on the time taken for ashing. This was checked at two furnace temperatures: a t 455" C. where no losses had been observed after a n itshing time of 20 hours, and a t 620' C. where appreciable losses were observed in this time.
90M70.
60-
s
RESULTS A N D I)ISCUSSIONS
The results are shown in Table I. The results for CsN03 are not shown; they were practically identical to those for CsC1. I n both csases losses became appreciable for furnace temperatures above 550" C. At 4150" C. any loss was less than the experimental error ( w 3%). A measurement of the activity on the crucibles used for ashing showed that some Cs137 was retained by the crucible even after cleaning with chromic acid. This amount was approximately the same over the tempclrature range used, and was estimated at about 2%. When a platinum crucible was used as container, no detectable Cs13' remained after removal of the ash. Gorsuch ( 2 ) has measured the retention of lead by used platinum crucibles and by both new and used silica crucibles after heating a t 630" C. for 16 hours. He found 1 to 2oj, adsorption by the platinum crucibles, 8 to 24% adsorption by the used silica ci~ucibles, and about 70Oj, adsorption by the new silica crucibles. I n this investigation, all crucibles were new and on the basis of these results it is rezisonable t o assume that no more than a few per cent of Cs will be retained by Vitreosil crucibles. The temperature measured by the
50-
40-
: 10-
perfectly safe for ashing purposes, even when samples with high organic content are being ashed. This confirms the International Atomic Energy Agency Panel recommendation ( 5 ) . The results indicate that in ashing a t this temperature the only loss to be expected is due to adsorption by the crucible used. This loss will depend on the type of crucible but a t most should be only a few per cent. ACKNOWLEDGMENl
Figure 1. Percentage loss of csI3' vs. time in furnace, a t furnace temperature of 620" C.
We are indebted to L.E. Smythe for helpful suggestions, and to Ian Caine who carried out some of the experimental work. LITERATURE CITED
(1) Blincoe, c . ,
~ A L CHEM. . 34, 715 (1962). thermocouple placed in the sample ( 2 ) Gorsuch, T. T., Analyst 84, 135 ranged from 50" to 100" C. above the (1959). furnace temperature and remained ( 3 ) Gorsuch, T. T., Zbid., 87, 112 (1962). ( 4 ) Harley, J. H., U.S.A.E.C. Health higher even after ashing was completed. and Safetv Laboratorv. New York. For example, when the furnace temperaprivate communication,"l963. ture was set a t 490" C., the sample (5) International Atomic Energy Agency. reached a temperature of 590" C. "Panel on the Methods for Collection and Analysis of Samples for the Deterduring the ashing process and remained mination of Trace Amounts of Radioa t a temperature of 540" C. after the active Substance in the Biosphere." ashing was completed. Vienna. 7-12 SeDtember. 1959. Figwe 1shows the time dependence of (6) Murthy, G. 'K., Jakagin, L. P., Goldin, A. S., J . Dairy Sci. 42, 1276 the loss of C S ' ~a t~ an ashing temperature of 620' C. The loss of C S ' ~ ~ (1959). increases to about 90% after 6 days, no ROY?VI. GREEN further loss being indicated after the RAYMOXD J. FINN seventh day. At 455" C. no Cs13' Health Physics Research Section was lost after the sample had been in the Australian .4tomic Energy Commission furnace for 3 days. Research Establishment It would seem, therefore, that a Private Mail Bag, Sutherland S.S.W., Australia furnace temperature of 450" C. is
Interpretation of Linear Carrier Gas Velocity in Gas Chromiatography SIR: Most equat ons for the plate height for a packed g:is chromatographic column contain the linear velocity of the carrier gas as a parameter. It would appear from the literature that the latter parameter is not always interpreted in the same 1vay (3, 4, 12, 20). This can affect the calculated values of the coefficients in the plate height equation (3, 10). The various ways in which the linear flow velocity of the carrier gas through a packed column may be described are most conveniently discussed by considcring the following model of a packed column. The granular support is considered as consisting of spherical porous particles. The pores may either be blind or pass right through the rarticles. The intraparticle volume is defined as that portion of the total Troid volume of the column which is occupied by the pores; the interparticle void volume is defined
as the total void volume minus the intraparticle volume. The packed length of the column will be designated by L, the fractional void volume of the packed column by C, and the fractional void volume excluding the intraparticle void spaces by E'. Finally, let ( p p.) be the pressure drop across the packed portion of the column, p , and p , being the inlet and outlet pressures, respectively. When a carrier gas flows through such a packed column, the flow velocities of the gas molecules in the void spaces will vary over a wide range, both transversely and longitudinally in the column, because of the differences in the diameters of the microscopic flow channels and because of the varying tortuosities of these channels. The overall flow velocity of the gas, therefore, must be considered as some sort of average value of the actual velocities of the gas
