Communications to the Editor
856
where dnldt is the desorption or decomposition rate in molecules/cm2 sec, A is the sample area, and V is the system volume. ( A l ) and (A2) can be combined to yield d p f + t L a -dn (A3 1 dt T dt where a = A / K V , T = V/S, the characteristic pumping time for the system, and p* = p - p e s . The development above is given by Redhead.13a T o evaluate the area under a pressure-temperature gas evolution curve, eq A3 can be integrated from the starting temperature To to any temperature of interest, if the timetemperature relationship is known. For linear heating rates (dT/dt = p) this produces
since p* = 0 at T = TO.If the integration is carried to infinite temperature
L;
#* dT =
Tp[
a1;dn
-
p*T.w]
(A4)
The left-hand side of the equation is just the area under the pressure-temperature gas evolution curve (measured from a baseline of p = &, and is proportional to the heating rate p. Redhead has shown13a that finite pumping speed causes the maximum in the P-T curve to shift toward higher tem-
perature. The shift (AT)is a fixed fraction of T Mfor fixed E A and 7. Redhead13a has treated the first-order case by numerical methods, and this allows us to estimate the temperature shift AT. The worst case is the smallest pumping speed or the greatest T. Assuming only diffusion pumping in our system T = 10-1 sec. For E A = 50 kcal and T = 10-1 sec, AT = 0.7%for first-order kinetics. We have corrected the T Mvalues in Table I for this effect and determined E A for the monetite conversion using the corrected temperatures. The corrected value is 49.0 instead of 49.4 kcal. The difference is not significant given the 1kcal standard deviation of the plot.
References and Notes (1) B. 0. Fowler, E. C. Moreno, and W. E. Brown, Arch. OralBlol., 11, 477 ( 1966). (2) A Boulll, and M. Dupont, C. R. Acad. Scb, 241, 42 (1955). (3) N. Newesely, Monatsh. Chem., 96, 379 (1967). (4) H. C. W. Skinner, Mater. Res. Bull., 5 , 437 (1970). (5) A. Gee and V. R. Dietz, J. Am. Chem. Soc., 77, 2961 (1955). (6) R. J. CvetanovlC and Y . Amenomiya, J. Phys. Chem., 67, 144 (1963). (7) J. M. Sedlak and R. A. Beebe, Thermochlm.Acta, 9, 261 (1974). (8) J. M. Sedlak and R. A. Beebe, J. Colloid Interface Scl., 47, 483 (1974). (9) C. W. Anderson, R. A. Beebe, and J. S. Kittelberger, J. Phys. Chem., 76, 1631 (1974). (IO) S. J. Jorls and C. H. Amberg, J. Phys. Chem., 75, 3172 (1971). (11) This laboratory, unpublished results. (12) X-Ray analysis was performed by Dr. H. C. W. Skinner of Yale Unlversity. The structure of monetite is discussed by Skinner4 or by W. L. Jones and D. W. H. Cruickshank, 2.Kristallogr., 116, 101 (1961). (13) (a) P. A. Redhead, Vacuum, 12, 203 (1962); (b) F. M. Lord and J. S. Kltteiberger, Surface Sci., 43, 173 (1974). (14) R. CvetanovlC and Y . Amenomiya, Catal. Rev., 6, 21 (1972). (15) See T. Ozawa, J. ThermalAnab, 2, 301 (1970), for example.
