Binary Infinite Dilution Vapor-Liquid Equilibrium from Adsorption Chromatography Tomoko Nakahara, Patsy S. Chappelear, and Riki Kobayashi" Department of Chemical Engineering, William Marsh Rice University, Houston, Texas 7700 1
Retention data were measured by gas-solid chromatography for helium, argon, methane, and tritiated ethane, each at infinite dilution, with ethane on Porapak-Q substrate. From these data the infinite dilution vapor-liquid equilibrium coefficients have been calculated. The Lennard-Jones, Modified Buckingham, Kirkwood-Muller, Slater-Kirkwood, and London potential functions were used to calculate enthalpy and entropy terms. Of these, the Kirkwood-Muller potential function gave the best results. Ethane showed a Langmuir type of adsorption isotherm; the perturbation gases in ethane showed non-Langmuir type. Continuity was observed from gas-solid, gas-monolayer, to gas-multilayer, which then approached gas-liquid equilibria.
Introduction The development of the use of gas chromatography for physicochemical measurements has essentially been in progress since 1952, as shown in the review article by Kobayashi, Chappelear, and Deans in 1967. This article should be consulted for a more thorough bibliography (ca. 200 references) than is given here. The absolute adsorption of a perturbation gas, at a constant temperature and as a function of increasing pressure, shows three regions of behavior: first, the perturbation gas adsorbs directly upon the solid surface of the adsorbent; then, as the pressure increases, the elution gas builds up a monolayer coverage of the adsorbent and the perturbation gas is adsorbed in this monolayer; and finally, as the pressure increases the monolayer becomes multilayer and the perturbation gas is adsorbed into the multilayered elution gas. This multilayered phase approaches a bulk liquid phase type of behavior; hence the partition coefficient for the perturbation gas continuously approaches the vapor-liquid equilibrium case. The limit on the pressure is the saturation pressure for the elution gas; this limit from the vapor-liquid equilibrium case is termed the infinite dilution, K " , of the perturbation gas in the elution gas. The evaluation of the adsorption partition coefficient depends critically upon the determination of the free gas volume, that is, the volume available for the perturbation gas in the column. This free gas volume, which is directly related to the adsorption energy, cannot be measured directly; hence it must be obtained by extrapolation with respect to some parameter proportional to the heats of adsorption. Chackett and Tuck (1957) and Tuck (lY58) reported that there exists a linear relationship between the heat of adsorption of the inert gases on charcoal and the polarizabilities of the inert gases. In earlier studies of the C H ~ - C ~ Hsystem G on Porasil-B in this laboratory, Masukawa and Kobayashi (1968a,b,c)calculated the free - Vf)] gas volume as a linear relationship between [log (VR.~ and the polarizability of i, where V R , is ~ the retention volume for gas i and Vf is the free gas volume. Hori and Kobayashi (1971) obtained the free gas volume by trial-and-error itera~ Vf)] and the tion for the highest linearity between [log ( V R , square root of the Lennard-Jones potential parameter ( e r / k ) I / * ;this assumed a proportional relationship between the heat of adsorption and the L-J parameter. In this study a different approach has been taken to find the free gas volume. The difference between a two-dimensional and a three-dimensional translation entropy of the ideal gas gives the entropy change of adsorption. Then, various intermolecular potentials are studied to yield the enthalpy 220 Ind. Eng. Chem., Fundarn., Vol. 16, No. 2, 1977
change of adsorption, by integration over all molecules on the surface. A trial-and-error procedure is then used to find the most linear relationship between the term [log ( V R , ~- Vf)] and the free energy change of adsorption; this yields the free gas volume, Vf, which is then used to calculate the adsorption partition coefficients. Experimental Section Apparatus. A schematic diagram of the equipment is given in Figure 1. The cryostat is a stainless steel Dewar with a volume of 20 L mounted in a Riki box.The bath fluid was isohexane, whose temperature was regulated by combined cooling/heating. Cooling was provided by a refrigerator, Cincinnati Sub-Zero Products Machine, Model No. PP-120. A ten-junction copper-constantan thermocouple with a thermostat, Thermotrol (Shell Development Co., Hallikainen Instruments), which regulated an immersion heater, provided temperature control within f O . O 1 "C. A Leeds and Northrup No. 8686 potentiometer with a