Determination of Brunauer-Emmett-Teller Monolayer Capacities by Gas-Solid Chromatography Peter R. Tremaine’ and Derek G. Gray” Pulp and Paper Research Institute of Canada and Chemistry Department, McGill University, Montreal, Que., Canada, H3C 3G 1
Brunauer-Emmett-Teller (BET) monolayer capacities can be obtained directly from net retention volume vs. vapor p/po 0.05 from the expressure data in the range 0.35 pression
>
>
v,po ( i - m 2 = r m + r m ( c - i ) x mRT
[l
(1
- m2
+ ( C - 1)X]2
where VN is the net retention volume at temperature T (K); X equals p/po, the relative vapor pressure; m is the weight of adsorbent In the column; r mIs the monolayer capacity (Mmol/g); and C is the BET constant. Thls approach avoids the calculation VNdp and thus requires much less extensive data.
The use of gas-solid chromatography to measure adsorption isotherms is well established. The accuracy of various experimental procedures has recently been critically evaluated ( I , 2) and, with appropriate precautions, isotherms obtained from gas chromatography agree with those obtained by classical measurements to within statistical error. This paper presents a procedure for fitting retention volume data, as obtained from the peak maxima elution method ( I ) , directly to an expression derived from the BET equation without the necessity of calculating the actual adsorption isotherm. This approach greatly facilitates the determination of BET monolayer capacities from gas-solid retention volumes. Although the procedure is presented from the point of view of results obtained by the peak maxima method, it is equally applicable to data obtained by frontal analysis or the minor disturbance technique ( 1 , 2 )
THEORY Calculation of Adsorption Isotherms. Briefly, the peak maxima elution method consists of measuring the retention volumes corresponding to the peak maxima obtained from a series of sample injections of various sizes, such as those shown in Figure 1. The recorder displacement a t each maximum is proportional to the partial pressure of the adsorbate vapor in the carrier gas if gas phase ideality is assumed. The detector is calibrated by injecting a known volume, V, of the adsorbate liquid and the vapor pressure is calculated from the expression ( 3 , 4 ) V XRT P = X F) Y
Each value of p , and the corresponding net retention volume, VN, must be corrected for the effects of the column pressure drop and the sorption effect on the local flow rate (1, 2 ) . The relationship between VN and the surface concentration of adsorbate r (@molg-l), is given by the expression (3, 4 ) .
where m is the weight of adsorbent in the column. The adsorption isotherm is then obtained by integrating Equation 2 so that (3) The application of Equation 3 requires that retention volumes be measured down to very low vapor pressures, preferably down to the Henry’s law limiting region. For heterogeneous samples, this may involve rather long retention times. Furthermore, base-line drift can seriously affect the accuracy of vapor pressures corresponding to very small injections (5). A second difficulty arises because there is no completely successful general expression to mathematically describe adsorption isotherms below monolayer coverage (6, 7 ) .As a result, the experimental values of p vs. VN cannot be statistically fitted to an analytic expression a t low coverages. Instead, a smooth curve is usually drawn by eye through the chromatographic data points, as shown by the broken line in Figure 1, and values of r are obtained by integrating the hand-drawn curve a t a large number of arbitrary values of p . Obviously, a great many experimental points a t small intervals of p are required to accurately define the hand-drawn curve. In addition, if some segment of the resulting isotherm is fitted to a theoretical model such as the BET or Frenkel-Halsey-Hill equations (6, 7 ) , a statistically distorted fit will be obtained since the model is fitted to results derived from the sketched curve rather than to the original experimental data. For these reasons, it is desirable to fit such models directly to the chromatographic data wherever possible. The BET Analysis. In practice, the surface areas of solid adsorbents are estimated almost exclusively from monolayer capacities obtained by fitting absorption isotherms to the BET equation (8).The equation takes the form:
(v
where p is the adsorbate vapor pressure corresponding to the recorder pen displacement y , is the molar volume of the adsorbate liquid in the injection syringe; T is the absolute column temperature; R is the ideal gas constant; S is the area on the recorder chart paper under the calibration peak; x and F are the chart speed and carrier gas flow rate, respectively. Present address, Atomic Energy of Canada Ltd., Whiteshell Nuclear Research Establishment, Pinawa, Manitoba, Canada, ROE 1LO. 380
r=
r,cx (1 - X ) [ 1
+ ( C - 1)X]
(4)
where X = p / p ~p, o is the vapor pressure of the pure adsorbate a t temperature T , rmis the monolayer capacity and C is a constant at T . The usual pressure range of the experimental fit is 0.35 > X > 0.05. The relationship between retention volume and vapor pressure over the range where the BET equation is obeyed can be derived by differentiating Equation 4 and substituting from Equation 2 to yield the expression [1+ (C - l)XZ] (-$)T=mRT= VNPO r,cx (5) (1- X)2[1 (C -1)X]2
ANALYTICAL CHEMISTRY, VOL. 48, NO. 2, FEBRUARY 1976
+
i
(7) I
where X B = p / p o a t point B. Equation 7 is readily derived from Equation 4 by assuming that point B and rmare identical. Unfortunately, Xg can only be determined accurately on plots of V Nvs. p when C is large. For example, in Figure 1 point B lies in the general vicinity of the B E T monolayer coverage, but its exact position is uncertain. Since the point B method itself has been shown to be accurate only for isotherms which display a very sharp knee (6, 7), and hence a large value of C, this is not a serious restriction.