molecules (24). The difficulty referred to above stems from the fact that this averaging process may be carried out in a variety of ways. Xot all of these would necessarily be suitable for the present purpose; those which include macroscopic parameters which are readily measured would undoubtedly be the most suitable in practice. Filter Flow Velocity. The filter flow velocity of the gas a t a distance x from the column inlet, uz', is defined as the ratio of the volume of gas which flows through the cross sectional area of the column a t a point z, per unit of time, to the cross sectional area of the column (23). This dimensionally linear velocity is, however, significantly different from the average value of the microscopic velocities in the flow channels. First, no cognizance is taken of radial variations of the linear velocity; second, no corVOL. 36, NO. 3, MARCH 1964
8
693
rections are made for the presence of the support; and third, u,!denotes only the axial component of the linear velocity. The tortuosity of the gas channels has thus not been taken into account. Pore Velocity. T h e pore velocity of a gas a t a distance 5 from t h e column inlet may be defined a s t h e average value of the magnitude of t h e linear flow velocities of t h e gas molecules at a distance 2 from t h e column inlet. According to the usually accepted Dupuit-Forchheimer assumption, the pore velocity is, however, given by the average value of the components of these flow velocities in the axial direction of flow ( 2 4 ) . The pore velocity so defined does not take into account the tortuosity of the flow channels and may simply be calculated as the ratio of the filter flow velocity, u z f ,to the porosity of the column. For columns packed with the materials usually employed in gas chromatography, this velocity is not uniquely defined. For these supports the average diameters of the interparticle channels are large in comparison with the diameters of the intraparticle channels if such channels are present. Consequently, irrespective of whether the intraparticle pores are blind or not, the gas in the intraparticle spaces may be considered approximately stagnant and the gas in the interparticle spaces moving. As a result of this, the pore velocity may be interpreted either as the effective linear flow velocity or as the interparticle pore velocity. The effective linear flow velocity may be defined as
Hence uZedenotes the average value of the velocities of all the carrier gas molecules, both moving and stagnant, in the axial direction of flow. The value 2 averaged over the length of the column, therefore, will be given by -
When the gas flow is assumed to be laminar, the flow velocity will be proportional to the pressure gradient (Darcy's law) ( 5 ) . From this, the average values of the different flow velocities over the length of the column may easily be calculated as (3, 4, 1618)
-
uf = u,'Y,(P)
where L is the length of the column and solute. The interparticle pore velocity may be defined as (3)
where uZadenotes the average value of the flow velocities of the gas molecules in the moving interparticle spaces in the axial direction of flow. Equation 3 is valid only when the moving part of the gas is restricted to the interparticle space. When the ANALYTICAL CHEMISTRY
B = 2yDI"
(10)
D1"is the gas diffusion coefficient at one atmosphere and y is a tortuosity factor. Since the diffusion takes place irrespective of whether the gas is moving or in this case, must stagnant, the use of be considered as correct. T h e n it is preferred to use P j however, the term should be written as
u',
(4)
€B ~
Ue ~
-
ua =
UPYP(P) u,aY,(P)
3 P(P2 - I ) 2 ( P a - 1)
Y,(P) = -
(6)
(7)
and P is the ratio of the inlet and the outlet pressure. The general plate height equation may be written as the sum of several terms which are independent of, proportional to, or inversely proportional to the linear carrier gas velocity. Several authors have used uzefor this purpose (1, 12, 20, 22). Although Bohemen and Purnell (3) have considered this parameter, in their own equation they use the interparticle pore velocity. Following the majority of authors, the plate height equation in terms of the average linear velocity ue may be given by
A , B, and C are coefficients in the mobile phase terms, CZ is a coefficient in the stationary phase term, lj is the pressure averaged over the length of the column, and Y,(P) is a pressure correction (65)
- 1) (P2 - 1) 8(P* - 1)'
9(P4
e'uap
(5)
where Y 2 ( P )is given by
YdP) =
t,,! the retention time of a nonretarded
velocity; t h e value of t h e coefficient A obtained will, consequently be independent of t h e particular choice of t h e flow velocity. The B Term. This term is determined by t h e longitudinal diffusion of the gas molecules and B is given by (8)
e.
