Temperature Limitations of Stationary Phases in Gas Chromatography STEPHEN J. HAWKES' and ERIC F. MOONEY Northern Polytechnic, london, N. 7, Great Britain
b The liquid mass transfer coefficient,
CL,
is proportional to viscosity: viscosity is of critical importance in the efficiency of glass bead columns and may b e so in other columns if C, is low and the column loading and k are high. Catalytic decomposition of the stationary phase by the support is common with polar liquids, especially if hydroxylic. Where this does not occur, the volatility of the stationary phase may b e determined by extrapolation of the vapor pressure of the bulk liquid, if the column loading is more Viscosities and volatilities of than 1%. stationary phases are tabulated.
T
MAXIMUM TEMPERATURE at which a stat'ionary phase may be used is determined by it,s volatility, since excessive volatility shortens the column life and contaminates the gas stream. The lower limit is determined by its viscosity which l i d s the efficiency of the column, or by its freezing point. Volatility and viscosity mill be discussed in separate sections of this paper. HE
VISCOSITY
T h e efficiency of a column is usually espressed as the height equivalent to a theoretical plate, H , which is given by the equation Theoretical.
H
=
+
Li B/u
+ C,U + Ciu
(1)
Alllformulas so far derived for Cl, the resistance to mass transfer in the liquid phaqe, show it to be inversely proportional to diffusivity, D l . According to the .\mold equation ( I ) ,
(which has been used by Scott (24)and Desty (4)to calculate C i ) , D1, 0: q-'/*. According to the Scheibel equation (26) Dl
=
17.5 X lo-''
T ~
TjV8'/3
(3)
in common with most others, Di 0: 7-l. Hence C l should be proportional either to or to 7 . In these equations only 1 Present, address, Chemistry Dept., IJniversity of Utah, Salt Lake City, Utah.
Table I.
Temperatures a t Which Stationary Phases Have Quoted Viscosityc
(Values in parentheses with no other comment are extrapolated) Viscosity 1 3 10 30 100 300 1000 3000 10000 Cnitsb (0.2" 2.54 cs, 20.2" = 1.68 CS) Acetonyl acetone cs Apiezori L 200 140 95 67 (45) cs 7,X-Benzoquinoline 74 37 (supercooled) cs 97 52 Benzvldiohenvl cs "1 : 2: 3-tris-(Cyanoethoxy) Propane"136 79 47 21 cs Dibutyl tetrachlorophthalate 119 66 37 16 cs Dibutyl o-tartrate 109 64 39 23 cs Diglycerol 223 147 106 79 60 42 cs 11.2 a t 0.2". 0 93 at 20.0") IXmethylformamide cs F,t,hvlene elvcol bis.~~ 83 25 8 C ~ ~ a n o - ~Ether t~yl cs 112 81 57 40 24 Glycerol (25, 28) 13 4 CP 91 26 n-Hexadecane ( 2 3 ) CP LAC-4-Ri76 (210) 135 92 67 cs p,p-Osydipropionitrile 69 22 -3 cs Poly(ethy1ene glycol adipate) 165 106 (75) cs (Polyethylene glycol 1500) adipate (50 at 160') 106 68 cs Polypropylene sebacate 115 78 52 Samnle 1 32 C8 Sample 2 187 126 80 52 cs Squalane 118 61 24 (0) cs Sucrose acetate isobutvrate 166 130 105 CS 98 80 75 55 (6) CP ( G 7 at 20 7", 10 25 a t 0 2') Tetraisobut ylene cs Tetraethylene glycol dimpthvl ether 13 7 at 20 4'. 6 72 at 0 2') cs Tetrahqdroxyethylethvlenediamine 142 105 77 59 cs Tricresyl phosphate (1,9, 31 1 77 38 19 7 -3 -11 19 cp Tri-2,4-uylenyl phosphate 135 85 63 46 (23) cs a J. F. Smith has reported that' the infrared spectrum of this substance contains a strong hydroxyl band and the authors confirm that this is true of the cornrnercial material. I t is presunied to be 1,3 bis-( 2-cyano-ethoxy)-2-hydroxy-propane. The distinction is unimportant f o r the purposes of this paper. b cs = cp/dl. c Other viscosities available in the literature are dL(2-ethyl-hexyl) phthalate and sebacate ( 2 ) ,dinonyl phthalate ( 1 5 ) ,n-hydrocarbons ( 2 3 ) ,ethylene glycol ( 7 ) . ~
