of decrease of d In y J d x , (or the curvature of the In yz us. x, curve) depends upon the values of the y’s at infinite dilution of both the solute and the solvent. At high concentrations, d In y J d x , can readily be expected to be less than 1/(1 - xJ. However at low concentrations this derhative may be either less or greater than 1/(1 - xt), depending upon the curvative of the yLus. x, curve. If d In yl/dxLis less than 1/(1 - 2,) at low solute concentrations, then K , will increase with increasing CLt at all concentrations, and a curved partition isotherm convex to the gas phase concentration axis will be obtained. If d In y J d x , is greater than 1/(1 - 5 , ) at low solute concentrations, K , will first decrease and then increase as CLt increases from zero. This results in a n S-shaped isotherm, concave at low concentrations and convex at high concentrations to the gas phase concentration axis. The S-shaped isotherm would be most likely to be obtained for systems exhibiting an extreme degree of nonideality with negative deviat ions-i. e., very small values of 7,-since the 7%us. x, curves for such systems are most likely to have a high degree of curvature. For systems with S-shaped isotherms, the concentration at which the minimum K , occurs also depends upon the curvature of the yz us. x1 curve. This concentration might be so high in some systems that it is never reached in gas chromatographic columns when relatively small samples are used. In such cases, the column would be operating only under the portion of the partition isotherm which is concave to the gas phase concentration axis. Multicomponent Liquid Systems. I n the actual practice of gas chromatography, multicomponent liquid systems, with two or more solutes, are more common than binary liquid systems. Equation 8 for a multicomponent system shows that K , is directly related to the concentrations of all the solutes present (total solute concentration) as well as indirectly related t o these concentrations through 7%. In other words, there is an interaction among solutes in a multicomponent system, even if the svstem 746
ANALYTICAL CHEMISTRY
xi
Figure 1. Graphic representation of y i = (1 - x i ) on semilogarithmic coordinates
?io/
is ideal. The relative volatility of any solute i t o any other solutej, a,,, is given by
vapor, at pressure P in the gas phase per unit volume of gas phase, and the liquid phase concentration for solute i, CL2*,is expressed in units of volume of i, as vapor, at pressure P in the liquid phase per unit volume of gas phase. The partition coefficient, K,*, is defined By assuming a perfect as C,,*/C,,*. gas phase, it can be shown that
(20) Pjoyi y j = 1.0)
Ccb* (RTt P ) C a (22) CL,* = ( R T / P ) ~ T ’ S / V G ) C L (23) ~
For a n ideal system (yi
=
and
P,O Oii
=
p,“
For a gas chromatographic column equivalent t o a fixed number of theoretical plates, it is the relative volatility for two solutes that determines the degree of separation between them. For a n ideal system under isothermal conditione, CY,] is independent of solute concentration. This is not true for a nonideal system except where the ratio y , / ~ ,is independent of concentration. In order t o illustrate the meaning of Equation 8 for K , and Equations 20 and 21 for c y l l , a few calculations for hypothetical two-solute systems were made with an I B M 7074 digital computer, using a Fortran program devised by Sumantri (15). The calculation method is based upon the plate theory and is an iterative stepwise method. The gas holdup of each plate is assumed t o be replaced during each time interval. The calculation method is therefore based on a discontinuous flow model. Since the actual gas chromatographic process involves continuous gas flow, the model used here only approximates the true process. The error due to the use of the discontinuous model results in somewhat better separations between two solutes than the continuous model (8, 12). In order to make the gas and liquid phase solute concentrations additive in the material balances, the gas phase concentration for solute i, CcI*, is expressed in units of volume of i, as
K,*
=
(T’s/Vc)K,
(24)
Combination of Equations 8, 23, and 24 gives
2 ..*I
(25)
i=l
For the calculations, the dependence of V s / V G and yi on concentration was neglected, and K , was expressed as a linear function of total solute concentration
K,*
=
B,
+ it,
CL,*
(26)
%‘I
where B, and A , are arbitrary constants. However, these constants were chosen such that K,* jncreases nith increasing total solute concentration. The ratio V S / V c would also increase with increasing total solute concentration in actual practice, resulting in a more extreme and nonlinear variation in K,* with concentration in the same direction. For a nonideal system, the variation of 7 % with concentration would also cause nonlinearity in the equation for K,*. However, a nonideal system can be approximated by choosing the constants in Equation 26 for two solutes i and j , so that the ratio K , / K i (= a r i ) varies with total solute concentration. Similarly, by choosing these constants SO that K J K , is independent of concentration, an ideal system can be approximated.
