Plate Height Theory of Programmed Temperature Gas Chromatography J. CALVIN GlDDlNGS Department o f Chemistry, University o f Utah, Salt lake City, Utah
b It i s the purpose of this work to expand the theory o f programmed temperature gas chromatography, particularly in relationship to the spreading of peaks as measured b y the plate height. The significance of the plate height in programmed temperature operations i s examined and is shown to agree with the developments of Habgood and Harris. The close relationship of this with the plate height in isothermal chromatography i s discussed. Two phenomena which are unique to the programmed methods are treated theoretically. First is the zone compression at the head of the column which i s a result of temperature being initially low followed b y continual heating. Second i s the effect of thermal lag which i s responsible for a higher zone velocity a t the wall than at the center of the column. This effect i s treated in terms of the nonequilibrium theory which has been developed for various plate height calculations in chromatography,
T
essential purpose of a plate height theory of programmed temperature gas chromatography (PTGC) is to investigate the origin and control of those factors responsible for peak spreading and the concomitant loss of resolution. Both Habgood and Harris (2, 13) and the present author (8) have demonstrated the comparability of PTGC and isothermal methods, and thus the usual isothermal technique will be taken as a starting point for this discussion. It will be necessary first to discuss the meaning of plate height since some controversy has arisen over the significance of this parameter in PTGC (14, 20). Second, the effect of condensing or freezing a solute sample as a very narroIv band in the initial part of the column (while the column is still cold) \vi11 be investigated. This effect is expected to be beneficial in comparison to the isothermal technique. Third, the effect of thermal lag (1, 12, 18) will be investigated. This effect will be harmful because the temperature lag within the column interior will cause a reduced zone velocity and thus a nonuniform velocity profile over the column cross HE
722
ANALYTICAL CHEMISTRY
section (1%). The additional contribution to the plate height caused by this phenomenon will be theoretically formulated. SIGNIFICANCE
OF PLATE HEIGHT IN PTGC
Various authors have discussed the significance of and means of obtaining theoretical plates in PTGC (6, 8, 13, 14, 16, 20). The early discussion by Habgood and Harris (13) is certainly the most complete of these. Very little can be added to the conclusions of these two authors since their deductions appear to be theoretically sound and experimentally consistent. The discussion of the next few paragraphs is intended to establish a firmer theoretical foundation for these conclusions. I n the investigation of PTGC it is important to keep in mind that nothing physically resembling a theoretical plate or an equilibration stage can be found in a chromatographic column (20). The plate height is no more than a convenient and widely accepted parameter for describing the extent of peak spreading. The justification for the name, “plate height” (formally: “height equivalent to a theoretical plate”) lies in the fact that each small increment to the zone variance, do2, is proportional to the distance traveled by the zone, dz, both in columns composed of hypothetical equilibrium stages (plates) and in chromatographic columns. This is not a compelling reason to relate chromatographic performance to theoretical plates since almost every physical process leading to zone spreading, including diffusion itself, obeys this relationship. [When a velocity gradient exists in a column ( I f ) , such as in the presence of pressure gradients in gas chromatography, d dlnv/dz)dz, one has du2 = ( H which somewhat stretches the theoretical plate analogy because the increment du2 now depends on the value of u? already evisting and thus depmds on the past history of the zone. The treatment of this problem, given elseIvhere, requires a rather wide departure from conventional plate height methods.] If one accepts the plate height convention then the plate height itself becomes the
+
proportionality constant in the foregoing relationship, Le., da2 = Hdz, or H
=
daz/dz
(1)
It is this plate height, or various contributions to it, which most theories of chromatography yield directly. This quantity may be termed the loca1 plate height (11) since i t is the value applying to a given point within the column as opposed to average plate heights, etc. If H is constant throughout the column, the integration of Equation 1 yields u2 =
HL
(2)
where L is the column length. If H is a function of the distance along the column (the variation caused, perhaps, by nonuniformity in the liquid distribution) the integration yields
where is the length-average plate height (this does not apply when velocity gradients exist as explained earlier). I n the case of PTGC, H varies continuously as a function of time. The variance is then given by
where the integration is n o w carried over the path traveled by the center of the zone (zl is the distance moved by the zone). Since the major part of the zone movement occurs a t temperntures near the elution temperature, will approach the plate height characteristic of isothermal chromatography carried out a t a temperature near the high end of the programmed run (8, I S ) . It is predicted quantitatively that E7 will approximately equal the plate height of an isothermal run if the temperature is held equal to the significant temperature, i.e., a temperature approximately 40” C. bel015 the maximum or elution temperature of PTGC (8). The value should realistically reflect the efficiency of a column because the bulk of the separation, where the “plates” are
a
actually effective, occurs as an average around the significant temperature. The experimental determination of l?, as pointed out by Habgood and Harris, requires a different procedure from that conventionally used in isothermal chromatography. Thus, g 2 can be written as R,2v2~2 R here R, is the ratio of the zone velocity to the gas velocity, v , a t the moment of elution. and r is the standard deviation of th: zone or peak in time units. Since H = u2,’L, Equation 4 becomes
where t ; is the retention time for an isothermal run a t the final or elution temperature. This quantity differs by a large amount from the usual definition, R = Lr2//tr2,where t, is the true retention time for PTGC, and hence the use of the latter is incorrect. The definition obtained in Equation 5 has been shown to yield values in substantial agreement with isothermal data (1 3). This definition of plate height is also consistent with the isothermal value when used in the formulation of resolution or separability (6,8).