~
~~
T and ~j vary appreciably wit,h temperak ) 2 D lthe only ture and since C i0: k / ( l further quantity in C l which is markedly temperature dependent is IC, t,he column capacity ratio. Hence either C l ( l k)z/kq1'2 or C L ( l k ) 2 T / k ~ jshould be independent of temperature and these relations have been used to evaluate the viscosity dependence of C l . The masirnum viscosity a t which a stationary phase should be used is then determined by the maximum acceptable value of H , subject to the further limitat,ion that there is little advantage in reducing Ci below C,. Two expressions are therefore derived for each type of column, one giving the viscosity where
+
+
+
Cl = C,, and t'he other that yielding an acceptable value of H , assuming C , to be negligible, The theories of column efficiency and of liquid diffusion do not permit8 the masimum viscositie$ to be detrrmined accurately, but fortunat'ely viscosity varies ,exponentially with temperature so that an error of a factor of five usually involves a temperature difference of less than 20" C., a margin of error which it is not usually difficult to allow for. The fact that a maximum viscosity has been suggested does not. necwsarily mean that a higher viscosity may not, be used, since a reduction in temperature VOL. 36, NO. 8, JULY 1964
1473
I 10
001,
k Figure 1.
Viscosities for which
CZ = C,
Figure 2.
(Carrier gas = He; P = 1 atm. (cap'y), 2 atm. (packed); r = 0.025 cm.)
may produce an increase in the retention ratio of a pair of substances which more than offseta the reduced efficiency. However, in such cases a more limpid stationary phase with similar chemical properties vi11 usually be an advantage. Since the viscosities of stationary phases have not previously been compiled, they have been collected or determined by the authors and are shown in Table I .
Experimental. A Pye Argon Chromatograph was used with a column 4 feet in length containing a mixture of equal parts by weight of polypropylene eebacate and 40-60 mesh Celite treated with hesamethyldisilazane. With so high a loading of stationary phase, A , B, and C, could be ignored, thus sirnplifyinp the analysis of the data. H was measured by the method recommended by the V. P C. conference, 1956 (3).
Table II.
+
+
Ci with
Viscosity
C(l
Sample Celite Benzene
T( K.) O
298 323 373 418
Glass Beads Methyl-pentane
Toluene p-Xylene
1474
dCP)
15,000 3,200 505 195
+ k)2
kq
0 0 0 0
063 050 018 011
0 117
,
,
,
,, 100
,
,
, , , ,, IOC
k Viscosities for which H,,,
=
0.1 cm.
+
16k d?DO (*) 1 6k llk2 C, The ratio D J D l may be obtained from t,he Gilliland equation for gaseous diffusivity (19)and the Scheibel equation for liquid diffusivity (22)
C[ -
-
+ +
0. -Di 2.46 X lo6 T1'2(.kf8-1M,-1)1/2Val'~t)
P(V,'/3
+ + V,1'3)2
(5) where P may be taken as the average column pressure. The following approximationq may be made: 20 f 4 from -2OOto 300" C. 3.1 f 0.1 for H2, He, Nz,X V a 1 / 3 / ( V 8 1-I-' 33.1)2 = 0.07 =k 0.01 for 25 < V , < 1000 ml. =
=
and hence for the conditions usual in gas chromatography
TC(1 k)2 kt) 2 2 2 3
14 83 91 42
302 334 373
51 4 12 2 4 6
0 040 0 019
4 93 3 83 3 2,5
334 353 373 334 353 373
12.2 6.9 4 6 12 2 6 9 4 6
0.066 0.057 0 037 0 137 0 087 0 057
6.30 7.63 6 33 13 1 11 7 9 7
ANALYTICAL CHEMISTRY
,/,
(Conditions as Figure 1 )
