>r. .T.
fln’i Gnlav.
E.. (‘Gas Chroma-
Cniversity of Pretoria, Pretoria, 1962. (12) Haarhoff, P. C., Pretorius, V., J. 8. Afracan Chem. Inst. 14,22 (1961).
(13) Klamer, K., Krevelen, van, D. W., Chem. Eng. Sci. 7,197 (19%). (14) Krige, G. J., Pretorius, V., ANAL. CHEM.37,1186 (1963.
1151 Ibid..
D.
Acta Chim. Acad. Sci. Huna. 22. 285
1191.
Ashley, J. W., Jr., ANAL 1198 (1962). (18) Schay, G., “Theoretische Grundlagen der Gaschromatographie,” Chap. 4, VEB Deutscher Verlag der Wissenschaften, Berlin, 1961. (19) Schay, G., Petho, A., Fejes, P.,
“Physicd Adsorption of Gases,” Chap. 1, Butterworths, London, 1962. (23) Ibid., p. 373. RECEIVEDFebruary 23, 1965. Accepted July 1, 1965.
Frontal Analysis Chromrtography Experimental Study of Stationary Fronts on Columns with Negligible Radial Permeubility Fluctuations G. J. KRIGE’ and VICTOR PRETORIUS Department of Physical and Theoretical Chemistry, University o f Pretoria, Pretoria, South Africa
b The existence of stationary fronts in frontal analysis at high sample inlet concentrations is experimentally demonstrated. Continuous introduction of samples of both pure methane and a mixture of methane and carbon dioxide, into a column packed with activated carbon, i s considered. A clear indication is given of the sources of error arising from limitations in the experimental technique, and their effect on the values of the observed front widths is estimated. The results are correlated with the theoretical expressions derived in an earlier paper by using an equation for the plate height in gas-solid chromatography which takes microporous diffusion in the adsorbent into account. Good agreement with existing data in the literature is obtained.
A
the assumption that stationary fronts are formed has been employed to determine distribution isotherms ( 7 ) , no attempt has been made to prove that the front width is actually independent of the column length. Furthermore, the interpretation of experimental front widths in terms of theoretical expressions for the effect of nonideality has attracted very little attention (2$), possibly because existing theories were too specific to be applied under the conditions normally encountered in practice. However, in a previous paper ( I S ) general theoretical expressions were derived for the widths of the stationary fronts formed when either a single solute or a binary solute mixture is continuously injected into a LTHOUGH
1 Present address, Department of Chemistry, University of British Vancouver, Canada. Permaner Atomic Energy Board, Pelindabs . of South Africa.
1202
ANALYTICAL CHEMISTRY
chromatographic column, assuming that radial variations in the permeability of the column packing are negligible. As a result, a meaningful experimental investigation of the phenomenon of stationary fronts is now possible. I n the investigation described the existence of stationary fronts is experimentally demonstrated on analytical scale columns. The mechanisms which determine the widths of such fronts are then analyzed in terms of the theory developed in a previous paper ( I S ) . The verification of theoretical predictions regarding the basic nature of processes, rather than exact agreement between theory and experiment, is stressed.
The general expressions for the widths of stationary fronts (IS) are written in the form which they assume when the sorbates exhibit Langmuir type sorption isotherms on the sorbent. This behavior has been observed for the methane- and carbon dioxide-activated carbon systems (12). Single, Undiluted Solute. The width of t h e stationary front formed when a single solute is continuously injected into a column is ( I S ) :
y
-
For a Langmuir type isotherm ( l a ) ,
k
--
1
+ bXp
-
pabQJnT
tT,(1
+ bXp)
A,.-A
dX.
’
V/
l+bAp
(2)
=
UO 1 + ki
Equations 1, 3, and 4, with Xi yield Wt’
THEORY
kX=--
The subsequent discussion is further facilitated by expressing the progression velocity of the front in terms of the flow velocity through the column once the last front has emerged, uo. This is achieved by using the relations derived for the rate of flow during frontal analysis (11). For the situation under consideration, the mass balance equations a t the first front of a binary solute mixture are used, t o obtain
+
=
2(1 kh(2
=
At*
C,h
+ bls)
(4) =
1,
+
+ Bt*/u02
for the width of the stationary front. Binary, Undiluted Solute Mixture.