1.6-
1.51.4
-
1.2
*~ 0 z. 0 2a I5
-
1.1-
1.0-
0.90.8-
0
z
s
0.7-
C
W
0.6A
0
*
0.5-
l9
z F
0.1 -
0.4-
3
0.3-
-
0.1 0
:K
0.2
0.2
K?
I
0
I
2 3 4 TIME INTERVALS X IO-'
5
I
Figure 2. Calculatetd chromatogram for a hypothetical two-solute system with K ' s independent of concentration Sample size: 5 ml. as vapor Sample composition: 80 mole % of solute 1 No. of theoretical plates: 100 Gas residence time (tc): i 00 time intervals Sample enters in 1 time interval with a rectangular concentration proflle
The chromatograms resulting from these calculations are shown in Figures 2 to 5. These results are not intended to represent accurate predictions of actual results, but illustrate the effects of partition coefficien Ls variations with concentration. Conditions were set for all the calculations represented by Figures 2 and 3. At infinite dilu ,ion of the solutes, K1* = 1 and Kz* = 2; for each calculation. The five calculrttions for Figures 2 and 3 differed only in the dependence of the K*'s on concentration. The K*'s a t infinite diluticn were limited to low values and the number of theoretical plates was limited to 100 to minimize computer time. Individual solute concentrations are represented by the broken-line curves, FT hile total solute concentration is shown as solid-line curves. Figure 2 shows the results for K1* = 1 and K2* = 2, both K*'s being independent of concentration. Symmetrical peaks are obtained as predicted by the simplified plate theory of gas chromatography a t infinite dilution (12). This theory gives the folk wing expressions for the solute retention times (tRJ and
0
I
I
I
2 3 4 TIME INTERVALS X IO-'
I
5
I
Conditions same os Figure 2
+ +
two
From Equation 27, t ~ l ' = lOO(1 1) = 200 time units and tR2' = 100 (1 2) = 300 time units. These values correspond t o the elution times for the peak maxima in Figure 2. -41~0from Equation 28, a1.2 = (300 - loo)/ (200 - 100) = 2 = Kz*/Ki*. Figure 3A shows the results for K1* = 1 CLI*and K2* = 2 2CL2*. In Figure 3B, K1* = 1 2CL1* while again K2* = 2 2CL2*. I n both these cases, the K* for a particular solute is dependent upon the concentration of that solute alone, and, as in Figure 2, there is no interaction between the two solutes. Therefore the individual peak shapes are the same as would be obtained if 4.0 ml. of solute 1 and 1.0 ml. of solute 2 had been injected into the column separately. Comparison of Figures 2, 3A, and 3B shows the peak
+
I
Figure 3. Calculated chromatograms for four hypothetical two-solute systems with K's as different functions of solute concentrations
the separation factor between solutes (a(,= relative volatility).
+
I
+
+
asymmetry and broadening due to the concentration dependence of the K*'s. I n Figure 3B, peak 1 is more asymmetrical and broader than in Figure 3A because of the stronger dependence of K1* on CL1*. The shape and position of peak 2 in both these figures is the same, since the dependence of K2* on CL2* is unchanged. Figures 3A and 3B do not present realistic results if the two solutes are injected into a column together because there is interaction between the solutes as predicted by Equation 8. The calculated results for simultaneous injection of solutes 1 and 2 are shown in Figures 3C and 3 0 , using Equation 26 for the K*'s. I n Figure 3C, KI* = 1 CLI* CLZ* and Kp* = 2 ~ C L I * 2C~2*. This represents an ideal liquid system, for K2*/KI* (= ai,2) is independent of solute concentration. Peak 1, instead of overlapping peak 2 as in Figure 3A, almost completely displaces the latter peak. Thus even though each peak is asymmetrical and broad, an almost complete separation occurs, equivalent to that in Figure 2. The separation would not appear surprising since a1,2= 2 at all concentrations.
+
VOL.
