VOl. 59
1184
COMMUNICATION TO THE EDITOR A MOLECULAR DYNAMIC THEORY O F CHROMATOGRAPHY
Sir: It is interesting to note the resemblance of the results obtained in the “Molecular dynamic theory of chromatography” [J. Calvin Giddings and Henry Eyring, THISJOURNAL, 59,416 (1955)l to those of a theory based on a transfer coefficient. Such a theory has been applied by many authors1 to the heating or cooling of a porous bed by a current of gas. The basic equations are
constants K1’and kl, so that the calculations should merge when Y = klto
z = kl‘t ( t o = time of passage of fluid)
(5)
(0)
The well-known solutions of (1) and (2) are To
= e - Y r e-uIo(21/%)du
+-
e-Y-ZIo(22/=
(7)
and
i
3 To = e - Y S0Z e - ~ I o ( 2 d / Y u ) d u
(8)
The results for the introduction of a peak of solute are found differentiating Tl and T2 according to time. dTz/bt is most conveniently found by the where T1= temperature of gas, T2 = temperature use of (lb) while deriving (T1- T z )from (7) and of solid, Y = length in dimensionless units, Z = (8). This yields Giddings and Eyring’s equation time in dimensionless units. (9). bTl/bt is found by differentiating (7). This In the case usually studied, the bed is initially gives Giddings and Eyring’s equation (5). a t a constant temperature, say T = 0, and gas is Giddings and Eyring have assumed the solute to introduced at another temperature, say T = To, start either in the liquid, their eq. (5), or in the so that the initial and boundary conditions read solid, their eq. (9), but always observed it in the Y e 0 Tt=To (2a) final liquid. I n the present derivation the heat is always z = o Tz=O (2b) The solid at the bed entrance is heated from T2 = 0 introduced with the inlet gas, but it is observed at to T2 = To with a gas constantly at T1 = To so the end in either the gas, eq. (7), or the solid, eq. (8) that in view of equation (lb) Giddings and Eyring’s eq. (5) and the derivative (2’g)y-o = To(1 - e - Z ) (3) of our eq. (7) are thus applicable to strictly idenLikewise, the first element of gas introduced at tical cases. From the identity of their eq. (9) with the derivaT I = TOwill constantly meet cold solid (Tz = 0), tive of our eq. (8) one concludes that for the result so that (la) gives it is immaterial whether the “extra” transfer is at (T1)z-o = Toe-Y (4) inlet or end. Similar exponential decay is observed by GidV. DE BATAAFSCHE PETROLEUM dings and Eyring upon desorption from initially N.MAATSCHAPPIJ filled or adsorption on empty surfaces, with rate (ROYAL DUTCH/SHELL GROUP)
-
(1) Anzelius (1926); Sohuman, Furnas, Nuaselt and others; for a review see A. Klinkenberg, Ind. En& Chem., 46, 2285 (1954).
a
THEHAGUE,HOLLAND A. KLINKENBERG RECEIVED OCTOBER 10, 1955