I n this case t h e region between t h e first and second fronts contains pure solute 1 (11), so that the front width is described by Equation 1. The progression velocity of the front at the
(2 - x.) moment the front breaks through may be obtained from a n existing expression ( 1 1 ) by setting sa = L:
ax
II
Uolli
I. Details of Columns Used for Gas-Solid Chromatography Column B C D Column length, L, cm. 100 58 30 Internal diameter of column, cm. 0.63 0.63 0.63 Column packing BDH activated carbon Mesh fraction of packing 60-80 80-100 80-100 Average particle diameter, d,, cm. 2.14 X 1.63 X 1.63 X lo-* Weight of packing per unit volume of column, pa, g./cc. 0.55 0.58 0.53 Table
with g = 1
+ kla + ( h i - kii)Xi'
(8)
Equations 1, 3, and 6, with v( v. and Xi XI. = 1, yield the following expression for the width of the first front:
for the width of the second front. EXPERIMENTAL
=
(Ata)a
Apparatus and Experimental Method. The experimental investigation was carried out on the chromatographic apparatus described in a n earlier paper (16). Columns B, C, and D (see Table I) were used.
+ (Bta)c/~oz
The width of the second stationary front may in general be written as
1 d (- [kZX(XZ - 1)l) x2 (2 XZ"d dXz { \
pi
+
minute was used to increase the accuracy of front width measurements. The response time of the recorder was determined, and did not introduce any error, provided that front widths exceeded 1 second. The effect of the time constant of the detector, and of the dead volume between its sensing element and the end of the column packing on the accuracy of the experimental results is discussed in detail below and in the appendix.
..
= '/ZXZ~
RESULTS AND DISCUSSION
(10)
(W4b =
ai
-
For a Langmuir type isotherm ( I d ) kzx
kz
1
-
+ (biX1 + b2Xz)p pobzammpnT cTn[l + (biXi + bzXz)FI
(11)
so t h a t
and d - [k2X(X2 - 1)l = dXz
+
when X1 Xz = 1. The progression velocity of the front is related to the mobile phase flow velocity by the expression (11)
(xii
and, for an equimolar mixture = X2i = Equations 10 and 12 to 14 yield
(2
- XZ')
Serious front distortion was observed in some experiments, mainly when relatively long columns-e.$., 245 em.or low temperatures-e.g., 60°C.-were employed. The recorded concentration a t the column outlet then increased to a maximum before assuming a constant value. No reason could be found for this unusual effect, but in some instances it disappeared after a number of runs had been performed. All the experimental results in this paper were obtained from smooth concentration profiles a t 80" C. The fact that the flow velocity a t the column outlet increases continuously as the front leaves the column will have an effect on the width of the front. These flow rate changes, however, also occur when the stationary front moves past a point in the column, and are taken into account in the theoretical treatment; in this respect the response of the detector is theoretically identical to that of a detector situat,ed within the column itself. The semidiffusion type thermal conductivity cell which was used was not measurably flow-sensitive, and the manufacturers claim that the time constant is less than 1 second; its approximate linearity for methane and carbon dioxide was verified for the concentration ranges employed. -4 flame ionization detector or throughflow type thermal conductivity cell would have yielded a smaller time constant, but the variations in flow rate which occurred would have led to changes in the base line (16, 17'). h recorder chart speed of 20 cm. per
The characteristic feature of a stationary front is that its width is indeDendent of the column length. F r o i t widths were therefore obtaked a t different column lengths and given flow velocities to ascertain whether or not stationary fronts are formed. Pure methane (X.