+ +
3 6 , NO. 4, APRIL 1964
+
747
A
c
l
B
0.1 T I M E INTERVALS X
Figure 4. Calculated chromatogram for a hypothetical two-solute system with K’s independent of concentration Sample sizei 5 mi. as vapor Sample composition: 20 mole % of solute 1 No. of theoretical plates; 100 G a s residence time h):100 time intervals Sample enters in 1 time interval with a rectangular concentration profile I
2
3 TIME
Figure 3 0 shows the results for K1* = 2CL1* ~ C L Zand * K2* = 2 ~ C L ~ *2CLz*. Here a nonideal liquid system is approximated, for K2*/K1* (= alIZ) is a function of solute concentration. I n this particular case, a1.2 = 2 a t infinite dilution and decreases with increasing total solute concentration. The peaks are more asymmetrical and broader than for the ideal liquid system, and only partial separation occurs because of the lower values of a t high total solute concentrations. I n Figure 3C and 3 0 the values of ( ~ K ~ * / ~ C L ~ * )and C L Z(6K?*/6Cm*)cLL,* * are the same as in Figures 3A and 3B, respectively, It is interesting to note that for both the ideal and the nonideal liquid systems (Figures 3C and 3D), a better separation is obtained than if there were no interaction between the solutes (Figures 3 8 and 3B). However, this is only true when the charge (sample) composition of the faster moving solute (solute 1) is high, relative to the slower moving solute (solute 2). Figures 4 and 5 show the calculated results for the same conditions as in Figures 2 and 3, except that the sample contains 20 mole per cent of component 1. The concentration dependence of the K*’s in Figures 4 and 5 are the same as in Figures 2 and 3, respectively. When the charge composition of the faster moving solute is low, the separations for both the ideal and nonideal liquid systems (Figures 5C and 5 0 ) are not as good as would be obtained if
1
+
748
+
+
ANALYTICAL CHEMISTRY
I
I
I
4
5
6
INTERVALS X IO-‘
Figure 5. Calculated chromatograms for four hypothetical two-solute systems with K’s as different functions of solute concentrations
+
Conditions same as Figure 4
there were no interaction between solutes (Figures 5 A and 5B). Furthermore, the separation obtained for the ideal liquid system (Figure 5C) is not as good as that obtained with partition coefficients independent of concentration (Figure 4), even though in both = 2.0 a t all concentrations. cases Thus for an ideal liquid system and a fixed number of theoretical plates, the degree of separation depends upon the charge composition. This is also true in distillation, but it i s not obvious whether the rekson is the same for both processes. DISCUSSION
The concentration dependence of the
K*’s for the calculated results shown in Figures 2 to 5 may be extreme when compared to many actual chromatographic systems. However, similar general results have been obtained with large samples of methyl esters on ethylene glycol succinate columns (IS, 16). It should be noted that similar calculations, carried out with the same equations for the K*’s as used in the examples but with a sample size of 0.1 ml. vapor (as compared to 5 ml.), result in elution curves similar to those obtained for K*’s independent of concen-
tration. That is, peak asymmetry and displacement are negligible. Thus, the theoretical concepts presented here in no way contradict the experimental fact that in many, if not most, cases the infinite dilution theory describes the chromatographic process quite well for very small sample sizes. This is generally ascribed to the approximate linearity of partition isotherms a t very low solute concentration. I n terms of Equation 8 for K,, if small concentration ranges are maintained in a column by the use of small samples, the variation in the K’s percentagewise will be small enough in many cases t o have only a negligible effect on peak shape and position. If Equation 8 for K , is multiplied and divided by p s / M s , then
since
Equation 30 for the variation in Ki as per cent of Kio between some finite value of Zxi and Zzi = 0 can be obtained from Equation 29.
i=l
For an ideal liquid syrkem Equation 30 becomes
a=
1
The maximum solu .e concentration occurs a t the column inlet. This concentration usually decreases rapidly as the solutes travel along the column under the operating conditions normally encountered in gas chromatography on an analytical scale. For a maximum Zz, of 0.1 a t the column inlet, an effective mean maximum Zx,of say 0.03 or lesb might be obtained over the length of column, resulting in an effective mean variation in the K's of only 3% or less if thl: system is ideal (Equation 31). Thct effect of this variation on solute peaks would be small. From Equation 30, less percentagewise variation in the IT'S and therefore less peak distortion a t finite concentrations would be expectelj for some, if not many, nonideal systems with negative deviations (rl