Data on glass beads were taken from Littlewood ( 1 7 ) . Results and Discussion. The results are summarized in Table 11. TC(1 k)'/kt) varies by a factor of 1.6 on the Celite column for a 75-fold change in t) and on glass beads by an average of 2 for any one solute for a 12-fold change in t). In contrast, C(1 k)2/kt)1/2 varies by a factor of 6 on both columns as t) varies by a factor of 12 and 75, respectively, or t ) l l 2 varies by a factor of 3.5 and 9. Hence Cl is proportional to t) within a good approximation but not to ~ " 2 . Accordingly, the Scheibel equation has been used for the remainder of this paper. Capillary Columns. The Golay equation ( I S ) for capillary column efficiency gives
Variation of
,
3.4 X lo6 (,Ifa-'
+
Mg-l)1/2t)
P
(6)
Scott's values (94) of Do and D ,for n-heptane in argon and dinonvl phthalate give D J D L = 3.3 X I O 5 where Equation 6 gives 1.6 X lo5in reaaonable agreement. This suggests that the uniform film model used by Scott I C approximately valid for the nylon capillaries he used. Equations 4 and 6 give 54 X 106k
-C[_C,
1
dj2
+ 6k + l l k 2 -r2x
Values of p when C l = C,, derived from Equation 7 , are plotted against' k in Figure 1. Even at moderate values of d, and IC, limpid liquids are needed to meet this requirement. For finer capillaries and for thick films Cl will usually be the dominant term. Ahxeptable values of p may be calculated by differentiating Equation 1, neglect'ing C , (=I is zero for a capillary column and negligible for most. others) giving
N,,, = 213"fCL"f
Experimental and Calculated C[/C, on 120-1 40 Mesh Chromosorb P
Table 111.
Carrier Sample
gas
C& C4Hio
IrT2
NZ
Hz Hz
CiH8
C4Hx C3H8 C4Hx
Nz
N2
Hz H?
C3H8
CIHIO
70
k 1 03 3 40 1 03 3 40 0 49 1 61 0 49 1 61
20 20 20 20
1-B 10 10 10
tained by combining Equations 6 , 8 , and 10 with
Hm,,nz= 18 X lo6
where y
+ ___________ P(l + k ) f
N,-l)%,?k
B
(9)
=
Hmln2 = 6.5
C,
= w
dP2/D,
(11)
where w > 0.8 for Chromosorb P ( I O ) . Combining Equations 6, 10, and 11 gives
+ +k
M - 1 ) l ' Z
2 2 3 7 9 9 6 8
0 0 1 1 0 0 1 1
Calcd./ Expti.
5 4 8 6 5 4 7 5
6 5 5 4 1 2 1 1
4 5 2 2 8 2 6 9
p
) 2 P
+
+
(14)
Values of 7 giving H = 0.1 em. are plotted in Figure 2. The limitation is not stringent. Glass Beads. For glass beads, Giddings ( I O ) gives w = 0.6, and shows (9) that =
Putting 0.7 for d, = 2.4 to 5.9 and dl = 0.7 to 1.0 f
The validity of Equation 12 can be cheeked using the data of Perrett and Purnell (20) a t 40.8" where q for noctadecane, the stationary phase, is 0.03646 poise (28). The results are recorded in Table 111. Calculated and experimental values differ on average by a factor of 3 which is reasonable in view of the assumptions made. Values of q which give Ct = C , are plotted in Figure 1. Cl is likely to be dominant for fine particles except at high values of k or loner values of p than can often he obtained in practice. For coarwr particles efficiency is usually limited by C, and high visco4ties are acoeptable. When C, is negligible, values of q which give acceptable efficiency are ob-
(16)
and combining Equations 6, 11, 15, and 16, using Giddings' value of m = 6.25 (9)
5 = 2.4 x 104 x
+
k(MM,-' Mo-1)1'2 P(1 kI2 (12)
UPPER TEMPERATURE LIMITS
(Jfs-l *lf,-1)1/* kq X P(l k)f
(d,/dL)1'2= 2.2
+
%l'Zq
(17)
Since this equation does not contain d,, the relative importance of C, and Ci is independent of bead size. Values of p giving C1 = C, derived from Equation 17 are plotted in Figure 1. C, is negligible except a t very high values of k or impossibly low 7. Values of p giving acceptable H,,, are obtained by combining Equations 6, 8, 13, 15, and 16 and using Knox and RfcLaren's value (16) of y = 0.60, obtaining ~
~
=
7,
(-If8-'
x
+
104 ~
x2 d D 2%'" kp
P(1
+ k)f
factory efficiency on large glass beads, and that even on small ones the viscosity needs to be carefully controlled.