COMMUNICATIONSTO THE EDITOR
Rotational Freedom of Adsorbed Molecules Publication costs assisted by the National Research Council of Canada
Sir: In a recent communication de Lara and Delavall suggested, on the basis of spectroscopic evidence, that the CH4 molecule retains appreciable rotational freedom when adsorbed on type A zeolites. This conclusion is supported by independent thermodynamic evidence. The Henry’s law adsorption equilibrium constant (defined by c = K p ) may be expressed as the ratio of the partition functions for adsorbed and gaseous molecules, or, in terms of the configuration integral (2): K = Z f f i n t / f ’ p kT (1) In this expression f is the rotational partition function and f i n t the internal vibrational partition function for an adsorbed molecule; f’g is the partition function per unit volume for a gaseous molecule, k is Boltzmann’s constant, and z is defined by 2 =
1
exp[(- u ( r ) / k T ] dF
The Journal of Physical Chemistry, Voi. 79, No. 8, 1975
(2)
where u(r) is the potential energy at a point specified by position vector F. For a monatomic species f = 1 and f P is simply the translational partition function per unit volume (f’trans) 50 that K =
Z/f‘trans1?T
(3 )
while for a polyatomic species P g = f’tranfrotfint and, assuming fint to be the same for gaseous and adsorbed species, we have
(4) K = z f / fI r a n s f r o t k r where frat is the rotational partition function for the gaseous (polyatomic) species. For nonpolar molecules the potential energy profile within a zeolite cavity arises from the sum of the dispersion, repulsion, and polarization energies and, in any given crystal framework, these energies are determined principally by the size and polarizability of the adsorbed molecule. For a monatomic molecule and a symmetrical polyatomic molecule of the same size and polarizability the potential energy profiles, and the configuration integrals for any particular zeolite, should therefore be essentially the same
057
Communications to t h e Editor
should be exchanged for the following treatment. Expand
TABLE I: Parameters for Sorption in 5A (Linde) Zeolitea Kr
CH,
Ar
02
n=O
3.44 3.5 3.6 3.8 16.3 16.0 l O Z 5 c y , cm3 molecule-' 25 26 1 71 froi(at 298°K) 1 72 K O , molecule cavity-' 1.26 X 1.9 X 1.04 x 1.48 X 10-6 10-6 Torrmi 10-6 10-6 3.36 3.30 q o , kcal mol-' 4.24 4.54 1.24 K p o i y I K m o n o (at 298°K) 2.56 22 15 f(at 2 98"K) The Henry constant is given by K = KOexp(qo/RT). Values of
where H , is the Hermite polynomina12 of order n obeying
K Oand q o for CH4, Ai-, and Kr are taken from the thesis of Derrah.2 The values for 0 2 are from unpublished data obtained in this laboratory. Values of u and a are from Hirschfelder et al.3
The conservation of total material gives
0,
A
H,," - 2zHn'
+
2nHn = 0
(33a)
LmHn(z)Hm(z) exp(-z2) dz = 2"n!GS,,
(33b)
Hfl'(z) = 2nHn-i(z)
(33c)
2zHn(z) = H,+~(z)+ 212Hn-,(z)
(334
and
Q
= -ac,(t) -
4-20 a-q(t) pz3
and, from eq 3 and 4,the ratio of the Henry constants will be given by fltrans mono
(5)
Kmono
Since the ratio of translational partition functions is simply the ratio of molecular weights raised to the three-halves power:
where
=AO = z,c/aii, a = Aexc/A1
Inserting (32) and (34) into (28) and utilizing (33a-d) one finds
(1) E. Cohen de Lara and Y. Delaval, J. Phys. Chem., 78, 2180 (1974). (2) R. I. Derrah, Ph.D. Thesis, University of New Brunswick, Fredericton. New Brunswick, Canada, 1973. (3) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids", Wiley, New York, N.Y., 1954.
Department of Chemical Engineering University of New Brunswick Federicton, New Brunswick, Canada
D. M. Ruthven
Received January 8, 1975
A Correction and Improvement of "On the Kinetics of Step-Wise Micelle Association" by E. A. G. Aniansson and S. N. Wall'
Sir: The part of ref 1 from eq 31b up to and including eq 37
(374
and
1
+
k - =k - a ( 1 + c,) o2 n3 The time dependence of the higher c,s' can be found similarly and will contain successively smaller time constants given by - ='T
l/Tn
References and Notes
(35)
has been used and we have introduced
= C,(O)
Comparative data for the sorption of CH4/Kr and On/Ar in 5A (CaA) zeolite are given in Table I. For these pairs of molecules the conditions of equal size (as measured by the Lennard-Jones U ) and polarizability are very nearly fulfilled and the limiting heats of sorption (40) for the species of each pair are almost the same, as is to be expected if the configuration integrals are similar. The rotational partition functions for the adsorbed molecules, estimated according to eq 6, are 15 for CH4 and 22 for 0 2 . These values, which may be compared with values of 71-72 for the freely rotating gaseous molecules, suggest that although rotation is somewhat restricted in the adsorbed state an appreciable degree of rotational freedom is retained, in accordance with the spectroscopic evidence.
(34)
= nk-/a2
(374 In relaxation experiments the quantity measured is & ( t )or its equivalent so that until new experimental methods sensitive to the finer details of the micellar size distributions are developed only co and c1 are needed. Under the present assumptions, then, only one relaxation time constant governs the fast rearrangement process even at very large deviations from equilibrium. At small deviations from equilibrium c o