L
Ue = tM
694
diameters of the interparticle channels become comparable to the diameters of the intraparticle pores, the gas in the intraparticle spaces must also be considered as moving (7) and uzabecomes identical with uze and should be calculated from Equation l. Furthermore, if nonporous support particles are used, uzaand uZebecome identical since E' =
(9)
It is evident from Equation 8 that the values of the coefficients A , B, C, and Cz calculated from experimental data are dependent on the choice of the linear carrier gas velocity and the question arises whether a particular choice, here 2, is correct or not. Because the flow velocities have been introduced in the different terms with a definite meaning, this question may best be answered by considering the different terms in the plate height equation separately. The A Term. This term is independent of t h e linear carrier gas
T h e C Term. This term should, on theoretical grounds, contain four contributions resulting from the resistance to mass transfer in the interparticle gas space (2. 8, 11, 15, IQ), the resistance to mass transfer in the intraparticle gas space ( 2 , 9 ) , the wall effect ( I S ) , and a nonequilibrium interchannel effect as proposed by Giddings (10). I t has been shown (1, 10) that C may be written as
c=2
31
x
10-3 6k 12k2)dP2 ( 1 - ~ ' ) ' ( 1 k ) * Di" k)*dp2 1 €(I - E ' / € 30 y( e - e ' ) ( I k)'Di0 a202d,' cr'd 1
+ +
+ + +
-YD'I
+ +
+ Di
(12)
The terms on the right hand side represent the contributions of these effects in the order given above. The average particle diameter is d,, IC is the mass distribution coefficient or the ratio of the amount of solute in the stationary phase and in the gas phase a t equilibrium, LY is the increment of the velocity of the carrier gas in the wall lager in units of the average linear flow velocity, e is the thickness of the boundary layer in terms of d,, CY' is a proportionality constant introduced by Giddings. I n the interparticle gas resistance term, the gas velocity considered is the average linear velocity of the moving gas in the interparticle space and, consequently, U" should be used instead of ?. The same applies to the intraparticle gas resistance term ]Thereby the velocity difference between the moving interparticle and the stagnant intraparticle space is considered. I n the wall effect and in the nonequilibrium interchannel effect, the carrier gas velocities in the interparticle spaces in the wall layer and the central cylindrical layer and in the interparticle capillaries of different diameters, respectively, have been compared. Therefore, the interparticle pore velocity should be used in
t h e corresponding terms. I n conclusion, the contributing term t o the plate height will probably be better described by C g p than by CU”j and when i t is preferred to use U”, the term becomes
The C2Term. Because t h e velocity i n this term represents t h e velocity of t h e gas phase wit’? respect t o t h e stationary phase, the effective linear velocity [leshould be used. It is interesting t o trace how far the experimental results obtained are in agreement with the above proposals. The A term need not be considered for this purpose. This also applies to experimental values obtained for the Czueterm since, until now, for columns packed with the usua porous support, t h e theory could not predict experimental results quantitatively. Therefore, the only terms in the plate height equation to be taken into account are the terms proportion tl and inversely proportional to u“p. Carman (6) has defined the tortuosity as the ratio of the tortuous path length, L,, between two points in the column to the length, L, of the straight line between these two points. It may be shown t h a t the tortuosity factor y as commonly used in chromatography is equivalent to y =
(:-)
(14)
and the higher value of y o is decreased to y =
(0.4/0.7)
=
- €’/e + k)’dP2 30 y ( s - E ’ ) (1 + kj2DT -k
c = -1
0.6
which is much closer l,o the theoretical expected value and I S in accordance with other experimental results (1, 20, 21).
It has been shown that, in general, the interparticle gas resistance term is negligibly small (1, 10) and t h a t the intraparticle gas resiritance term, al-
e(1
in terms of the intraparticle gas resistance term and the wall effect. From experimental values of C obtained from Equation 8, the value of (ae) may be calculated. The (cue) values so obtained, however, are apparent values and the corrected values, by taking the actual linear velocity ua may be calculated from -
((.e)
Table 1.