(13)
*o)
c, k(M-1 (1
D,
27
(
ci
The const,ant 4 X lo-' is for 100-140 Chromosorb 1' but can probably be used for other mesh sizes as a first approximation. I t s use for other supports would be speculative. Ciddings gives the C , (IO) term as
=
= 0.6 ( I C ) , giving
1)
Values of p giving If,, = 0.1 are plotted In Figure 2 Reaionable efficienciey can evidentlj be obtained at fairly high vi>coyities irrespective of C,. Chromosorb P. Pretorius et al. ( 1 4 ) gire the formula for C i for 120-140 mesh Chromoeorb I' as
CI
3 2 9 6 0 0 2 2
Ct/C, Exptl.
(8)
The expressions for B and C l in the Golay equation ( I S ) combined with Equations 6 and 8 give
(M-1
ClIC,
Calcd.
(18)
Values derived from this equation, plotted on Figure 2, show that it is virtually impossible to obtain satis-
Limitation Due to Decomposition. Gerrard, Hawkes, and Xooney ( 8 ) have shown that polypropylene sebacate and the adipate of polyethylene glycol I500 are dehydrated on the column and that dinonyl phthalate is decomposed. This has been further investigated. The dehydration has been shown to be catalytic. Fifty grams of polypropylene sebacate was boiled with xylene in a Dean and Stark apparatus a t 150" C. A few milligrams of wvat'er distilled in t,he first few minutes, but no more in 8 hours. Since Gerrard el al. found substantial dehydration a t 150" this could not have been purely thermal but must have been catalytic. Pure polypropylene sebacate decomposes only at much higher temperatures, decomposing rapidly at 350" to 400" C. and leaving a residue with no infrared hydroxyl band. The decomposition of diglycerol and polyethylene glycol 400 by Celite is demonstrated by their behavior in the Pye Argon Chromatograph. When supported on acid- and alkali-washed Celite a t 100" C., polyethylene glycol gives a wavy baseline following the cycles of the heater. The instrument instruction manual gives a test for excessive stationary phase volatility based on the change in background current in the detector with change of voltage, which the polyethylene glycol just passes a t 100" when supported on Celite, probably assisted by the elution of water reducing the det'ector sensitivity. However, when supported on glass beads it passes the test comfortably a t 140" C. and gives a stable baseline. Diglycerol passes the test at 100" and 150" C. when supported on glass beads although it passes only a t 50" when supported on Celite. Giddings and Hawkes (11) have found that the apparent vapor pressure of dimethyl phthalate when supported on Chromosorb P at 100" C. is initially many times higher than that of the bulk liquid and after prolonged elut'ion drops VOL. 36, NO. 8, JULY 1964
1475
Table IV. Calculated Volatilities at Tuey’s Recommended Limits
Substance Benzyldiphenyl Diglycerol Dinonyl phthalate n-Hexadecane Squalane
T
( ” C.)
120 150 130 75 160
Volatility (gram/ml.) 1.13 X 0.2 x 10-6 0.27 X 10-6 2 06 X lo-@ 0.9 x 10-6
to a steady value about twice that of the bulk liquid, indicating substantial decomposition. Extrapolation of Vapor Pressure Data. T h e maximum permissible volatility of a stationary phase has been discussed by Gerrard, Hawkes, and Xooney (8). The temperature a t which it reaches that volatility can be obtained by extrapolation of vapor pressure data provided it is not decomposed by or strongly adsorbed on the support. These effects may be considered in the light of Tuey’s data (18) which show the temperaturei: a t n hich stationary phases lose a weight which is equivalent under his conditions to 1.1 X gram of stationary phase per ml. of carrier gas. Table IV shows these temperatures with the volatilities obtained by extrapolation of vapor pressure. Results for squalane and benzyldiphenyl agree well with theory and hexadecane shows a low volatility indicating adsorption which is
Table V.