= 1) and a mixture of carbon dioxide and methane (XI%= X2" = were used as samples on columns B , C, and D a t 80" C. It is evident that coefficients A and B in Equations 5 , 9, and 15 should be independent of the flow velocity and column length, provided that variations in p and p are not important. dlthough it is not possible to keep all other operating conditions constant when the column length is being varied, under the experimental conditions employed in this study p t / p o varied from 1.0 to 1.4, and it was found by calculation that the corresponding fluctuations in the front widths did not amount to more than about 2%. To avoid greater variations in the widths, larger particles were used in column B than in columns C and D to keep p J p 0 < 1.4. C, and hence p, is not significantly affected when d, is changed (see below). The dead volume a t the column outlet, and the detector's response time, introduced a constant error, and were therefore not factors of importance. Experimental front widths obtained at different column lengths are plotted against 1/u,2 in Figures 1 and 2. NO results could be obtained for the first front on column B, because of distortion of the concentration profile (see above). It is clear from Figures 1 and 2 that stationary fronts were observed in the experiments. To provide further proof of this the velocity-independent coefficients, A and B in Equations 5, 9, and VOL 37, NO. 10, SEPTEMBER 1965
1203
tion, when the present experiments at 80' C. are considered. When X . = 1, Equations 1 and 4 then yield
10.01
with (IS)
for the frontal analysis of pure methane. The front width, corrected for the theoretical approximation employed, is (13)
-
0 0
Figure 1. Experimental front widths obtained for methane at 80" C. and various flow velocities on columns 6, C, and D Column
0
0.3
0.1 0.2 l/u$sec2/cm2
Let e denote the fraction of the total front width due to the velocity-independent term. When Equation 16 is employed,
B
100
X A
C
58
D
30
0
and i t is found from Equations 18 and 19 that
15, were evaluated for each column length, by performing a linear regression analysis on the data in Figures 1 and 2. The results are given in Table 11. As a point of interest the minimum column length, L,, required for the formation of stationary fronts which correspond to the results in Figure 1, was calculated from the simple expression proposed in a previous paper (IS). It was found that L, 5 5 cm. for all the experimental flow velocities in the figure. Accuracy of Results. Deviations between the linear dependence of wta on 1/uO2,as predicted by Equations 5 , 9 , and 15, and the experimental curves in Figures 1 and 2 may be introduced by the approximate expression employed in the theoretical treatment to describe the departure from equilibrium (IS), by the finite time constant of the detector, and by the dead volume at the end of the column packing. It is evident from the adsorption data of methane and carbon dioxide on the activated carbon ( l a ) that nonlinearity may be neglected, to a rough approxima-
Equation 20 is also applicable t o the first front of the methane-carbon dioxide mixture. A corresponding expression for the second front may be obtained by methods similar to those employed above. If it is assumed that (Y( = 3, and p = p1 ~ ~ 2 p2 . 5(see below), it is found that
1204
ANALYTICAL CHEMISTRY
X A
Col- Length, umn cm. Carbon dioxide-methane B 100 Carbon dioxide-methane C 58 Methane-helium c 58
rd may be regarded as approximately velocity-independent. It is apparent from Equation 22 that 2 T d must then be less than or equal to the intercept observed in Figures 1 and 2, which may be taken as e w . Equation 23 then becomes
By combining Equations 20, 21, and 24, it follows that in general, The effect of the time constant of the detector, T d , on the measured front width, ( w ~ * )is~ ,investigated in detail in the appendix, and it is shown that (wta)n2'v (wta)czf 4
Td2
(22)
i.e.,
Values of Velocity-Independent Coefficients in Equations 5, 9, and 15, Calculated from Data in Figures 1 and 2 (Standard deviations also given) B C D Column 30 58 100 L, cm. 1.08 f 0.14 1.44 =k 0.25 1.09 f 0.19 A sec. 17.59 f 1.53 13.74 f 1.54 15.00 f 1.14 B1*, sq. cm./sec. ... 1.27 =k 0.17 ... (AI*),,,sec. ... 14.52 f 0.70 ( B t * ) s sq. , cm./sec. ... 1 . 8 0 f 0.18 1 . 7 4 % 0.14 ( A t d ) b sec. , ... 17.93 & 0.75 17.85 f 0.50 (Bt')b, sq. cm./sec.
Table II.
0.4
Figure 2. Experimental front widths obtained for equimolar mixture of methane and carbon dioxide at 80" C. and various flow velocities on columns 6 and C
n
Length, cm.