null detector was used in the temperature circuit. Two detection systems were used. A Gow-Mac Instrument Co. Model 405-01 filament-type thermoconductivity cell and pressure-reducing valves were mounted in a thermostat, insulated with flake-type insulation to prevent temperature fluctuations caused by heat convection effects and by adiabatic expansion at the reducing valves. A Cary, Applied Physics Corp., ionization chamber with a Cary Vibrating Reed Electrometer Model 31 was used for detection of the tritium labeled ethane. A two-pen (one to timer and one to the sample injection valve) Leeds and Northrup Speedomax W/L recorder was used with each detector. . The two rotameters shown downstream of the sample line and the reference line were used only for establishing steady-state conditions. Actual flow rate measurements were made with the soap-film flowmeter. The chromatographic column was a 6-ft length of ?&-in.0.d. copper tubing packed with 0.791 g of Porapak-Q. The elution gas was Phillips Petroleum Co. pure grade, 99% minimum purity, ethane. Perturbation gases were Matheson Chemical Co. ultra-high purity helium, argon, and methane, Union Carbide Corp. research grade neon and krypton, Matheson Chemical Co. research grade xenon, and New England Nuclear Corp. tritiated ethane CzHsT, diluted to 4 pC/mL concentration at STP. Heise gauges which have an accuracy of 0.1% of full scale were used for the pressure measurements. The sample injection valve had a volume of 0.25 mL. Procedure. Approximately 10 h was required to establish
J 7
I
r
8
L ---- ---_ SAMPLE
VENT
VENT
I
I
r
- - --- SYSTEM
Figure 1. Experimental apparatus: V.P., vacuum pump; P.R.l, pressure regulator (for over 150 psi); P.K.2, pressure regulator (for under 150 psi); F, filter; D, desiccating agent (magnesium perchlorate); G, pressure gauge (Heise gauge); V.G., McLeod vacuum gauge; S.I.V., sample injection valve; the inside ,volumeis 0.25 mL; B.P., bypass; C, column; S.V., switching valve; N.V., needle valve (reducing valve); R.M., rotameter; F.M., flow meter (soap-film type); T.C., thermal conductivity cell; I.C., ionization chamber.
steady-rate conditions in the system. During this time, the bath temperature was inonitored and the elution gas flowed through the column and occasionally through the by-pass, with the T C cell and IC turned on. To check for equilibrium, the recorder was set a t the highest sensitivity and the switching valve was quickly turned from column to by-pass or vice versa. If the pen shifted, an adjustment in the pressure gradient was made by the needle valves. Desired pressure was established by use of the two pressure regulators, with the lower scaled one (150 psi, 10.2 atm) used exclusively for the lower pressures. After steady state was established, the sample system was evacuated and the perturbation gas w a introduced. ~ When the sample injection valve was opened into the column by pushing the top of the valve, one of the recorder pens made a mark on the recorder chart paper, and the timer started. Just after the peak appeared, the sample injection valve was disconnected from the column system to stop the timer and make another mark on the chart. The retention time varied from 20 to 60 min; it was obtained as the difference between the total time on the timer and the time as measured by the chart speed from the best estimation of the peak maximum to the second mark on the chart paper. This method minimized any errors arising from nonhomogeneous chart speed and variations in the chart paper. Duplicate runs were made both through the by-pass and the column, to check reproducibility. The flow rate was constantly monitored by the soap-film flowmeter. The detector was switched from the TC cell to the IC for C2HsT. A sample size effect was noted for all perturbation gases except C,H,T; therefore, three different sample sizes were investigated by setting pressures of 6 , 4 , and 2 psi in the 0.25 mL volume sample injection valve. The true retention time was obtained by extrapolation to zero sample size. For C2H5T sample pressure was the same as the system pressure. No sample size effect appeared for the by-pass. Results From the measurements of the retention time, flow rate, temperature, and pressure, the retention volume a t 25 "C and 1atm may be calculated from
