J . Phys. Chem. 1986, 90, 2300-2302
2300
HF+H++e
4.9
:
W
- 3s,lAl
Figure 4. Energy level diagram for H2F showing the experimentally determined ground state scaled to H2F+. Several theoretically calculated2 levels for H2F are also shown.
Our experimental energy for H,F falls about 1 eV below the lowest level calculated. This is a strong indication that our experiment accesses the ground state of the molecule. The locations of Na and K on the diagram (dotted lines) correspond to hypothetical product levels in H,F that would be formed by resonance electron transfer. It may be noted that by scaling the calculated levels
Qr= ( 0 , ) tanh
in H2Fdownward to the experimentally determined ground state, the H2F+/K reaction falls into near resonance with the excited 3p, 1 Bl state of H2F. This suggests the possibility that a radiative process may be competing with direct electron transfer to the ground state of H2F. The enhanced metastability of D2F relative to H D F indicates the presence of a dissociation barrier on the potential surface that is less penetrable by deuterium than by hydrogen. This same type of kinetic behavior has been observed for ND4/ND3H and D , 0 / D 2 0 H pairs where the effect is attributed to differences in the tunnel rates for H and D along the dissociation c ~ o r d i n a t e . ~ ~ ~ The mass spectrum for H D F shows that this molecule fragments slightly faster by H loss than by D loss (DF+/HF+ = 1.3). This is a reasonable isotope effect for a process with a small dissociation barrier where the overall rate is dominated by a mass effect on frequency factors ( v H I v D= 2'12). The data in Figure 2 shows that relaxation of the precursor ions, D2F+and D30+,prior to neutralization, is essential for formation of the metastables. This relaxation mechanism is most probably due to the increasing rate of collisional interaction between the ion-dipole pairs, D2F+-DF and D30+-DF as the pressure is increased. These considerations should be important in the design of any beam experiment to study D,F or D 3 0 . Acknowledgment. We are grateful to the National Science Foundation for support through the Materials Science Center (NSF Grant DMR-82-17227A03), Cornell University and N S F Grants CHE-8215184 and CHE-8314501.
(Bt''*): A Descriptive Equation for Diffusion/Sorption Data
Lloyd Abrams* and Aaron J. Owens Central Research & Development Department,? Experimental Station, E.I. duPont de Nemours and Company, Wilmington, Delaware 19898 (Received: January 14, 1986)
Diffusion processes are often analyzed by separately treating the short- and long-time events. We have found that the relationship for the amount adsorbed, Q, = (Q-) tanh [B(t'/2- A ) ] , provides an alternative method for calculating a diffusion parameter, B , and the limiting amount sorbed, Q-,for processes at constant temperature and pressure. A single, constant diffusion parameter, B, is sufficient to describe the entire range of data, i.e. at short and long times.
Introduction Isothermal sorption experiments provide data from which diffusion coefficients, D, may be calculated by using Fick's laws. For different geometries, such as spheres and cubes, the general solutions are similar; spheres:' Q,/Qm
= 1 - ( 6 / r 2 ) x ( 1 / n 2 ) exp(-n2r2Dt/r2)
cubes:2
Ql/Qm
= 1 - (S/r2I3
kmn
exp[-(k2
1 / ( k 2 ) ( m 2 ) ( n 2X)
+ m2 + n 2 ) ] ( r 2 D t / w 2 )
(odd k , m, n )
where Q, and Qm are the amounts sorbed at time t and at equilibrium, respectively, into spheres of radius r or cubes of side w. The equations are somewhat unwieldy to use over the entire range of Dt/r2 and approximations are used for long and short times. For example, p t long times, the solution for spheres has the form 'Contribution No. 3871.
0022-3654/86/2090-2300$01.50/0
while for cubes, it has the form (Qm
- Q,)/Qm = (8/r2I3 exp(-3r2Dt/w2)
At short times, the general solutions converge slowly and the often used Qt vs. t'l2 relationship is used to calculate diffusion coefficients. Plotting Q, vs. t 1 / 2yields a straight line whose slope is ( 6 Q - / r ) ( D / ~ ) for l / ~spheres and (12Qm/w)(D/r)'/' for cubes. The values for diffusion coefficient for long and short times, t , are very often not the same and this difference suggests a concentration dependence of the diffusion coefficient. As pointed out by Weisz3 and others, D should not be concentration-dependent but should include another term (in Fick's law) which carries (1) Crank, J. Mathematics of Diffusion; Oxford University Press: London, 1956. (2) Mathews, J.; Walker, R. L. Mathematical Methods of Physics: W. A. Benjamin: Reading, MA, 1970; 2nd ed. ( 3 ) Weisz, P. B. Chemtech 1981, 134. Smith, D. M.; Keller, J. F.Ind. Eng. Chem. Fundam. 1985, 24, 499.
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 1I, 1986 2301
Letters
Chabazite
1.oo-
Gmeiinite
Heulandite
0.90-
0.800.708 0.60..
?&
..
