Modifications to the van Deemter Equation for the Height Equivalent to a Theoretical Plate in Gas Chromatograp hy WALTER L. JONES Engineering Research laboratory, Engineering Department, E. 1. du Pont de Nemours & Co., Inc., Wilmington, Del.
The van Deemter equation for the H.E.T.P. of a gas chromatographic column is modified to include the effects of mass transfer and velocity variation in the gas phase; the results are applicable to columns of light liquid phase loadings. A limitation on the path term proposed by multiple Giddings is equivalent to the velocityvariation term derived in this study. Equations derived by Golay for tubular (capillary) columns are special cases of the modified equations for simple, calculable geometries.
-
T
van Deemtcr equation (8, 11, 14) for the height equivalent to a theoretical plate (H.E.T.P.) has mct with considerable fiuccess in explaining the efficiency of gas chromatographic columns. This cquation relates the H.E.T.P. of a column to the c:arric!r gas vrlocity, u. In simplified form, HE
H.E.T.P. = A
+ U/u + Cu
(1)
The first term on the right has Lcen assumed to express the effcct of multiplc paths in the gas, the second the effect of longitudinal diffusion of the sample in the gas, and the third the effect of resistance to mafia transfer, prcsumcd to he controlled by diffusion in the liquid phase. Recent experiments ( I , $ , 10,12) have indicated the need for a n additional term or term8 similar to the liquid phasc mass tranfifer term, but controllcd hy gaseous diffusion. Van Dwmtcr, Zuidcrweg, and Klinkcnhcrg (14) originally derived such a tcrrri to dwcrih: mass transfer in the gas phanc. It was discardcd on t1icort:tiaal grounds with the conclusion that it was scvc:ral ordws of magnitude smah!r than the liquid diffusion term. Golay (6) dwivcd “dynamic diffusion” terms far tubular columm (capillarics), taking into account tranrjfer not only through the gas to the liquid hut between rapidly and slowly moving portions of the g a ~ . Giddings (3) has a h o considercad the Prescnt address, 25 Woodland St., Belmont, Mass.
effect of transverse diffusion between fast and slow moving strcams in so far as it affects the A , or multiple-path term. This work follows the approaches of Giddings (2) and van Decmtcr, Zuidcrweg, and Klinkenberg (14) in a fiimplified fashion to derive a generaliacd expression for the H.E.T.P., including the effects of gas-phase mass transfer and of the vrlocity distribution in the carrier gas. The Golay “dynamic diffusion” terms are 8hown to be specific cases of this equation for simple calculable gcomctries. The Giddings equation for the multiple-path term i8 considered a8 a limit to the velocitydistribution term of the equationfi developed.
of the individual factors by the expresfiion (13) u2ry
=
u*s
+ + u*y
2PzYuruY
(3)
The factor p Z y is the correlation coefficient for the variables z and y. If the variables are indepcndent, p = 0. If there is a strict‘ correspondoncc between x and y, such as x = ay, then p = 1. Intermediate degrees of dependence lcad to valucs of p bctwccm 0 and 1. The factors entering into the 1I.I~C.T.I’. equation-for example, longitudinal diffusion and mass transfw-have becn assumed to be ind(:pc:ndcnt v:LriablcM. This is probably truc in all CILHOH cxccjjt for interaction hctwccn gas pltasc maw transfer arid vclocity distribution variations. Equation 3 may bc dividcd H.E.T.P. EQUATIONS by L to show that the components of H.E.T.P. add arithniotically. Considor a sample molecule a~ it proThe framc: of rcfcronce for this disc e d s down a column, alternately recussion will ha chosen to movc with thc siding in the gas and liquid phases. center of thc pcak h i n g corisidcrcd. If Excluding multiplc-path effects, there the average carricr gas volocity is u arid are three factors which cause it t o the ratio of tirnc in thc liquid j,hasc to emerge more or less rapidly than otlicrs time in the ‘gas p h n e in k, tlicn in thc of its kind. It may stay longer or frame of rcf(wncc cli~sen,the g a ~will shorter times per step in the liquid phasc:, and it may do the same in the g a ~ move forward with a valucity __ arid phasc:. Finally, its vc:locity in the gas the liquid will movc hackwttrd with :L phasc may ~ J Cgrcatcr or ICSH than the U velocity I k. A molcciilc ha^ m average. This