scarcely credible if squalane is unadsorbed. Dinonyl phthalate and diglycerol are both substantially more volatile than theory, indicating decomposition. Giddings and Hawkes (1I) have found that the vapor pressures of ethylbenzene and ethylenediamine support’ed on Chromosorb P a t loadings greater than 1% are substantially those of the bulk liquid. It is therefore concluded that extrapolation of vapor pressure data is satisfactory so long as there is some security against catalytic decomposition. Volatilities. The volatilities of a number of stationary phases are compiled in Table V, calculated in grams of stationary phase per milliliter of carrier gas. Except where a reference or note is shown, these were obtained by extrapolating from boiling points a t various pressures given in the literature. T h e accuracy is not high but fortunately the needs of gas chromatography in this matter are not exacting. GLOSSARY A 1 ~
A , B, c
B
abnormality constants in the Arnold equation for liquid diffusivity constants in the van Deemter eauation constant in ‘the Arnold equation for liquid diffusivity
Volatilities (gram per mi.) of Stationary Phases
ANALYTICAL CHEMISTRY
H,,,
k
P r
(Literature citations are to data from which the volatilities are calculated) Same 10-6 10-6 10-7 10 -* 10-9 34 1 Freezes Acetonyl acetone 118 72 Freezes 7,8-Benzoquinoline 161 121 85 58 Freezes Benzyldiphenyl 92 55 24 Freezes Dibenzyl ether (6) 89 48 16 - 10 - 34 Dibutyl maleate6 125 97 72 52 34 Dibutyl phthalate (26) 177 142 111 85 61 Dibutyl tetrachlorophthalate ( 2 1 ) 200 165 135 108 85 Di-n-decyl hthalatec 232 179 126 100 70 Di-iso-decyP phthalatec 212 151 134 104 77 Di-(2-ethyl-hexyl) sebacate (21 ) Di-(2-ethyl-hexyl)phthalate(21,26, SO) 178 145 116 90 68 117 81 50 24 3 Diethyl sebacate (27) Diglycerol 203 171 142 118 96 Dimethylformamide -6 -32 -49 Freezes 175 145 119 95 75 Dinonvl phthalated Di-n-octyl phthalate ( 2 1 ) 194 158 127 100 78 182 142 110 81 57 Di-iso-octyl phthalate ( 3 0 ) 212 179 150 125 102 Di-t,etradecyl phthalate (21 ) Dodecane (27) 47 13 Freezes Ethylene glycol (27) 69 24 5 Freezes 136 101 70 44 22 Glycerol (27) Glyceryl tristearate ( 2 1 ) 230 203 179 157 36 n-Hexadecane (27) 99 -64 25 Freezes 143 111 83 59 37 p,p-Oxydipropionitrile Squalane 196 160 130 103 80 211 169 135 105 Sucrose acetate iso-butyrate” 169 127 93 63 39 n-Tetracosnne (27) Tetraethylene plycol dimethyl ether 98 62 32 7 - 13 58 22 -6 - 29 -49 Tetrrt-iso-hut,ylene Tri-p-cresyl phosphate (29) 198 163 134 104 85 a Determined by the signal in an argon detector as described by Gerrard e t al. (24)s using squalane as standard. 6 By extrapolation from the methyl, ethyl, and n-propyl esters. c By interpolat,ion between n-octyl, iso-octyl, and tetradecyl phthalates. d 2,2,4-Trimeth?;l-hexyl phthalate.
1476
H
s T
centipoises centistokes resistance of the sample to mass transfer in the gas and liquid phases (seconds) density of glass film thickness (em.) density of stationary phase particle diameter (em.) diffusivity of the sample in the gas and liquid phases second) height equivalent to a theoretical plate (em.) minimum value of I€ as u changes (cni.) column capacity ratio (amount of sample in liquid phase,’amount in gas phase) molecular weights of stationary phase, sample, and carrier gas viscosity (poise) average gas pressure (atm.) vapor pressure of sample
(mm.1
radius of capillary (em.) sum of molecular diameters of sample and stationary phase temperature of the column (” K.)