where f is the flow rate, t~ is the retention time, Ta is the ambient temperature in K, Pais the ambient pressure in atm, is the vapor pressure of water a t ambient temperand Pwater ature in atm. Calculated results are listed in Table I. The retention volume through the column extrapolated to zero sample size minus the retention volume through the bypass, when divided by the weight of the packing material, gave the results in Table I for retention volume a t 25 "C and 1 atmlg of adsorbent. Experimental Errors and Uncertainties. An example is given in Figure 2 for the sample size effect. This type of behavior was observed for all perturbation gases except C2HsT through the column; the true retention volume for the column was obtained by extrapolation as shown. The shape of the adsorption isotherm implies non-Langmuir type of adsorption. The characteristics of the solid adsorbent packing material are important in this experiment. No degradation should be observed during the course of the experiment. The ideal adsorbent should be relatively inert, with no chemical or sorption effects. It should have an approximate uniform pore diameter and pore structure, have a firm consistency, and resist fragmentation during normal handling. The Porapak-Q used here fairly well satisfies these requirements. Some degradation was observed in the packing material during the course of the three months of the experiments; however, no quantitative evaluation could be made of this degradation. Theory for Adsorption Equilibrium The governing relationship for the adsorption of a perturbation gas i eluted through a packed column under equilibrium conditions is vR,I =
Cad,i vf + v a d -
(2)
Cgi
where V R , ~is the retention volume of the perturbation gas, Vf is the free gas volume, or volume in the column available to Ind. Eng. Chem., Fundarn., Vol. 16, No. 2, 1977
221
Table I. Experimental Data of Retention Volume of the Gases Retention volume of perturbation gases (mL at 25 O C and 1atm)/g of adsorbent Ar Kr Xe
Temp, "C (K)
Pressure, atm
He
Ne
-20 (253.15)
1.63 3.10 4.39 5.71 7.10 8.48 10.82 11.97 12.53 13.28 1.69 2.72 3.73 4.75 5.78 6.63 7.14 1.35 2.03 2.71 3.39
8.22 14.89 20.62 27.04 33.55 39.91 50.78 57.40 57.36 56.86 8.79 13.58 18.63 23.74 28.93 30.99 30.47 7.12 10.32 13.36 17.02
8.34 15.02 20.82 27.29 33.92 40.05 51.17 57.45 57.56 57.45 8.86 13.75 18.65 23.94 29.32 32.17 31.15 7.38 10.43 13.55 17.29
-40 (233.15
-60 (213.15)
10.29 18.03 24.70 32.49 40.08 49.12 63.48 69.81 72.77 77.55 11.00
16.92 23.19 29.37 36.75 40.09 44.53 9.49 13.24 17.09 21.26
15.31 25.79 35.35 44.51 54.88 65.64 86.54 99.42 106.22 125.36 17.56 26.21 35.18 44.93 56.59 65.59 79.92 15.75 21.47 27.46 34.25
41.53 71.63 90.11 110.43 131.98 153.38 201.93 236.25 256.33 326.36 62.72 86.78 111.07 138.28 174.67 213.19 283.03 63.50 79.84 110.81 119.58
CHI
CoHcT
14.05 23.49 31.85 40.48 49.79 59.30 78.12 80.11 94.74 107.70 15.56 22.97 31.OO 38.74 48.75 55.98 67.00 13.74 18.76 23.18 29.91
71.71 100.24 121.83 146.71 170.57 183.26 255.99 300.43 332.21 437.26 99.95 129.22 161.34 197.86 248.53 339.42 426.60 108.20 127.02 149.80 181.02
Since K* is always unity, eq 4 and 5 give
K' =
- vf VR,i - vf vR*
If the pressure is very close to the saturated vapor pressure of the elution gas, Ki becomes the vapor-liquid equilibrium coefficient at infinite dilution, K, The relationship between the free energy change of adsorption, AFi, and the equilibrium coefficient Ki is
Equations 6 and 7 give 02
01
0
C g , ~ e p, m o l e / m l ELUTDN GAS
Figure 2. Sample size effect: perturbation gas, He; temperature, -40 "C; pressure, 5.78 atm; retention volume through bypass: 51.46 mL (at 25 "C, 1 atm); weight of packing material, 0.791 g.
At constant temperature and pressure, VR* and Vf are constant, or In ( V R , ~ Vf) = AFJRT
+ constant
(9)
and the perturbation gas molecules, v a d is the volume of the adsorbed phase on the surface of the packing material, and Cad,' and Cg,, are the concentrations of the perturbation gas in the adsorbed and gas phases, respectively, in moles per unit volume of each phase. The ratio of the concentrations, Cad,l/Cg,r, is termed the partition coefficient. The adsorption equilibrium coefficient, K,, is related to the partition coefficient by -=-=-1 XL
KL
Y1
Cad,, Pad Cg,1
(3)
Pg
where Pad and pg are the molar densities of the adsorbed and gas phases, and x, and y, are the mole fractions of the two phases. Substituting eq 3 into eq 2, we obtain VR,I = vf
+ Vad(Pg/Pad)(l/Ki)
(4)
Perturbation with an isotopically labeled gas, denoted by
*, is governed by an analogous equation (5) 222
Ind. Eng. Cham., Fundam., Vol. 16, No. 2. 1977
In ( V R , ~- Vf) =
(mi- TASi)/RT + constant
(10)
These are the thermodynamic equations for adsorption of component i where the initial state is the adsorbed phase and the final state is the gas phase. Entropy Change of Adsorption. Relatively few determinations or theoretical calculations have been made for the entropy of adsorption. A major consideration is the state of mobility of the adsorbate. For our calculations here, we shall apply the simplest theory and assume that the adsorbed molecule behaves as an ideal two-dimensional gas, with unrestricted freedom of movement in two directions and no movement a t all in a direction perpendicular to the surface and no movements of vibration or rotation. According to Kemball (1950), the entropy is given by