kist variation may be brought ahout hy diffusion in the direcopportuniticn to diffusc out of a j)Iirtno tion of flow or ~ J Ymotion iri a fitrcam of in the course of its travel down the other than average gas velocity. (ICven column- Le., it tliffuscs to thc intcrfacc in a ninglc uniform tuhc, thore is a crossrn times. If it travc:ln a distancc 1 hoscctional velocity dititrihution.) Variatwccn such opportunitics, the variance tion in velocity will rcwlt in a correin the distance travclod in that jiIittfii(: in sponding vnrintion in the number of (12 = m 1 2 (4) transitions botwc:c:n phases. Ilcforo c:x:imiriing thew factor8 in The nurnbcr uf trarisfcr olij)orturiitic:n more dctail, it iti n w w a r y to consider is q i i a l to the total timc nl~!ntin tho the statistical t m i x for thc: 1I.E.T.P. of a pl~asedividcd by the time the rnolwiilc column. ’l‘hc 1I.E.T.P. ifi related to rcquiren to diffunc! to z~ndfrom the intcrthe variance, u2, ( J f a pcak and to L, the face. In the liquid phase the total time kL The diffusion tirnc: is CldP length fJf column travel, by the equsifi -, - D where ~ U tion : C1 iH a gwmetric Constant, d l in the H.E.T.P. = u 2 / L (2) liquid film thickn(:nx, rind fj, in thct diffusivity of thc fiamiilc in t h liquid The variance is proportional to the film. ‘Jhrcfor(!, Equarc of pwik width mc:asurcd US a length along the column. If t w o factors contribute to a vari:inco, thc total variancc can be relatcd to the variance8 +
VOL. 33, NO. 7, JUNE 1961
829
to the time per step multiplied by the relative velocity of the phase with respect to the peak. In the liquid phase,
tudinal diffusion, a molecule travels an average distance, d, away from the peak center in a time, t, related by the equation :
From Equations 2,4,5, and 6: H.E.T.P. = u 2 / L =
This is the van Dccmter mass transfer term where Clis equal to 8 / n 2 . (Golay has derived a liquid diffusion term (5) which, when put in terms of film thickness rather than tube radius, is of the same form. However, it has a coefficient of 2/a. The difference is discussed later.] In the gas phase: m
= DY czdo2
u
and
where C2 is again a geometric constant, dois the diffusion path length in the gas, and Do is the diffusivity of the sample in the gas. It follows that
Van Deemter et al. assumed that do was half the diameter of the fine pore structure of the packing, or about 10 microns. This is comparable to the liquid film thickness (for a 30% by weight loading) as calculated from observed data and Equation 7. Since D I is about lO-'D,, the gas term was assumed to be negligible in comparison to the liquid term. However, the number of times a sample molecule diffuses to the wall can be much more than the number of times it has access to the liquid phase, if the walls are only partially coated. Firebrick has a surface area of about 3 sq. meters per gram, so that if the layer is 10 microns thick, a 30% by weight liquid coating will cover only 1% of the surface area. This means that Cz is 100 times as great as it would be if the entire wall were coated. A reduction in liquid loadings to less than 10% will decrease the liquid phase mass transfer term (Equation 7) by a factor of 10. At high values of k, the gas phase mass transfer term may thus be comparable to the liquid term, and should not be discarded without experimental verification. I n the case of tubular columns, the diffusion path is of the order of the tube radius and the term can be important. The variance of the velocity of a molecule does not directly affect the ratio of times in the two phases. The peak variance due to this factor is thus proportional to the velocity variance and independent of k. In the case of longi-
830
ANALYTICAL CHEMISTRY
where n is the number of dimensions in which the diffusion is measured ( 7 ) . In this case only the component of diffusion parallel to the direction of flow is of interest and n = 1. Over a length of column L: t = L/u = d2
(12)
effect of diffusion is to decrease that term. Where van Deemter et al. (If, 14) derived a term H.E.T.P. = 2Xdp (19) in which A is a geometric constant, Giddings has derived a corresponding term H.E.T.P.
=
2Xdp
1
+ 4XDolB2d,u
where p2 is the geometric constant related to the distance between flow paths. At very low flow velocities, Equation 20 is approximately
and H.E.T.P.