U UOPt
% Y w
linear gas velocity (em./ second) linear gas velocity for (em./ minimum H second) molecular volumes of carrier gas, stationary phase, and sample (ml.) grams stationary phase in 100 grams of loaded packing constant in equation for B constant in equation for C, ACKNOWLEDGMENT
The authors gratefully acknowledge the assistance and encouragement of Villiam Gerrard, and helpful discussions with J. H. Knox and J. C. Giddings. S. J. H is indebted to the management and directors of 11‘. J. Bush and Co. and especially to their chief analyst. X. J. 11. Bailey, for time, opportunity, and encouragement to undertake the research. LITERATURE CITED
(1)Arnold, J. H., J . Am. Chem. SOC.52, 3939 (1930). (2) Bried, E.AI., Kidder, H. E., llurphy, C. M..Zisman. W. A,. Ind Ena. Chem. 39,487(1947).’ (3) Committee on Nomenclature of the First International Symposium on Gas Chromatogra,phy, “Vapour Phase Chromatography, D. H. Desty, ed., p. xii, Butterworths, London, 1957. (4)Desty, D. H., Geach, C. J., Goldup, A , , “Gas Chromatography 1960,”R. P. W. Scott, ed., p. 46, Butterworths, London, 1960. (5)Dreisbach, R. R., Martin, R. A , Ind. Ena. Chem. 41, 28i5 (1949).
(6) Eastman Kodak Ltd. Bulletin, SAIB. ( 7 ) Fordham, S., Research (London)Suppl. 1 , 3 3 6 (1948). (8) Gerrard, W.) Hawkes, S. J., Mooney, E. F., “Gas Chromatography 1960,’’ R. P. W. Scott, ed., p. 199, Butterworths, London, 1960. ( 9 ) Giddings, J. C., ANAL CHEM.34, 1026 (1962). (10) Zbid., p. 1190. ( 1 1 ) Giddings, J. C., Hawkes, S. J., unpublished data, University of Utah, 1964. (12) Gilliland, E. R., Ind. Eng. Chem. 26, 681 (1934). (13) Golay, 11. J. E., “Gas Chromatography 1958,” I). H. Desty, ed., p. 36, Butterworths, London, 1958. (14) Gord(in, Y. M,,Krige, G. J., Pretorius, I.,,A s . 1 ~CHEM. . 36, 750 (1964). (1.5) Houghton, G., Kesten, A. S., Funk, .J. E.. Coull., J.., J . Phus. Chem. 65. 653 (1961).
(16) Knox, J. H., McLaren, L., ANAL. CHEM.36,1477 (1964). (17) Littlewood, B., “Gas Chromatography 1958, D. H. Desty, ed., p. 27, Butterworths, London, 1958. (18) May and Baker Ltd., Dagenham, Essex, Gt. Britain, “Materials for Gas Chromatography,” p. 15 (1958). (19) Meissner, W., 2. Angew. Physik. 1, 75 (1948). (20) Perrett, R. H., Purnell, J. H., A x . 4 ~ . CHEM.35,430 11963). (21) Perry, .E. S., Weber, W. H., J . A m . Chem. SOC.7 1 , 3727 (1949). (22) Reid, R. C., Sherwood, T. K., “Properties of Gases and Liquids,” p. 286, McGraw-Hill, New York, 1958. (23) Rossini, F. D., “Selected S-alues of
,p.
Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,” Table 20c, (Part 1 ) p. 3, Carnegie Press for American Petroleum Institute, 1953.
(24) Scott, R. P. FV., “Gas Chromatography 1960,” R. P. JV. Scott, ed., p. 144, Butterworths, London, 1960. (25) Segur, J. B., Oberstar, H. E., Ind. Eng. Chem. 43, 2117 (1951). (26) Small, P. A., Small, K. W., Cowley, P., Trans. Faraday SOC.44, 810 (1948). (27) Stull, D. R., Ind. Eng. C‘hem. 39, 532 (1947). (28) Yand, V., Research (London) 1 , 44 (1947). (29) T’erhoek, F. H., Marshall, A. L., J . A m . Chem. SOC.61,2741 (1939). (30) Werner, A. C., Ind. Eng. C‘hem. 44, 2736 (1952). (31) Williamson, I., Suture 167, 316 (1951).