0 0.500.40-
0.30
0.200.10--
0.006 t%
tilz
(set"')
(set'")
tl/z
(Sec")
Figure 1. Intrinsic diffusion of water into zeolites. Experimental values of Q,/QB vs. from Barrer and Fender.4 (a, left) Chabazite: (V) 30.8 O C , tanh [0.0196(t1/2- 2.027)]; (A)75.4 "C, tanh [0.0366(t'/2 + 1.966)]. (b, center) Gmelinite: (V) 31.7 "C, tanh [0.0356(t1/2- 1.284)]; (A)62.5 OC, tanh [0.0790(t'/2 - 1.135)]. (c, right) Heulandite: (V) 37.4 "C, tanh [0.00776(t1/2- 0.442)]; (A) 77.4 OC,tanh [0.0176(t1/2- 1.842)].
information about the sorption isotherm. We have found that the relationship Q, = (QJ tanh (Bt112)
(1)
completely describes diffusionally dependent isothermal sorption processes over a range of times where B is a constant related to the diffusion coefficient and particle geometry. The experimental validity of eq 1 suggests that a single diffusion coefficient satisfies the entire range of data (at long and short times) while providing a term to account for sorption effects. In this Letter, we show that eq 1 provides an identical fit to the exact series solution at both long and short times. Some examples are provided to demonstrate the utility of this approach.
Results and Discussion From eq 1, the derivative of Q,with respect to t'12 provides the relationship dQ,/d(t1/2) = (Q,B)[l
- tanh2 (Bt'lZ)]
1111.:
!8 7.
4.
(2)
At short times, QJ must equal the slopes obtained from the general solutions for specific geometries such that B = (6/r). ( D / r ) * I 2for spheres and B = ( 1 2 / ~ ) ( D / r ) ' /for ~ cubes. In effect, eq 1 satisfies certain criteria describing the diffusion process: (i) The equation is a continuous function with an initial slope (nonzero and noninfinite) and an asymptotic (equilibrium) value. Furthermore, few data points are needed to provide a reasonable quantitative description of the sorption process. (ii) Only two parameters are needed to define the entire sorption process, i.e., the equilibrium value, Q-, and a constant, B, related to the diffusion coefficient via particle geometry. (iii) Equation 1 provides the correct, exact functional form to the diffusion data for short as well as long times. Shown in Table I is a comparison of the expansions of the general solution (based on the series summation discussed above) for diffusion into cubes vs. values obtained from eq 1 for different (Dt/d)'/2. The largest deviation from the exact solution is 10%. The fact that eq 1 describes the entire sorption process implies that the diffusion coefficient is constant for the entire process and is independent of the amount sorbed. Equation 1 was modified to the form
-
Q,/Qm = tanh [ B ( t ' l 2- A ) ]
9.0-
(3)
where A is a constant representing a time offset from the origin. Applications of eq 3 to fit experimental data are shown in Figure la-c. As noted above, the value B is related to the diffusion
1.o
0.01
5
15
10
20
25
$0
(sec"2) Figure 2. Sorption of water into H-ZSM-5. Experimental values from Hill and S e d d ~ n .(A) ~ 20 OC,10.0 tanh [0.0460(t1/2+ 0.053)]. (V)50 O C , 6.37 tanh [0.0656(~'/~ + 1.248)]. ( 0 )100 "C, 5.08 tanh [0.100(t1/2 + 2.350)]. '12
coefficient, D, Le., B = (c/r)(D/r)1/2, where c and r a r e constants depending on particulate geometry. Assuming that the diffusion is activated and the temperature dependence of D can be represented by the Arrhenius equation, such that D = Doexp(-E,/RT), then the activation energy of diffusion may be calculated directly from B,Tvalues, E, = 2R(ln B , - In B2)/(1/T2 - l / T l ) . The activation energy is independent of particle geometry as the terms that account for geometry cancel. For the data in Figure la-c, activation energies of 5.1, 10.5, and 8.8 kcal/mol were obtained for chabazite, gmelinite, and heulandite, respectively. The values are in reasonable agreement with those previously r e p ~ r t e d . ~ (4) Barrer, R. M.; Fender, B. E. F. J . Phys. Chem. Solids 1961, 21, 12.
J. Phys. Chem. 1986, 90, 2302-2305
2302
TABLE I: Comparison of Solutions for Diffusion into Cubic Particles vs. General Solution tanh (6.7703{(Dt/~')'/~1) general ratio (Dt/w2)'1* solution tanh (tanh/general) 0.006 1.001 0.001 0.006 0.036 0.056 0.076 0.096 0. I26 0.166
0.186 0.226 0.256 0.296
0.224 0.333 0.43 1 0.519 0.633 0.755 0.804 0.88 1 0.923 0.960
0.239 0.361 0.473 0.571 0.692 0.808 0.850 0.910 0.939 0.964
1.064 1.086 1.097
1.100 1.093 1.070 1.057 1.032 1.017 1.004
vs. In the preceding application, experimental values of time were available. Generally, values of weight gain (or loss) vs. time at constant temperature and pressure are obtained experimentally. Typical data are shown in Figure 2. Equation 3
was used to fit the data; values of B at the different tempeatures yield a calculated activation energy of 4.21 0.12 kcal/mol which agrees well with the reported value.5 In addition, values of Qm as a function of temperature decrease with an increase in temperature as expected. Furthermore, the calculated values of Q m are obtained from relatively short-time experimental data whereas, experimentally, much longer time periods would be needed to obtain reasonable equilibrium values. In conclusion, the relationship
Qt= Qm tanh [B(t'i2- A ) ] provides an alternative method for calculating a diffusion parameter, B, and the limiting amount sorbed, Qm,for processes at constant temperature and pressure. A single, constant diffusion parameter is sufficient to describe the entire range of data, i.e. at short and long times. (5) Hill, S. G.; Seddon, D. Zeolites 1985, 5, 173.