=
20 U
This is the van Deemter B term for longitudinal diffusion. In practice, path tortuosity reduces the effect by a factor y . The interchange of gas between two streams of different velocity is analogous to the interchange between discrete phases. The time for a molecule to diffuse between streams of different velocities is
where dp is the packing particle size and C, depends on the geometry and the gas velocity distribution. I n this case,
and
Combining Equations 4, 15, and 16, we arrive a t the contribution to H.E.T.P. caused by velocity distribution of the gas, H.E.T.P.
=
Csdp2U
Do
(17)
where C8 = C4CS2
(18)
I n packed columns, a large portion of the gas is in the relatively stagnant interior of the porous packing materials. There will be a large velocity difference between the gas in these regions and that between particles, and diffusion paths will be of the order of particle size. In open tubular columns, velocity differences will be smaller and diffusion paths will be less than tube dimensions, so that it is reasonable to expect that the velocity-distribution component of H.E.T.P. should be less than the gas phase mass transfer component. Giddings (3) has calculated the effect of such diffusion between paths of different velocities on the velocityindependent or multiple-path term of the H.E.T.P. for packed columns. The
which is the same as Equation 17 if of = 2 cs. Giddings has estimated that the velocity at which diffusion reduces multiple-path effects by a factor of 2 lies between 10 and 100 cm. per second for conventional packed columns. It seems likely that the higher value is the more probable one. Glueckauf (4) has measured this effect with glass bead packings and water as a carrier. The critical velocities for reduction of the term by a factor of 2 are of the order of 0.01 cm. per second in his tests. The critical velocity is proportional to the diffusivity, and the diffusivity of the sample (HI) in water a t room temperature is about lo-' that of the sample in helium. This would indicate critical velocities of 100 cm. per second in a helium carrier. The critical velocity may also be calculated if the values for fl and A can be determined. Glueckauf (4) found 2X t o be of the order of 5. From the data of Kieselbach (IO), in 100- t o 120-mesh packings, C3d,2 E 2
x
lo-'
(22)
This mesh size corresponds to particle sizes of about 1.4 X 10+ cm. so that Cs is of the order of 1 and p2 is of the order 4x of 2. Therefore, 1: should be of the B order of 5. This compares with the value of 10 estimated by Giddings. If helium is used as a carrier gas, diffusivities of samples of molecular weights in the order of 50 to 100 will be around 0.3 sq. cm. per second. The critical, or transition, velocity will be reached when - 4XDo B2d,
--
5 X 0.3 N 100 cm./sec. 1.4 x lo-* (23)
Coarser packings and nitrogen or argon carriers would lower the critical velocity, possibly to within normal operating ranges. Thus, the velocity-distribution term may well reach limiting values a t high velocities normally outside the operating range of packed columns. This limit is imposed by multiple-path effects, which
are not, as previously supposed, important a t normal carrier gas velocities. A new explanation of the velocity-independent term sometimes observed seems to be required. Up to this point it has been assumed that each peak-broadening factor is independent of the others. However, a molecule which stays closer than average to the walls will spend shorter times in the gas phase and will also travel a t a lower average velocity. Conversely, high velocities will tend to be associated with long gas residences. A correlation term must, therefore, be included in the equation for the H.E.T.P. From Equations 3, 10, and 17, this term will be
The total equation for the H.E.T.P., including the experimentally observed velocity-independent term, is
If the term for resistance to mass transfer in the gas phase of the packed column is small compared with the velocitydistribution term, the equation becomes : H.E.T.P. = A
~ D o+ ___ Ctk + 2u (1 + k)2
I Cadp2u (26)
DI
Do
In the case of tubular columns of circular cross section, d, = do = TO and Equation 25 becomes:
where (7’
=
c,
Golay (5) has assumed a parabolic velocity distribution in open tubular columns and utilized the differential equations of flow and diffusion to calculate a gas phase “dynamic diffusion” term. This term includes the effects of gas phase mass transfer and velocity distribution. It is, as would be expected, of the form of the last term of Equation 27. Specifically, it is H.E.T.P. =
(1
+ 6k + Ilk*) 24(1 + k ) l
r& DO
(29)
for a tube of circular cross section. The implied values of Cn, CS,and p may be calculated from Equations 27, 28, and 29.