+
zSt = R In ( M T A s ) 65.80
(11)
where the subscripts 2 and t on the entropy S denote twodimensional and translational, the units of entropy are cal/ deg-mol, and A s is the area available for a molecule in the standard state. A s is characterized by the two-dimensional
Table 11. Potential Functions Name of the potential function (abbreviation)
Integrated form of potential energy per moleculeU
Formula
Notation potential parameter of ro Lennard- Jones E
cy = aro E , c y : potential
parameter of Modified Buckingham KirkwoodMuller ( K-M1
@ij = ---6mc2
CYicyj
r -‘
1
CriCYj
@ i j (= 2 -NMnmc2---) ai
aj Z 3
Xi
xi
-+-
(;)+(;)
m = mass o f the electron c = velocity of light cy = polarizability =magnetic susceptibility
-
x
the electron fi = Planck’s cons tan t m = mass o f the electron CY = polarizability NE = number of electrons of outer shell Ii Ij
3
4.. = - - - c y ’
London (L)
1 cyJ
2
IJ
Ii + l j
r -6
e
= charge of
CY =
polarizability
I = ionization
potential a
N M : number of molecules per unit volume of adsorbate
pressure F s = 0.0608 dyn/cm (de Boer, 1953, p 115) in the equation of state for an ideal gas
AsFs = R T
(12)
For rare gases at atmospheric pressure, the entropy of the gas phase is given by the translational entropy in three directions gSt =
R In (M3/2T5’2)- 2.30
(13)
The entropy of adsorption is the difference between eq 11 and 13 Asad
= 3St
- ZSt
= R In T M
+ R In T3/2 - R In A s - 63.50
At constant temperature, As is a constant (eq la), and is a function only of the molecular weight, or A s a d , [ = Ai!
In
a + constant
(14) ASad
(15)
Enthalpy Change of Adsorption. For an isothermal system, the enthalpy change AH is given by
AH=AU-PAV
(16)
where AU is the internal energy change of adsorption and AV is the volume change of adsorption. The enthalpy change of adsorption often corresponds to the heat of adsorption, where the heat of adsorption is (a differential heat defined by de Boer (1953, p 43) as
where U , and U,d are the molecular energy of the gas and the adsorbate per mole and N a d is the number of moles in the adsorbed phase. However, in isothermal adsorption, there is also a work term in the heat, or Qisothermal
=
Qdiff
+P s V
(18)
If the energy of the adsorbed molecule is independent of the amount of adsorbed molecules, and AH are equivalent. The work term PAV in eq 16 and 18 is in most cases small with respect to either AU or Qdlff. The internal energy, S U , in eq 16 is composed of the potential energy and the kinetic energy. In a liquid or adsorbed phase, the potential energy is the major term. We can closely approximate AH by the potential energy per mole, obtained by integrating the potential function over all pairs of molecules of the adsorbate and adsorbent. Young and Crowell (1962) reported this as the interaction energy at potential minimum, or = N A h , (ZOi])
(19)
where N A is Avogadro’s number, &](ZO~,) is the potential minimum, and Zol, is the distance between two molecules on the surface. They also reported the relation of Zo to the potential parameter of VOas
20 = 0.765Vo
(20)
Calculation of Interaction Energy AH. Crowell and Steele (1961) showed that the interaction energy of simple nonpolar molecules absorbed on a nonionic adsorbent can be solely attributed to dispersion forces. Five potential functions Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
223
Table 111. Numerical Values of Various Potential Parameters for Perturbation Gases and Ethane
Name of parameters (unit)
He
Ne
Perturbation gases and ethane Ar Kr Xe CH4
C~HF,
ro (Lennard-Jones)“(A) 2.63 2.75 3.41 3.60 4.10 3.82 3.95 158.0 221.0 148.2 243.0 35.6 119.8 c/k (Lennard-Jones)”(K) 6.03 14.0 12.3 13.0 LY (Modified Buckingham)b 12.4 14.5 14.0 158.3 231.2 152.8 38.0 123.2 d k (Modified Buckingham) (K) 9.16 4.21 4.06 4.45 3.14 3.15 3.87 ro (Modified Buckin ham)b (A) 4.00 2.70 4.47 1.63 2.46 LY (Polarizability) (~3/molecule) 0.204 0.393 11.6 31.6 48.1 71.5 29.9 44.5 x (Magnetic susceptibility) ( X 1030cgs/molecule) 3.15 2 8 8 18 18 8 14 N E (Number of electrons of outer shell) 24.6 21.6 15.8 14.0 12.1 13.1 12.8 I (Ionization potential)e (eV) 0 From second virial coefficient data: Hirschfelder, Curtiss, and Bird (1954). Mason and Rice (1954). Pierotti and Halsey (1959), for C& from Hirschfelder, Curtis, and Bird (1954). Selwood (1956); for He and CzHs from “Handbook of Chemistry and Physics.” e From “Handbook of Chemistry and Physics,” for CH4 from Shinoda (1967), and for CzH6 from McDowell and Warren (1951).
