Plate Height in Programmed Temperature Gas Chromatography SIR: Recent work on programmed temperature (PT) chromatography has directed attention t o the meaning and usefulness of the plate height concept in these procedures ( 3 ) . It has been observed that plate height, H, is a function of temperature (1-3). The value of H calculated in the usual manner from elution curves of PT chromatograms, H o b s d , represents a n averaging of the plate height over the temperature range in question. If we consider the Keulemans ( 4 ) expression for H, H =A
+ B / u + Cu
(1)
as appropriate in the normal range of operation, there are two types of information available, the importance of which depends upon the investigator’s interest. The analyst, interested in resolution, desires minimal elution plate height. He seeks the velocity a t m-hich maximum resolution is attained and he is concerned mith observables and how they vary with the temperature. From Equation 1 we h a r e the minimizing condition on the flow velocity, u: Umln
=
(B/C)1’2
(2)
‘I‘hc temperature dependences of B and C have not been determined exactly and, in fact, have b e w shown to be a strong function of the system employed ( 2 ) . It is known t h a t B increases with temperature and C decreases with temperature and, therefore, u,,, must increase with T. If we choose d e a e t and Pretorius’ expression for the temperature dependence of H ( I ) , H =A
+ B‘T/u + C‘u/T
where TBc is the temperature a t which
B and C are evaluated from Equation 1 and T,,is the time average temperature of the program. The averaging of the temperature should, in practice, be carried to the highest retention teinperature, as error on the high velocity side of umlnis less critical in view of the decrease of C with temperature. T o the theoretician interested in the column parameters A , B, and C of Equations 1 and 3 and, in particular, in their temperature dependence, a local is the term of inplate height. Hloeall terest ( 5 ) . The value of Hlocal a t each time or temperature within the column must be deduced from the average Hohsd which has been obtained by the equation Hobsd = L(T/t)*. L is the column length, t is the retention time, and T is the standard deviation in retention time, or one fourth the peak width. [The use of thc isothermal retention time a t the retention temperature, tTR, recommended by Habgood and Harris (3) in calculating H o b s d is the form of observed plate height consistent n-ith the definition employcd here (see below).] I n the practice of PT chromatography, no temperature gradient exists along the column during the program. The variance in distance of the zone, u2. a t any time is the sum of the variances obtained a t each position along the rolunin; u2 = f H l o cdL ~ ~( 5 ) . The distribution of the zone is determined upon elution as a standard deviation in time and the conversion factor between u and r is the RJ value, RTl at the elution temperature times the gas velocity.
(3)
the minimizing condition becomes (5)
The integration may be carried out over the temperature range employed.
Here, r = dT/dt, and both Iiloeal and R, are functions of temperature. If a n appreciable pressure gradient exists along the column, this may be included in the integral, as shown elsewhere ( 5 ) . With Equation 5 , the relation between Habsd and Hlocal can be formulated.
The local height in the integral may be replaced by the individual terms from Equation 1 or 3. The usefulness of this expression is somewhat dependent on discovering integrable forms for the temperature dependence of both Hlacal and R,; however, the integral may be evaluated graphically from isothermal data ( 3 ) . LITERATURE CITED
W. J., Pretorius, V., ANAL. CHEM.30,325 (1958). (2) Duffield, J. J., Rogers, L. B., Zbid., 32.310 11960). (3) Habgdod, H. R., Harris, W. E., Zbid., 32,450 (1960). (4) Keulemans, A. I. XI., Kwantes, A., “Vapor Phase Chromatography,” D. H. Desty, ed., p. 15, Rutterworths, London, n.m (1) deWet,
3
IYOi.
(5) Stewart, G. H., Seager, S. L., Giddings, J. C., ANAL. CHEM. 31, 1738 (1959). GEORGEH. STEWART Department of Chemistry Gonzaga University Spokane 2, Wash. RECEIVEDfor review April 18, 1960. Accepted May 13, 1960.
VOL. 32, NO. 9, AUGUST 1960
1205