0
0.2 0.3 1/u%,sec2/cm2
0.1
where 141 5 1, although its precise value is uncertain. e increases as the flow velocity is increased, and Equation 25 suggests that a plot of the experimental front width against 1/uO2should deviate from a straight line in the region of the origin, if the corrections considered above are important. Strong curvature is not evident in Figures 1 and 2, but as a result of the scatter of the experimental points, it cannot be concluded that 4 is negligible. The most important conclusion to emerge from the above discussion is that the discrepancy between (wta), and wtb decreases rapidly as 1/u,2 increases. The lines of best fit to points in Figures 1 and 2 for which 1/u,2 2 0.15 sec.2 per sq. em. were therefore determined. Their slopes and intercepts are compared in Table I11 with the corresponding values obtained by using all the experimental points. It is
evident from the results that the errors under consideration do not amount to more than factors of 1.3 and 1.5, for the slopes and intercepts, respectively. The dead volume a t the column outlet consisted of the volume of the glass wool plug in the column, which was always less than 0.3 cc., and the volume of the detector, which was less than 0.4 cc. The effect of the dead volume of the apparatus on the accuracy of the experimental results was investigated by varying the size of the glass wool plug a t the outlet of column B. The front width was measured a t different values of the flow velocity, using pure methane (Xi = 1) as an adsorbate. The values of i l l s and Bt8 with their standard deviations, as calculated from the regression analyses, are given in Table IV. The data in the table indicate that any error due to the dead volume is less than the experimental error. On the basis of the above discussion it may be concluded that the slopes and intercepts in Table I1 are definitely accurate to within factors of 1.4and 2.0, respectively. This accuracy is adequate for the present purpose, and the values of A and B determined from all the experimental points obtained for the width of a solute front (see Table 111) are used in the remainder of this paper. Additional inaccuracies in the values of -4 and B arising from errors in the values of the mass distribution coefficients, for example, are neglected. It is clear from the above discussion and the experimental results presented in a previous paper (12) that errors arising from the above-mentioned effects are likely to be far greater than those arising from uncertainties in the isotherm experiments and the Langmuir fit to the adsorption data, for example. Contribution of Longitudinal Diffusion in Mobile P h a s e to Front Width. The coefficients in Equations 5, 9, and 15 describing longitudinal diffusion in the mobile phase may be analyzed as follows. The effective diffusion coefficient is related to the molecular diffusion coefficient, Do, by
D=yD,
(26)
where y is a tortuosity factor. The value of y may be obtained from Equations 5 and 26 as
for a front obtained when pure methane is injected, whereas Equation 9 yields
from Equations 15 and 26. Dlc and D12 denote the values of D, for the methanehelium and methane-carbon dioxide systems, respectively. The values of the parameters applicable to Equations 27 to 29 are given in Table VI. The mass distribution coefficients were calculated by assuming that all the experimental results could be characterized by an average pressure p = 70 em. of Hg. They were then estimated a t 80" C. from Equations 2 and 11 with E = 0.8 ( 2 ) , using the data in Tables I and V. ga was calculated from Equation 7 ; the viscosity coefficients were determined from tabulated values (15) and Wilke's equation (29). The diffusion coefficients a t 80°C. and 70 em. of Hg were calculated from the Hirschfelder equation (20). The values of Bt5, (B18)., and (Bt)a are those given in Table I11 for all points. The values of y calculated from Equations 27, 28, and 29, using the data in Table VI, are 0.66, 0.51, and 0.55, respectively. Values of the tortuosity factor ranging from 0.4to 1 .O are usually reported (1, 5, 6, 20, 1 8 ) , and the results obtained here are therefore in excellent agreement with existing data. Contribution of Resistance to M a s s Transfer to Front Width. The mobile phase contributions to CTAin gas-solid chromatography are the same as those in gas-liquid chromatography ( 2 , 8). Two proposals have been made to account for the magnitude of the stationary phase contribution to CTh when an adsorbent is used. Giddings (8) has recently suggested that it may be ascribed to the nonuniformity of the adsorbent surface, but did not succeed in deriving an expression for Crh in terms of parameters of known order of magnitude. Van Berge and Pretorius (2) have assumed the surface to be approximately uniform, and have included a term due to Knudsen diffusion of the solute molecules in the microporous structure of the solid. They have used their expressions to estimate the average pore length, d,, from euperimental results, and obtained realistic values for the parameter. The validity of the present theory may be gauged by analyzing the experimental results presented above by a similar method, and comparing the calculated values of d, with those obtained by Van Berge and Pretorius. According to Van Berge and Pretorius ( 2 ) the expression for C,h may be written as
Table 111. Comparison of Slopes and Intercepts Determined from All Experimental points in Figures 1 and 2 with Those Determined from Points Corresponding to l / u O 2 2 0.15 Sec.2 per Sq. Cm.
(Standard deviations indicated) l/U,Z 2 0.15 All points sec.2/sq. cm. A t a ,see. 1.25 i 0.11 1.96 f 0.50 Bt', sq.
crn./sec. 14.68 f 0.77 11.20 =k 2.22 (Ata)a,sec. 1.27 f 0.17 1.04 f 0.08 (Bta)a,sq. crn./sec. 14.52 i 0.70 15.15 =t0.26 (At*)b,sec. 1.78 i 0.10 2.12 =k 0.26
(Bt')h. sa.