Table IV. -bjj*(Zoj;) (erg X cm3/molecule2) X
Name of the gas
(In Parentheses the Ratio to Helium Value, dj)
Modified Buckingham“
Lennard-Jones 6.22 16.46 39.52 48.25 65.97 49.76 68.71
5.28 10.12 25.13 32.36 43.83 31.76
(1.00) (1.92) (4.76) (6.13) (8.30) (6.02)
5.65 13.79 38.64 54.66 74.77 44.17 65.38
04
06
08
(1.00) (2.15) (5.26) (8.10) (10.34) (6.69) (11.56)
3.96 7.26 20.20 27.08 35.59 21.17 41.06
Av std dev X 10’
Lennard-Jones Modified Buckingham Kirkwood-Muller Slater-Kirkwood London
8.17 7.18 0.27 4.40 7.55
(21)’
Figure 3. Correlation coefficient for eq 24. (@,from Kirkwood-Muller potential).
are examined in this study to evaluate the most suitable one for the adsorption energy of gases on Porapak-Q. The potentials with their formulas are given in Table 11. The L-J and M-B potentials have both dispersion and repulsion terms; the K-M, S-K, and L potentials have only dispersion terms. The numerical values of the potential parameters for the various gases are listed in Table 111. These are used in the various formulas given in the third column of Table I1 to calculate the interaction potential between i-j molecules, where the j component is the adsorbed molecules of the elution gas C2Hs and the i component is each perturbation gas, He, Ne, Ar, Kr, Xe, or CH4. For the M-B potential, the i-j form is not available; hence we must use (21) Ind. Eng. Chem., Fundam., Vol. 16,No. 2, 1977
Potential function
Since the M-B potential parameters for ethane are not known, we make the approximation
IO
Vf ”JR,H~
224
5.59 12.03 29.44 45.30 57.81 37.29 59.41
London
Table V. Arithmetic Mean of the Standard Deviation of Eq 24 for All Runs; 21 Conditions
- 2 0 “C , I63 atrn
-5* ,0
(1.00) (2.44) (6.83) (9.67) (13.22) (7.81) (11.56)
Slater-Kirkwood
(1.00) (1.83) (5.10) (6.84) (8.99) (6.61) (8.00) (11.05 (10.37) For Modified Buckingham potential the numeric values of V‘&*Z~,~) [erg’/? cm’~2/molecule]X 1 O l y are listed. The values for ethane d i l * ( 2 ~ , ,are ) not used for calculation of free gas volume. He Ne Ar Kr Xe CH4 C2Hfih
(1.00) (2.65) (6.35) (7.76) (10.61)
Kirkwood-Muller
In the equations in Table 11, the number density of the adsorbate molecules N M is sometimes considered to be the same as its density in the liquid state. However, in this case N Mis common to all of the gases at the same temperature, so we may write &; in another form as
For better ease of calculation and comparison, the ratio of of the terms &;* for the gas to +i;* for helium is computed and tabulated in Table IV, along with &j* for each gas. Definition: +i = 4 i j * ( Z o i j ) / 4 H e i j * ( Z o ~ , j ) .For the M-B potential, eq 21’ is used without including the constant for the quantity &;*. Calculation of Free Gas Volume. Equations 15 and 19 substituted into eq 10 give
Since the derivation of eq 23 has assumed the simplest model for the behavior of the adsorbed molecules, and the V R , are ~
7.-
->
E-
0
calculated
I
Ell/:;:;
0
I
Xe
IO
5 I
I
II
I
;
PRESSURE, otm
Figure 5. Free gas volume at column conditions.
15
ai
( f r o m K-M
potential
formula)
Figure 4. Linearity of eq 26 with the calculated free gas volume (Vf = 18.20 mL).
experimental values, a most probable value of Vf can be obtained from a linear form In [(VR,~ - V f ) 4 K 1 = 0 4 ~+ P
(24)
The degree of linearity is obtained by the correlation coefficient (c.c.) from statistical analysis
C.C.
=
-
1 -Cxy-Ty n
(25)
for y as a function of x , where n is the number of data, X and $ are the mean values of x and y , respectively. For a perfect
linearity the C.C. would be unity. From the experimental data at each equilibrium condition, the C.C. is calculated for various values of Vf. An example is shown in Figure 3 for the K-M potential a t -20 OC and 1.63 atm. The maximum value of the C.C. then defines the best value for Vf, which is then used in eq 24 with the experimental data, and the two constants CY and (J’ are obtained by the least-squares method. The standard deviation os for eq 24 is given by J
C
as =
r=l
(In [(VR,~ - Vf) 4KI
___
J-1
-
- 81 (26)
where J is the number of perturbation gases (in this work, six). At any equilibrium condition, the standard deviation is the smallest for the K-M potential. Table V gives the arithmetic mean of the standard deviation for all 21 experimental conditions for each potential function. The K-M potential is clearly the best for repyesentation of the interaction of the adsorbed molecules on this adsorbent, Porapak-Q. An example is shown in Figure 4 for the K-M potential a t -40 “C and 4.75 atm.