Cz =
’ / , C,
= ‘/u p = (‘/a)‘/’
=
0.817 (30)
For a tubular column of rectangular cross section and half width ,,z, Golay (6) has shown that H.E.T.P. = 4(1
+ 9k + 51/2k2) 105(1
+ k)*
,to% DO (31)
from which the implied values of C2,C3, and p are
Cs = 2/3 Ca
= 4/105 p =
7 (2/35)’/* 2 (32)
The values of the coefficients listed above are required to fit the dynamic diffusion terms to the gas diffusion term of Equation 27. The values of Cz and C3 may also be calculated independently to show that they are as given in Equations 30 and 32. If k = 0, the mass transfer component of H.E.T.P. and, hence, also the correlation component must vanish. The velocity-distribution term is thus the corresponding Golay dynamic diffusion term with k = 0. For circular and rectangular tubes, they are H.E.T.P. =
roau 240,
(33)
4202u 10500
(34)
and
where Pa is the probability that the molecule will go to the wall in ,the z direction. If two diffusion paths are of equal areas, the flow of molecules will divide in inverse proportion to the lengths of the paths; therefore (7) :
p I -- z o+z 220
Combining Equations 37 and 36 and simplifying,
The average value fort. is E. and can be found by integrating over the length of the tube. It is assumed that the probability of finding a molecule at a given position of the gas is uniform over the gas volume.
Since the molecule must diffuse into as well as out of the gas, the time between collisions with the wall is
from which C2 = H.E.T.P. =
respectively. The values of C3 are thus those required by Equations 30 and 32. The C2 coefficients for the mass transfer term may be derived by substituting a uniform velocity distribution for the parabolic distribution in the differential equations and solving for the modified dynamic diffusion term. When this was done, resulting terms were of the form of Equation 10 with Ct having the required values. A less formal proof may be derived on the bask of Equation 11 and used to explain certain differences in coefficients as calculated here by Golay on the one hand and by van Deemter et al. on the other. The case of a rectangular column will be taken up first. As previously pointed out, the time for a molecule to diffuse a distance d is related to the diffusivity by Equation 11. Therefore, the average time required for a molecule initially midway between two walls to difiuse to a wall is (35)
where zo is half the distance between walls. If the molecule is a distance z from the mid-plane,
(37)
2/a.
A median plane that is reflective instead of transparent to the molecule would make no difference in the above calculation. The diffusion time constant for a layer of thickness 2% and open on both sides to phase transfer is thus the same as that of a layer of thickness za open on one side and bounded by a reflecting surface on the other. If the liquid phase is assumed to be a uniform filmand dl = 20, then C1 = 2/8. This, as noted previously, agrees with Golay’s liquid diffusion term, but not with that of van Deemter. If molecules in a n arbitrary distribution in a rectangular tube diffuse to the walls and are not re-emitted, the distribution will rapidly take on a cosine form. Once this has occurred, the value of 2i. will differ from that calculated above and, in fact, will become E This does not occur in columns 7 2
Di
because the time of passage of a peak past a point is normally long compared to i,. I n the case of a cylinder, the components of molecular diffusion are important in two directions instead of one, and n = 2 in Equation 11. From an argument similar to that presented above,
and VOL. 33, NO. 7, JUNE 1961
831
where ro is the radius in the cylinder.