1 1
2 0.2
\ &
0.I
EXPERIMENTAL, RESULTS, AND DISCUSSION 0 VN / m l Flgure 1. Gas
chromatographic data for dioxane on cellulose at 35.0
OC
(0) correspond to peak maxima and some of the peak profiles are omitted for clarity. (-
- -) was used in Equation 3 to calculate the adsorption isotherm.
A potentially more useful form of Equation 5 can be obtained by separating the variables so that VKPO mRT
-x (1 - k ) =~rm+ r,(c - 1) x
(1 - X)2
[l
+ (C - 1)X]2
(6) Equations 5 and 6 both require nonlinear techniques to estimate the least squares values of rmand C which correspond to a set of V N vs. p data. T h e Point B Method. In favorable cases, the necessity of subjecting the data to a nonlinear curve fitting procedure can be avoided by locating point B, where B is defined (6, 7 ) as the point a t which a type I1 isotherm passes from a Langmuir-like region to an apparently linear region as the coverage is increased. For isotherms characterized by values of C somewhat greater than 9 ( 9 ) ,the values of rm derived from the BET equation have generally been shown to be experimentally identical to those a t point B (6, 7 ) . I t follows from Equation 2 that B corresponds to the point a t which a plot of V Nvs. p becomes vertical for increasing p . If the position of B can be located with some accuracy, then the values of V N and p a t that point can be used to calculate I?,,, directly from Equation 5 by means of the expression
The most rigorous test for Equations 5 and 6 is to apply them to isotherms characterized by low values of C, that is, to type I1 adsorption isotherms whose knees are poorly defined. The data used for this purpose have been reported elsewhere (10). Briefly, values of VN vs. p were measured for a number of nonswelling adsorbates on water-dried cotton cellulose fibers by the peak maxima elution method using a Hewlett-Packard 5711A gas chromatograph equipped with flame ionization detectors. The sorption mechanism was demonstrated to be physical adsorption, and a typical set of chromatographic data is shown in Figure 1. Data for the adsorption of carbon tetrachloride and decane on 60/80 mesh glass beads were obtained by identical techniques. Initially, the chromatographic results were converted to adsorption isotherms by applying Equation 3 to a hand-drawn f i t of each set of V Nvs. p data and the isotherms were fitted to the BET equation in the usual way (10). The resulting values of rmand C are listed in columns 1 and 4 of Table I. Two methods of deriving the B E T monolayer capacity directly from the VN vs. p values were adopted. The first approach consisted of fitting data in the range 0.35 > X > 0.05 directly to Equation 5 using the Share Program No. NLIN 3094, which is based on Marquardt's algorithm for the least squares estimation of nonlinear parameters ( 1 1 ) . The second approach was to fit a plot of V N P (1 ~ - X)2/ mRT vs. (1 - X)2/[1 (C - 1)XI2 to a least sqaures straight line for 0.35 > X > 0.05 using a trial value for C and assuming for curve-fitting purposes, that X is known exactly. From Equation 6, the slope and intercept yield rm and a new value of C. The procedure was repeated until successive iterations produced values of C which agreed to within 0.1%. The latter method is illustrated by the plots in Figure 2. The initial trial values of C = 10 and C = 3 do not
+
Table I. BET Parameters for Given Sets of V, vs. p Data I'm/(kmoi g-1)
-
Sample
c
4
1
2
Integrated isotherm
Eq 5 , Marquardt algorithm
Eq 6 , i t e r a t e d linear least squares
Integrated isotherm
6.00 6.03 9.83 9.62 8.91 8.51
5.99 5.90 9.72 9.60 8.68 8.05
6.01 5.92 9.73 9.63 8.68 8.09
0.088
0.092 0.103
0.093 0.103
3
N o . of p o i n t s in Eq 5 , Eq 6 , t h e range h l a r q u a r d t i n t e r a t e d linear 0 . 3 5 > algorithm least s q u a r e s X >0.05 5
6
4.04 4.25 8.07 8.30 11.62 12.91
4.00 4.34 6.27 5.98 8.30 9.20
3.98 4.31 6.49 5.94 8.28 9.32
11 10 10 9 5 8
4.75 4.22
3.63 3.57
3.55 3.53
11
I. Cellulose 1. Decane, 56.0 " C 2. Decane, 45.8 "C 3. Dioxane, 35.0 "C 4. Dioxane, 25.0 "C 5. 1-Butanol, 45.0 "C 6. 1-Hexanol, 6 4 . 8 "C 11. Glass 1. Decane, 4 5 "C 2. Carbon tetrachloride, 5.0 "C
0.101
12
ANALYTICAL CHEMISTRY, VOL. 48. NO. 2, FEBRUARY 1976
381
sorption (6, 7, 12) and thus little physical significance can be attached to the exact numerical values of C.