Discussion of Results Table V and Figure 4 show that the K-M potential function gives the best results for this study. Pierotti (1963) applied the K-M formulation to the interaction between the dissolved gas
and the liquid. Pace (1957) and Crowell (1957) and Crowell and Young (1953) used the K-M potential to calculate the dispersion force for the interaction energy between a graphite adsorbent and adsorbed simple nonpolar gas molecuIes. The K-M potential has been applied here to obtain the free gas volumes and to calculate the partition coefficient and adsorption isotherms. Figure 5 shows the gas volume isotherms as a function of pressure. The larger fluctuations at low pressures are due to the small difference between the free gas volume and the retention volume of helium. , Equation 2 may be rewritten to define N A , the absolute adsorption in moles per gram of adsorbent
NA= VadCad,i = c g , i ( VR,I - v f )
(2)’
Using the molar density of the gaseous phase for Cg,r,the absolute adsorption for the elution gas ethane has been calculated with the results shown in Figure 6. The adsorption isotherm is the Langmuir type and shows both the monolayer and multilayer adsorption regions. Figure 7 illustrates the decrease in the free gas volume as the absolute adsorption increases. At a fixed temperature and pressure, eq 2’ can be used to obtain the adsorption isotherm of the perturbation gases in the atmosphere of the elution gas, by setting Cg,L as the sample concentration and V R ,as ~ the retention volume at that concentration. Figure 8 shows that results for the adsorption of helium in ethane a t 5.78 atm and -40 “C as a function of the sample concentration. For a pure component, such an adsorption isotherm as Figure 8 would be termed the nonLangmuir type, which occurs when the heat of adsorption is less than the heat of liquifaction. In this two-component system, a qualitative explanation is that the adsorption energy between the helium gas and the adsorbent is less than the interaction energy between helium gas and the elution gas, ethane. The adsorption type can be obtained from the BET equation
where
and V, is a monomolecular layer of adsorbate on the adsorbent, P is the system pressure, PS is the saturation vapor pressure, Vad is the total volume of adsorbed gas, and ( E a d El) is the “net” heat of adsorption, or the difference between the heat of adsorption in the first layer and the heat of liquefaction. The adsorption type is defined by the relationship Ind. Eng. Chern., Fundam., Vol. 16,No. 2, 1977
225
I
I
I
I
2 15 W
m iT
SI n a
E,
10
\
W
z
2+
w u 5
EE
ADSORBENT PORAPAK 0
zCI 0
15
5 10 PRESSURE, a t m
0
Figure 6. Absolute adsorption of ethane on Porapak-Q. 40
I
I
I
0
3.51
A
5
A
1
0
o
0
03
02
01
C9,He, prnole / mi ELUTION GAS
Figure 8. Adsorption isotherm of perturbation gas in the atmosphere of elution gas.
A
SI 0
=-'
A
-2ooc -4OOC -6O'C
25
;
ZOO
5
10
15
NA , rnmole ETHANE / g m ADSORBENT
Figure 7. Absolute adsorption of C2Hs and free gas volume.
between V& , ,' and PIPS. For ethane the heat of liquefaction at -40 "C is 2920 callmol and the saturation pressure is 7.57 atm. Figure 9 gives adsorption isotherms for ethane with the heat of adsorption less, equal to, and greater than the heat of liquefaction. Absolute Adsorption of Perturbation Gas. Since the retention volume of the perturbation gas is obtained by extrapolation to infinite dilution, a direct numerical calculation of the absolute adsorption of the perturbation gas cannot be made. However, we can assume that the small amount of perturbation gas is constant and calculate the adsorption isotherm. At 1 psi in the sample injection valve of 0.25 mL volume, this is 0.696 pmol, and the retention volume at this sample size is quite close to that at infinite dilution. Figure 10 shows that adsorption isotherms for methane as a function of the pressure of the elution gas. At low pressure the adsorption is relatively large, most of the adsorption occurs directly onto the surface of the adsorbent without any effect from the elution gas, and no temperature effect is seen. However, as the elution gas adsorption increases to complete monolayer coverage, the surface of the adsorbent becomes covered with the elution gas which has the larger adsorption energy. This corresponds to the minima seen in Figure 10. Then an increase in the absolute adsorption of 226
Ind. Eng. Chem.. Fundam., Vol. 16, No. 2,1977
P / Ps
Figure 9. Adsorption isotherm of ethane from BET equation.