Cz is l/4 as required. The author knows of no simple way to calculate the values of the correlation coefficient, p . The values required are less than the maximum limit of 1 and are reasonable for a fairly strong correlation between the velocity distribution and mass transfer effects. An equation previously derived by the author has recently been used by Kieselbach (10) t o analyze data from several columns by a least squares technique. Unfortunately, this equation was in error, in that the velocitydistribution term was thought to be a 1 fu‘nction of (1~k2 and the correlation term was not included. Thus, the equation corresponded to Equation 27 with C” = 0. However, since C’ is determined principally by early peaks where C’ is much larger than C“k and C”’ is determined largely by late peaks where C”’k2 is much larger than C’k, the values for these coefficients should be reasonably accurate. Kieselbach found experimentally that C’ is of the same order as C”’ or even a little larger (8). Equations 28, however, indicate that this condition is im-
possible. His results may reflect incomplete compensation for system dead volume. In any case, his results suggest that the C3 or velocity-distribution term is large compared with the mass transfer term. It is also notable that the gas diffusion terms observed by Kieselbach vary approximately as the square of particle size. This would be expected for a velocity-distribution term, but not for a mass transfer term, which is related to pore size. The combined gas diffusion terms were found to be larger than the liquid phase mass transfer term when liquid loadings were less than 10% by weight and k was not near unity. In an early work, Jones (6) found gas diffusion dominated liquid diffusion in his columns even a t higher liquid loadings. However, if values of Cs of unity in Equation 26 are assumed, the values of d, to explain Jones’s data are between 1 and 2 mm., an order of magnitude larger than particle size. Channeling in the column packing occurring over dimensions of this order would lead to a large velocity-distribution term that could explain the data. ACKNOWLEDGMENT
The author expresses gratitude to E. I. du Pont de Nemours & Co. for permission to publish this works. The helpful comments of Richard Kiesel-
Gas Chromatography of the C, to
bach, of that company, and of Marcel Golay have been found invaluable. Ann Sheldon of Du Pont carried out the computations of the Cz coefficients in the modified Golay equations. LITERATURE CITED
(1) Bethea, R. M., Adams, F. s., A N A L . CHEM.33, 832 (1961). (2) Giddings, J. C., J . Chem. Educ. 35, 588 (1958). (3) Giddings, J. C., Suture 184, 357 f 1959). (4) Glueckauf,,,E., “Vapor Phase Chromatography, D. H. Desty, ed., p. 29, Butterworths, London, 1958. (5) Golay, M. J. E., “Gas Chroma\ - - - - ,
tography-1958,” D. H. Desty, ed., p. 36, Butterworths, London, 1958. (6) Jones, W. L., Southwide Chemical Conference, A.C.S.-I.S.A., Memphis, Tenn., Dec. 6, 1956. (7) Kenyrd, E. H., “Kinetic Theory of Gases, p. 287, McGraw-Hill, New
York. 1938. (8) Keulemans, A. I. M., “Gas Chromatography,” p. 96, Reinhold, New York, 1957. (9:) Kieselbach, R., ANAL. CHEM. 32, 880 (1960). (10) Zbid., 33, 23 (1961). (11) Klinkenberg, A., Sjenitzer, F., Chem. Ena. Sci. 5. 258 (1956). (12) Loyd, R. J., Ayers, B. O., Karesek, F. W., ANAL.CHEW32, 618 (1960). (13) Ospensky, J. V., “Introduction to
Mathematical Probability,” McGrawHill, New York, 1937. (14) van Deemter, J. J., Zuiderweg, F. J., Klinkenberg, A., Chem. Eng. Sci. 5, 271 (1956).
RECEIVED for review November 28, 1960. Accepted March 10, 1961.
C, Nitroparaffins
Isothermal vs. Linear Temperature Programming R. M. BETHEA and F. S. ADAMS, Jr. Chemical Engineering Department, Iowa State University, Ames, Iowa
b The van Deemter equation has been modified to show qualitatively the effect of resistance to mass transfer in the gas phase on the component band width-retention time ratio, as represented by the height equivalent to a theoretical plate. Qualitative predictions as to the effects of flow rate, heating rate, and isothermal level on the separation and resolution of the Clto Ca nitroparaffins have been made and demonstrated experimentally. Optimum resolutions were obtained at a constant temperature of 50” C. and flow rates of 90 and 60 ml. of hydrogen per minute with a heating rate of 2.9” C. per minute starting from
40’ C. 832
ANALYTICAL CHEMISTRY
M
investigators (3-6, 7, 11, 14) have contributed to the knowledge of gas chromatographic separation by linear time-temperature programming. The results reported here are from a study made to determine the optimum operating conditions for the quantitative separation of the C1 to Ca nitroparaffins with a reasonable analysis time. I n previous work on this problem (@, qualitative separation of all eight nitroparaffins was obtained, but quantitative separation was possible only for nitromethane, nitroethane, 2-nitropropane, and I-nitropropane. The present study includes a comparison of constant temperature operation with linear temperature programming. Variables ANY
studied were carrier gas flow rate, heating rate, and constant temperature level. THEORETICAL
In this report isothermal and linear temperature programmed gas chromatographic analyses are compared. A. modified version of the equation proposed by van Deemter, Zuiderweg, and Klinkenberg (6) was used to explain the effects of the process variables on retention time, t i . In van Deemter’s development, all mass transfer resistances are included in OL, where CY is the over-all mass transfer coefficient per unit volume of