Thermal Collision Rate Densitles of Small Clusterst William H. Marlowl Department of Applied Science, Brookhaven National Laboratory, Upton, New York 11973 (Received: January 28, 1986)
Model calculations of the thermal collision rate densities between C02 molecular clusters consisting of 1 to 13 monomers are presented. Cluster morphologies are chosen according to the minimal-energy configurations of Hoare and Pal and the attractive part of the intercluster potential energy is calculated by summing dipole-induced dipole interactions to all orders. To model the collision rate density for each pair of clusters, an average is taken over the rates for four orientations computed in body-fixed coordinates. Results of these calculations have implications for the interpretation of experimental results on nucleation, cluster growth, and cluster mass-frequency distributions.
Gas-phase atomic and molecular clusters are important in numerous areas of science and technology. They are involved in vaporization, deposition, and nozzle-beam expansions and constitute an important stage in the formation of condensed matter. Collisions lead to the formation and evolution of clusters in both critical',2 and activationless nucleation3 and in heterogeneous growth processes. In deriving classical nucleation theory, the cluster collision rate density is initially assumed to have a simple form independent of composition and related to morphology only via cluster surface area. Treatments of cluster coagulational growth4 assume collisions occur according to a liquid drop model (with unit sticking probability) that incorporates no compositional or morphological characteristics of the colliding species. Despite their apparant successes in accounting for monomer-dominated ,~ theory of nucleation), processes ( C 0 2cluster g r ~ w t hclassical liquid drop and spherical, square-well potential models raise several questions for both monomer and multimer collision rate densities because such models are known to be inadequate for molecular collisions. Current practice gives no guidance as to why these methods are successful, how far they can be extended, or when they should fail. In addition, these methods give no hint of how the collision rate densities involved in cluster coagulational evolution are related to those for free-molecule aerosol coagulational evolution where significant size and composition dependences of 'A preliminary version of this work was presented at the First International Aerosol Conference, Sept 17-21, 1984, Minneapolis, MN. f Address for Sept 1985-Aug 1986: Environmental Engineering Division, Civil Engineering Department, Texas A&M University, College Station, TX 77843-3136. Permanent address as of Sept 1, 1986: Department of Nuclear Engineering and Applied Science, Texas A & M University, College Station, TX 77843-3133.
0022-3654/86/2090-2302$01.50/0
the aerosol particle coagulation rate densities have been identified both theoreticallySand experimentally.6 This Letter will show that including intercluster attractive energies and cluster morphologies into a cluster collision model can help address these questions and the details of cluster size distributions as well as provide physical insight elsewhere. While the system of reference for this sutdy is C 0 2 cluster^,^ the objective is to derive general properties of the collision rate densities which are fundamental to describing all gas-phase cluster collisional growth processes. Therefore, the interaction potential energy used below for C 0 2 arises solely from the dipole-induced dipole interaction (the "van der Waals" interaction common to all matter). Inclusion of the quadrupole energy might sharpen the results for C 0 2 but would render questionable any general conclusions regarding other substances. The C 0 2 molecule is assumed to be effetively spherically symmetric, a not unreasonable assumption above the dimer dissociation temperature.' For the sake of definiteness, cluster morphologies are selected according to the maximally stable structures for spheres with pairwise interactions as determined by Hoare and Pal.* Molecular cen~~~
~~~~
~
(1) Abraham, F. F. Homogeneous Nucleation Theor); Academic Press New York, 1974. (2) McGraw, R.; Marlow, W. H. J . Chem. Phys. 1983, 78, 2542. (3) Friedlander, S. K. J . Colloid Interface Sci. 1978, 67, 388. (4) Soler, J. M.; Garcia, N.; Echt, 0.;Sattler, K ; Rechnagel, E Phys Rev Lett. 1982, 49, 1857. (5) Marlow, W. H. J. Chem. Phys. 1980, 73, 6288. (6) Okuyama, K.; Kousaka, Y . ;Hayashi, K. J. Colloid Interface Sei. 1984, 101. 98. (7) Etters, R. D.; Flurchick, K.; Pan, R. P. Chandrasekharan, V. J. Chem. Phys. 1981, 75, 929. ( 8 ) Hoare, M. R.; Pal, P. Adu. Phys. 1971, 20, 161.
0 1986 American Chemical Society