Curve
Ead
(cal/mol)
El =
2920 cal/mol
the perturbation gas occurs with the pressure, as multilayer adsorption takes place. In this region the perturbation gas primarily interacts with the multilayered elution gas rather than the adsorbent, and the effect is dissolution of the perturbation gas into the bulk liquid of the elution gas. Equilibrium Coefficient. Equation 6 has been used to obtain the partition coefficient for methane in ethane shown in Figure 11.As the pressure increases, the adsorbed elution gas becomes more and more similar to bulk liquid, and the partition coefficient extrapolated to the saturated vapor
Table VI. Vapor-Liquid Equilibrium Coefficients a t Infinite Dilution for He, Ar, CH4, and CzH6 in the Binary System with CzHs -40 "C
-20 "C From this Pierotti He Ar CH4
249.4 32.1 27.4
Nakahara
work"
Pierotti
Nakahara
92.0 15.0 6.6
666.5 62.0 52.5
132.7 33.9 18.7
57.2 18.9 11.2 6.0b
-60 "C From this worka 180.0 23.0 10.0
Pierotti
Nakahara
From this work0
1908.1 120.9 99.5
323.4 61.5 32.3 14.66
300.0 31.0 12.0
1.1 (1.0)b
1.0
8.9b
C2H6 4.7 1.0 1.0 6.84 1.1 (1.0)b (C2HgT) (1.0)b " Calculated from eq 6 and data in Table I. From GPSA Data Book (1972).
9.37
1.0
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THIS WORK PRICE 8 KOBAYASHI NGPSA DATA BOOK
w
z PRESSURE, arm Figure 10. Adsorption isotherm for CH4 with a fixed amount of sample in the elution gas.
pressure is the vapor-liquid equilibrium coefficient at infinite dilution. Shown on Figure 11 are also the vapor-liquid results of Price and Kobayashi (1959) and values from the GPSA Data Book (1972), based 011 Price and Kobayashi (1959) and Wichterle and Kobayashi (1972). The extrapolation of this work gives values for K" slightly less than K" extrapolated from the vapor-liquid experiment a t -60 O C , and values slightly greater than a t -40 and -20 "C. Similar calculations 'were made for K, " for the other two perturbation gases helium and argon. Table VI gives these values with predicted values by the methods of Pierotti (1963) and Nakahara and Hirata (1969). The Pierotti values are much larger (several orders of magnitude) than this work. Values from the Nakahara and Hirata method are reasonably close to the values obtained in this work, with some values lower and some values higher.
Conclusions This study has shown that gas-solid chromatography may be used to obtain the infinite delution vapor-liquid equilibrium coefficient as the limiting value of the partition coefficient a t the saturated vapor pressure of the elution gas. The adsorbed elution gas on the adsorbent surface, as multilayers build up, approaches bulk liquid behavior as the pressure approaches the saturation vapor pressure. The adsorption isothlerm for the perturbation gas at constant temperature and pressure exhibits a non-Langmuir type of behavior, while the adsorption of the elution gas ethane is a Langmuir type. The relationship among the interactions between the elution gas, the perturbation gas, and the adsorbent is evident from the adsorption isotherm for a fixed small amount of perturbation gas. The Kirkwood-Mullei~potential function was effective and accurate for the modeling of the adsorption behavior, in the
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PRESSURE, atm Figure 11. Adsorption and vapor-liquid equilibrium coefficient for CH4-CzHs.
calculation of the free gas volume with consideration for the enthalpy and entropy of adsorption. The free gas volume is an essential quantity for the calculation of the absolute adsorption and the partition coefficient, or K; ". The adsorbent Porapak-Q was effective in this study since it produced reasonable retention time differences for a short length of column. A little degradation was observed during the course of the three months experimental work; no suitable solution has been found for this problem.
Literature Cited de Boer. J. H., "The Dynamic Character of Adsorption." pp 43, 115. Oxford, 1953. Brunauer. S., Ernmet, P. H., Teller, E., J. Am. Chem. Soc.. 60, 309 (1938). Chackett. K. F.. Tuck, D. G., Trans. Faraday SOC.,53, 1652(1957). Crowell, A. D., J. Chem. Phys., 26, 1407 (1957). Crowell, A. D., Steele. W. A., J. Chem. Phys., 34, 1347 (1961). Crowell, A. D., Young, D. M., Trans. Faraday Soc., 49, 1080 (1953). Gas Processors Suppliers Association, "Engineering Data Book," 9th ed, pp 18-99, 1972. "Handbookof Chemistry and Physics," 51st ed,Chemical Rubber Co.,Cleveland, Ohio. Hirschfelder, J. O., Curtiss. C. F., Bird, R. B., "Molecular Theory of Gases and Liquids," Wiley, New York, N.Y., 1954. Hori, Y., Kobayashi, R.. J. Chem. Phys., 54, 1226 (1971). Kernball. C., Adv. Catal.,2, 233 (1950). Kobayashi. R.. Chappelear, P. S., Deans, H. A.. Ind. Eng. Chem., 59, 63-82 (1967). Mason, E.A., Rice, W. E., J. Chem. Phys., 22, 522, 843 (1954). Masukawa, S.,Kobayashi, R.. J. Gas Chromatog., 6, 257 (1968a).