Reference
(L”L2)
Carman has adopted a value of 4 for (L,/L), which cori*esponds t o y = 0.5. The experimental values of y which show the largee t deviation from 0.5 have been repori ed by Bohemen and Purnell (3, 4 ) . They have found the average value of y t o be unity. These authors should, however, have used the effective liiear velocity 2 instead of 2,and the apparent value y‘ when using the latter may be related t o y as =
though of more importance for porous particles, only amounts a small fraction of the coefficient C in Equation 12. The biggest contribution is made by a k independent term proportional to d,2/D10 which may be a result of the wall effect and/or the nonequilibrium int,erchannel effect (IO, I S ) . The two effects are very similar in nature as well as in the form of the contributing terms. Since the present arguments are not affected by attributing the whole effect to t h e one or the other, the following equation for C may be used
= (ae)a
Comparison of Values
Support
( 4 ) Firebrick (20) (22) (22) (18) (14) (1)
-& (ore)a 0.9 0.9 0.5 0.4 0.8 0.4 0.9
be)
(16)
and
(4
0.7 0.7 Chromosorb 0.5 Glass beads 0.3 Chromosorb 0.6 Firebrick 0.4 Glass beads Celite 0.6 Average values 0 . 7 0.54 Standard zt0.24 f 0 . 1 5 deviations
In Table I the ( d )and (@e). values from the different sources have been given. The value of y is taken as 0.7. For the experimental results obtained from glass bead columns the ((re) and values are identical because E‘ = E . The (ae)values from Kieselbach’s, Norem’s, and Giddings’s results and the (&)“ values from Bohemen’s and Purnell’s results have been obtained taking E = 0.7 and E’ = 0.4. The ( m e ) a value obtained in this laboratory ( 1 ) has been corrected taking the experimentally determined E value of 0.82 and E’ = 0.4. It is apparent from Table I that, except for Norem’s
results, the (ae) values obtained from the glass bead column experiments are smaller than the ( ( y e ) and values obtained from columns packed with a porous granular support. The mutual agreement between the ( d ) values is, however, better as is illustrated by the smaller standard deviation. This suggests that in this case u“ should be preferred. LITERATURE CITED
(1) Berge, P. C., van., Haarhoff, P. C., Pretorius, V., Trans. Faraday SOC.58,
2272 (1962’1. (2) Bethea, R. M., Adams, F. S., ANAL. CHEM.33, 832 (1961). (3) Bohemen, J., Purnell, J. H., J . Chem. SOC. 1961, 360. (4)Ibid.. D. 2630. (5) Carman, P. C . , “F,l,ow of Gases through Porous Media, p. 1, Butterworths, London, 1956. (6) Ibid., p. 12. ( 7 ) Ibid., p. 39. ( 8 ) Deemter, J. J. van., Zuiderweg, F. J., Klinkenberg, A., Chem. Eng. Sci. 5, 271, (1956). (9) Giddings, J. C., ANAL. CHEM. 33, 962 (1961). (10) Giddings, J. C., Ibid., 34,1186 (1962). (11) Giddings, J. C., J . Chromatog. 5, 46 (1961). (12) Giddings, J. C., Seager, S. L., Stucki, L. R., Stewart, G. H., ANAL.CmM. 32, 867 (1960). (13) Golay, M. J. E., “Gas Chromatography,” p. 11, Academic Press, New York and London, 1961. (14) Gordon, S. M., Krige, G. J., ANAL. CHEM.35, 1537 (1963). (15) Haarhoff, P. C., “Contributions to the Theory of Chromatography,” D.Sc thesis, University of Pretoria, 1962. (16) Haarhoff, P. C., Pretorius, V., J . S. African Chem. Inst. 13, 97 (1960). (17) James, A. T., Martin, A. J. P., Bioch. J . 50, 679 (1932). (18) Keulemans, A. I. M., “Gas Chromatographv,” p. 135, Reinhold, Sew York, 1957. (19) Khan, M. A , , ,Vattire 186,800 (1960). (20) Kieselbach, R., ANAL. CHEM. 33, 23 (1961). (21) Littlewood, A. B., “Gas Chromatography, 1958,” p. 23, Butterworths, London, 19.58. (22) Sorem, S. D., ANAL. CHEM. 34, 40 (1962). (23) Scheidegger, A. E., “The Physics of Flow-Through Porous Media,” p. 60, Macmillan. Ne-- York, 1957. (24) Ibzd., p. 92. (25) Stewart, G. H., Seager, S. L. Giddings, J. C., A N ~ LCHEM. . 31, 1738 (1959). PIETER C. VAN BERGE VICTORPRETORIUS Department of Physical Chemistry University of Pretoria Pretoria, South Africa Work supported by African Explosives and Chemical Industries, Ltd., South Africa.
VOL. 36,
NO. 3, MARCH 1964
695