RECEIVED for review February 3, 1964. Accepted May 15, 1964. Presented a t 2nd International Symposium on Advances in Gas Chromatography, University of Houston, Houston, Texas, Rlarch .23-26. 1964.
A New Gas Chromatographic Method for Measuring Gaseous Diffusion Coefficients and Obstructive Factors J O H N H. KNOX’ and LlLlAN McLAREN Departmenf of Chemistry, University o f Edinburgh, Edinburgh, Scotland
b A new elution method is described for the determination of gaseous diffusion coefficients and obstructive factors ( D o and y). It has a standard A narrow deviation of about band of an unsorbed solute is injected into a column and eluted part way down. The flow is arrested and the band is allowed to spread by diffusion for different times. The band is then eluted from the column at a precisely known velocity and its concentration profile is determined by a suitable gas chromatographic detector. Do or y D , is determined from the gradient of a plot of (peak widthI2 against time of residence in the column. Experiments with open tubes gave D, = 0.1 65 sq. cm. second-’ for ethylene i n nitrogen at 18’ C. and 750 mm. of Hg. Experiments with conventionally packed columns gave the following values of y for glass beads, firebrick, and Celite, respectively: 0.60, 0.46, and 0.74. These values are interpreted in terms of the tortuosity and constriction of the diffusion paths. The reality of the two factors contributing to the obstructive parameter is verified b y comparison of the calculated and measured values of y for a series of columns composed of a single row of spherical beads in a closely fitting glass tube. Agreement is within 2y0 for all columns studied.
270.
HEX
AN
IKFINITELY
THIN SLICE
of a gas I3 spreads by diffusion into anothrr gas h the concentration Iirofilc of 13 in X after a finite time t is
gaussian and the rate of spreading is given by the equation
duZ2’dt = 2 0 ,
(1)
where us is the standard deviation of the gaussian concentration profile measured as a distance, and D, is the diffusion coefficient of B in 4. Equation 1 holds for asial diffusion in a uniform empty tube, but if the tube contains obstructing, but not sorbing, material, diffusion is hindered and
duZ2/dt = 2yD0
(2)
where y is less than unity and in gas chromatography has usually been called the “tortuosity factor”. The term is unfortunate since the tortuosity of paths through a packed bed is only one factor contributing to the obstruction. I n this paper, following Giddings (8),we shall call y the “obstructive factor.” D , could evidently be determined by inserting a slice of B into a tube containing rl and measuring u,* as a function of time, but this is not generally possible in practice and an elution method must be used where the time variance, uC2,of the eluted hand rather than the length variance, ur2, is measured. If the spreading band moves slowly down the column a t a constant linear velocity, u, longitudinal diffus’on w 11 occur a t the same rate as if the band were static. I n an empty tube, if u is sufficiently low, other sources of spreading may be made insignificant. Since U ? / U , = u , diffusional spreading is given by
dut2/dt
=
2D,/u2
(empty t,ube) (3)
d u t 2 / d t = 2-yD,,/uZ (packed tube)
(4)
The speed of movement of the band along the column is u = f!A,e, where f = volume flow rate (cc. second-’), -1, = area of tube when empty (sq. em.), and e = total porosity of the packing = (volume of gas in packed column)/(volume of gas in empty column). If the gas 13 were sorbed by the column the apparent diffusion coefficient would be y D J ( 1 12) where k is the column capacity ratio; u would still be the band velocity but would be (1 IC) times lower than velocity of an unsorbed band. There are two elution methods for determining D , and y, the continuous elution method hitherto used and the arrested elution method now described. The continuous elution method has a number of distinct variants. In esperiments specifically designed to determine D , or rD,,. a band (3. ?’) or front (12) of an unsorbed gas is driven slowly down a column at a series of sufficiently low gas velocities that spreading from slowness of mass transfer in the gas phase is either negligible or can easily be allowed for. An interesting modification is the pulsed flow method of Carberry and Bretton (6) in which the rate of degeneration of a sinusoidal concentration profile is measured. These methods are applicable to both
+
+
1 Corresponding author. rlJdress until September 30, 1964, Department of Chemistry, Universitr of Utah, Salt Lake City, IJtah.
VOL. 36, NO. 8, JULY 1964
0
1477