INITIAL ZONE CONDENSATION IN PTGC
The attempt to keep chromatographic peaks as narrow as possible is made more difficult by the spreading processes occurring within the sample introduction and detection volumes. In well designed instruments, this additional spreading is negligible, but in a large number of units in current use it can cause a significant loss of resolution. The standard deviation in time, r , of 3n isothermally eluted peak can be written approximately as
uhere the three terms are the column, detector, and injection system contributions to over-all T * . If a programmed run is made a t the same gas flow velocity, the zone \\-ill go on the column uith a variance in nidth of u , ~= X o 2 v 2 ~ ,nhere 2 R,. the ratio of zone to gas velocity a t the initial or starting temperature, is yery low. Except for rionideal solution effects, the zone qreading represented by us2 will be ,mpagated unchanged through the olumn (although the column effects nil1 :)e adding their own contribution). At the end of the column the ut2 contribution ,rill be trarslated into a time varinee equal to a,’/R,2v2. Using the ibove value of ut2, the time variance oecomes RO2r,*/Rr2.For the total contributions to T *
For all but the early components in PTGC, the ratio of initial to final zone velocity, R J R , is essentially zero, and the last term can be effectively ignored. Thus, very large mixing volumes can be tolerated in the injection chamber of an apparatus designed for PTGC. These large volumes would have the same effect as a volume R,,’R, times smaller in an isothermal unit providing the total amount of solute remained the same. This factor varies from about 0.5 to 0.01 and thus provides a wide margin of convenience in most PTGC operations. This may be particularly useful in preparative scale work where large samples are required.
THERMAL LAG AND PLATE HEIGHT
As mentioned previously, thermal lag within the column should be responsible for a plate height contribution which does not exist in isothermal columns. This contribution depends upon the exact form of the temperature profile existing radially across the column. This can be obtained through integration of the heat conduction equation
here b T / & is the rate of increase of temperature with time, K the thermal diffusivity (conductivity/density x specific heat), v2 the Laplacian operator, and T the temperature. Shortly after the temperature program is started a steady rate of temperature increase will be found throughout the column with the temperature of the interior regions lagging a constant amount behind the wall regions. Thus, bT/bt = p is the same everywhere and is equal to the instrument setting for the rate of temperature increase. This fact combined with the cylindrical symmetry of the column leads to the following modification of Equation 8 p = - K- d r dr
dT
dr
where r is the radial distance from the center of the column. The integration of this equation using the boundary condition, T = T , (wall temperature) a t T = ro, and the symmetry condition, d T / d r = 0 a t r = 0, leads to
8
The nonuniform temperature profile described by Equation 10 is responsible for a more rapid zone migration near the wall than a t the column center. It is also responsible for the presence of a greater quantity of solute per unit volume of packing near the center than near the outside. Thus, both zone ve-
locity and solute concentration vary laterally across the tube. Previous theories (6) of the chromatographic process have treated the problem of continuously varying velocity profile but have not accounted for continuously varying concentrations. The general problem is not only of importance here but is also important in preparative scale columns where a variation in packing density can lead to lateral concentration as well as velocity variations (9, 17). The formulation of the nonequilibrium and hence the plate height characteristics of a column with thermal lags proceed along the same general lines as previous nonequilibriuin treatments (3, 4,6, 7 ) but differ in detail because of the concentration variations. The rate of change of solute concentration within a small volume element of the column is given by = s - - R v - +dCR D , - - , at
a2
a 2C az