17.86 i 0.40 16.81 k 0.79
cm:/sec.
Table IV. Effect of Volume of Glass Wool Plug on Values of At. and Bta
(Standard deviations indicated) Volume of Plug, Bta, cc. A f, sec. sq. cm./sec. 0.37 1.11 h 0.23 15.42 f 1.25 0.19 1.09 i 0.19 15.00 i 1.14 0.06 0.81 i 0.12 15.68 i 0.65
Table V. Adsorption Data for Methane and Carbon Dioxide on BDHActivated Carbon at 80" C. (72)
Adsorbate
Methane
Carbon dioxide
adsorbent urnrn,ml. at NTP/
29.02
34.16
31.59 4332
31.59 5007
g. adsorbent Q, cal./mole
Table VI. Values of Parameters Required for Determination of y from Equations 2 7 to 29
ParamPararnValue, sq. eter Value eter cm./sec. Injection of pure methane ICh 5.63 D I , 0.84 IC' 5.11 Bt8 14.68 Injection of mixture klh 5.63 kzn 14.73 kz' 13.65 Dio 0.84 CY' 2.87 Diz 0.21 84 4 . 2 0 ( B t s ) , 14.52 ( B f g ) b 17.86 ~i 1.692 X 10-4 poise
tlC
2.254 X 10-4 poise
for the first front obtained when a mixture of methane and carbon dioxide is injected. For the second front,
+ r ) d,' + 3 D,) + C-2 + CijA+
2 kh(6,/8 (1 Ciph
+
k"2(6,2
7
Cmph
(30)
These coefficients, taken in order, describe diffusion in the intraparticle void VOL. 37, NO. 10, SEPTEMBER 1965
1205
,;, and Cijh from Equations 5 Table VII. Estimation of Values of C,h, CipA,C and 30 for Frontal Analysis of Pure Methane on Column B at 80' C. Param-
eter
Value 70 cm. Hg
f
Table
3 . 1 9 x 10-3 (cm. Hg)-' 4332 cal./mole 2714 X cm. i . 2 5 sec. 0 . 8 4 sq. cm./sec. 5.63
0
;i,
A 't Do kh
...
Parameter Et
Y
V
Q
I
C;h Ciph
v
7"
IT1 \'I VI
CWh Ci/h
Reference
I'alue 0.4 0.7 0.7
(2) (2)
E1
sec. 0:178 sec.
\ T
0.4 X sec. 5.5 X sec. 1 . 2 X 10-11 sec.
Values of Parameters Required for Determination of d, from Equations
Table VIII.
32 to 34 Parameter
70 cm. Hg 20 X 10-8 cm. 6 . 8 3 X lo4 cm./sec. 4 . 1 2 X 104 cm./sec. sec. 4332 cal./mole 5007 cal./mole 4.83 X sec. 1.26 X 10-lo sec.
71 6,
O,(.~ = 0). 82
t2 Q2
TI(
=
Value
7)
72
spaces, the wall effect, interfacial adsorption, and microporous diffusion. e ' denotes the interparticle fractional void volume of the column and Q is a coefficient determining the magnitude of the wall effect. 6, and d, denote the pore diameter and average pore length, respectively. D, is a surface diffusion coefficient, and 0 is the average thermal velocity of a molecule in the gas phase. The time, T , may be written as (4) 7
= T,,eQ/ROT
(31)
where is a convenient parameter. i\n indication of the relative magnitudes of the contributions to Crhfor the present systems may be obtained from a n analysis of the results in Table 111 for pure methane. Crh was calculated from Equation 5 for pure methane on column B at 80" C., and the corresponding values of C,,h, Cut,and C,jh were estimated from Equations 30 and 31. Table VI1 contains all the necessary data, and the results. It is clear that for this particular case C,h z Cmph,and i t may be shown by calculation that this conclusion is generally valid for the systems considered in the study. I n order to estimate the value of d, from the experimental results, values must be assigned to 6,, D,, and r0. Although the precise magnitude of these parameters is uncertain, the values used by Van Berge and Pretorius are employed, and their values of d, are compared with those obtained. It is assumed that 6, = 20 X cm. (9), that 370,