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Masukawa, S., Kobayashi, R., J. Gas Chromatog., 6, 266 (1968b). Masukawa, S., Kobayashi, R., J. Gas Chromatog., 6, 461 (1968~). McDowell, C. A., Warren, J. W., Discuss. faraday SOC., IO, 53(1951). Muller. A,. Proc. Rov. SOC.(London). Ser. A. 154. 624 (19361. Nakahara: T., Hirata, M., J. 'Chem. Eng. Jpn.; 2, 137 (1969). Pace, E. L., J. Chem. Phys., 27, 1341 (1957). Pierotti, R. A., J. Phys. Chem., 67, 1840 (1963). Pierotti, R. A., Halsey, G. D., J. Phys. Chem., 63, 680 (1959). Price, A. R., Kobayashi, R.. J. Chem. Ens, Data, 4, 40 (1959). Selwood, P. W., "Magnetchemistry," Incrscience, New York, N.Y., 1956. Shinoda, K., "Solution and Solubility," (in Japanese), Maruzen Pub., Inc., Tokyo, 1967. Tuck, D.G., J. Chem. Phys., 29, 724 (1958).
Young, D. M., Crowell, A. D.,"Physical Adsorption of Gases," Butterworths, London, 1962. Wichterle, I.,Kobayashi, R., J. Chem. Eng. Data, 17, 9 (1972).
Received for review January 26,1976 Accepted July 21,1976
The National Science Foundation provided financial support for this work.
Multicomponent Gaseous Diffusion in Porous Media in the Transition Region. A Matrix Method for Calculation of Steady-State Transport Rates Rajamani Krishnal Department of Chemical Engineering, University of Manchester Institute of Science and Technology, Manchester M60 700. England
Isothermal steady-state diffusion of n-component gaseous mixtures through porous media in the transition region is considered in the absence of viscous flow. The extended Maxwell-Stefan equations accounting for both molecular and Knudsen diffusion processes are represented in compact matrix notation and a general analytic solution obtained for the transfer rates Ni. Both the equimolar ( ~ ~ = ' = = , N0) i and the Graham diffusion ; = ' = , N i f i = 0) restrictions on the transfer fluxes are considered in the analysis. The results generalize published analytic solutions for two components.
(E
Introduction Gaseous diffusion in porous solids occurs by two mechanisms. When the diameter of a pore is less than the mean free path of the gas, collision a t the wall controls and Knudsen diffusion predominates. The diffusion flux is given by
Ni = -Ca)Ki- dYi (i = 1,2 , . . . , n) (1) dz where a ) ~is; the Knudsen diffusivity of species i in the porous solid. On the other hand, when the pore diameter is much greater than the mean free path of the gas, the collisions are mainly between gas molecules, and bulk diffusion prevails. In this diffusion regime the fluxes are related to the composition gradients by the Maxwell-Stefan equations
Only n - 1 of the composition gradients in (3) are independent for (4) and therefore the determination of the n fluxes N , requires an additional "determinancy" condition. Two determinancy conditions are normally used in practice (Dullien and Scott, 1962; Rothfield, 1963): (i) equimolar counterdiffusion, valid for a closed system at constant total pressure, requiring n
ZNN,=O
(5)
r=l
and (ii) the Graham diffusion relationship
,fd\/M,N,=O
(6)
1=1
For steady-state conditions (dN,/dz = 0), eq 3 together with eq 5 or 6 may be solved for specified boundary conditions j#i
where a ) i j represent the gas phase diffusivities of the binary pairs i-j in the mixture. Many practical systems operate in the transition region, which can be described by a combination of the two mechanisms. Neglecting viscous flow phenomena, we may thus write the diffusion relationships in the transition region as (Feng and Stewart, 1973; Feng et al., 1974)
j#i 1
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at z = 0, y ! = y t o (i = 1 , 2 , . . . , n) (7) at z = 6 , y , = yla to yield the transfer rates N,. Analytic solutions for the binary ( n = 2) case are easily obtained and are available in the literature (see, for example, Geankoplis, 1972).For the ternary case, solutions are available in parametric form (Cunningham and Geankoplis, 1968; Remick and Geankoplis, 1970). No general analytic solution for the n -component diffusion problem has been presented in the literature. It is our object here to consider the general n-component problem and to obtain convenient analytic expressions for the transfer fluxes. The results of this study may be expected to