(11)
where the concentration, c, gives the total amount of solute (including both that in the gas and in the liquid phase) per unit volume in the column. The quantity s is the rate of accumulation of solute per unit volume due to lateral diffusion, R the ratio of the zone velocity to the gas velocity, v , 2 the axial coordinate, and D, the binary diffusion coefficient for solute in the carrier gas. Assuming that the departure from lateral equilibrium is small and ignoring the diffusion term in Equation 11, the accumulation rate becomes s =
bc *
-
at
* + RUacaz
where c* is the “equilibrium” concentration of solute, Le., the concentration existing a t a point in a cross section providing all the solute in the cross section is allowed to diffuse until a steady condition is reached. (Since a thermal gradient exists, true equilibrium is not reached, but a quasi-equilibrium with respect to diffusion processes can be attained.) I n view of the thermal gradient, c* will vary from point within a cross section. I t is convenient to define the term E = c * / c , giving the ratio of the concentration at a given location to the mean concentration, e, averaged over the cross-sectional area. (From this point on, a11 average values, indicated by a bar, signify an average over cross-sectional area.) The average value of (, 5, is, of course, unity. To evaluate &*/at in Equation 12 it is noted that &/3t = - b J / & E dRvc*/bz, where 7 Rvc“ is the average flux of material due to flow into a unit cross section of column. The quantity &*/& = (a?/& since the variation of 5 with time (due solely to the programmed VOL. 34, NO. 7, JUNE 1962
723
temperature) is minor compared,to the iation of c (due to the passage of the
Using the definition of [ and the fact that neither f nor Rv is a function of z (normally velocity gradients are entirely negligible), B becomes
-
8
E (Ru
- m)& / b
(15)
ent expression for k e d in terms of lateral diffusion, is needed to eliminate it as an unknown from Equation 15. It is convenient to assume under the qunsi+quilibrium conditions discussed above that the amount of solute in the gas phase per unit volume of packing, c,+, is constant laterally across the column. In actual fact i t is the partial pressure of solute which will be constant a t quasi-equilibr i m if thermal -'.iision is negligible. However, the difference between the two is insignificant compared with the exponential variation of R. Assuming, in addition, that lateral diffusional transport occurs only in the gas phase, B may be written as s
-
D~v'c~
(16)
The concentration c, may be written as
Rc,and c can be expressed as c
where term. yields
i=
c" (1
4- 4 )
=
E
C (1 f c )
(17)
is the equilibrium departure titution into Equation 16
e
(22)
The evaluation of 1/R is dependent on the molar heat, AH, and entropy, A8, of solute vaporization from the stationary liquid in the following manner (6): I / R = exp ( A H / W l ' ) / a f 1
(23)
where is the gas constant, a = (V,/Vl) exp (AS/@, and V,/Vr is the ratio of the column volume opupied by gas and stationary liquid, respectively. Tho exponential temperature dependence shown in Equation 23 is particularly clumsy, and i t is desirable to convert this to a more tractable, approximate form. Writing T = T , - 6T, where dT can be identified by comparison with Equation 10, the following expansion can be made exp ( A H / R Z ' ) = exp I(AH/(RTw) 11 &T/Twf (6T/'Tu)' f .I) (24)
+
.
If the average bT is small such that T w / 8 T > PAH/(RT, or, approximately, bT < "25 C., the square term in the above series is negligible to within 10%. A small 6T also makes it possible to
=I
D,CZR@(I f
c)
Since ? becomes
=
and thus E = c,"/RE. = E = c,*[-) the latter
= c,'/R
~
= 1/R
(VI, RE
*i
When these values are substituted back into Equation 27 and an average taken, the rcsul t is
To formulate the plate height contribution of the thermal lag phenomenon, HT,in terms of IJal Fogare, S., Chem. Eng. S e w s p. 102, June 26, 1961. (20) Stewart, G. H., rls.41,. CHEJf. 32, 1205 11960). RECEIVED for review January 25, 1962. Accepted April 12, l9ti2. Investigation supported by the U . S. Atomic Energy Commission unaw Contract No. AT( 11I
\ - I
1)-748.
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