CONCLUSION
"0
0.4
(I
- x ?/ ( I (C- I x )*
0.8
+
Figure 2. V, vs. p data for dioxane on cellulose at 35.0 O C plotted according to Equation 6 using different values of C. (-)
-
are the linear least squares fits. For C = 6.49, the slope equals r m ( C
1)
satisfy Equation 6 since rm(C - I), as calculated from the intercept and the trial value of C, does not equal the slope of the least squares line. However, five and six successive iterations, starting from C = 10 and C = 3, respectively, yield the value of C = 6.49 which satisfies Equation 6. The BET parameters for each isotherm, as obtained by the two procedures, are listed in Table I. In every case, both methods converged very quickly over a wide range of initial values for C. From the table, values of rmcalculated from Equations 5 and 6 deviate from the rmvalues derived from the integrated isotherms by less than 6%; in most cases, the agreement was considerably better. The standard deviations for the results in columns 2 and 3 were of about the same magnitude and the differences between the three columns can thus be accounted for by the experimental scatter. I t is, of course, difficult to estimate the uncertainty of the results in column 1 since there was some scatter about the hand-drawn curves from which the isotherms were calculated. In general, the C values obtained from the integrated isotherms are somewhat higher than those obtained by fitting the GC data directly, probably because the integrated isotherms are so heavily dependent on data corresponding to relative vapor pressures below 0.05. The discrepancies in C are not important, however, since the BET equation is commonly regarded as a useful algebraic tool for locating Tm, rather than as a realistic model of ad-
382
BET monolayer capacities obtained by fitting V N vs. p data directly to Equations 5 or 6 agree with those calculated from the integrated adsorption isotherms to within experimental precision. The use of these equations requires GSC data which meets the usual criterion for the determination of adsorption isotherms (1, 2). Furthermore, the range of validity of the BET equation must be established for the adsorbate-adsorbent pair under study before these nonlinear least squares procedures can be applied. This can easily be accomplished by determining one or two adsorption isotherms in the conventional way (1-5).The direct fit approach avoids the statistical distortions caused by fitting data to a hand-drawn curve, and eliminates the need for collecting data below -0.05 PO. Meaningful values of rm can be rapidly determined from only a few data points. Although the sample conditions required by the GSC technique ( I , 2) are more exacting than those for classical nitrogen adsorption measurements ( 1 3 ) , GSC has the advantage that it is not restricted to cryogenic temperatures and is thus particularly useful for surface area determinations which must be made over a restricted temperature range.
ACKNOWLEDGMENT We are grateful to E. Koller for suggesting and running Marquardt's algorithm program. LITERATURE CITED J. F. K . Huber and R. G. Gerritse, J. Chromatogr., 58, 137 (1971). J. R. Condor, Chrornatographia, 7, 387 (1974). A. V. Kiselev and Y. I. Yashin, "Gas Adsorption Chromatography", Plenum Press, New York. N.Y.. 1969, Chap. IV. H. W. Habgood, in "The Solid-Gas Interface", E. A. Flood, Ed., Arnold, London, 1967, pp 611-846. D. Dollimore, G. R. Heal, and D. R. Martin, J. Chromatogr.. 50, 209 (1970). A. W. Adamson, "Physical Chemistry of Surfaces", 2nd ed., Interscience, New York, N.Y., 1967, pp 584-625. D. M. Young and A. D. Crowell, "Physical Adsorption of Gases", Butterworth, London, 1962, pp 137-214. S. Brunauer, P. H. Emmett, and E. Teller, J. Am. Chem. SOC., 60, 309 (1938). L. White, J. Phys. Chem., 51, 644 (1947). P. R. Tremaine and D. G. Gray, J. Chem. SOC., Faraday Trans. 1, 71, 2170 (1975). D. W. Marquardt, J. SOC.Ind. Appl. Math., 11, 431 (1963). G. D. Halsey, Discuss. Faraday SOC.,8, 54 (1950). F. M. Nelson and F. T. Eggertsen, Anal. Chem., 30, 1387 (1958)
RECEIVEDfor review April 25, 1975. Accepted October 1, 1975. Thanks are due to the National Research Council of Canada for awarding a Postdoctoral Fellowship to P.T.
ANALYTICAL CHEMISTRY, VOL. 48, NO. 2, FEBRUARY 1976
