July, 1962
ADAPTATION OF LATTICE
V A C A N C Y T H E O R Y TO
GASADSORPTION PHENOXENA
1305
ADAPTATION OF LATTICE VACANCY THEORY TO GAS ADSORPTION PHENOMENA BY J. M. HONIG Lincoln Laboratory, Massachusetts Institute of Technology, Lexington 73, Massachusetts AND
C . R. MUELLER
Department of Chemistrg, Purdue Universitu, Lafayette, Indiana Received January 10, 1908
The Flo ry-Huggins polymer-monomer solution theory is adapted to obtain a lattice theory of gas adsorption in which the fractional hole size concept is utilized. Expressions for the zero-order isotherm and for the partial molecular configurational entropy are derived, in which the hole size parameter r makes its appearance. It is pointed out that certain symmetry properties of the lattice theory no longer obtain when r > 1. The theory is compared with experimental work on certain types of adsorbent-adsorbate systems, and experiments for providing further checks against the theory are suggested.
Introduction I n the lattice vacancy theory proposed by Mueller and Stupegia2 the liquid is regarded as a quasi-binary lattice assembly of particles and of fractional sized holes. Despite its simplicity, the t)heory is quite successful in reproducing such quantities as the critical constants, vapor pressure curves, and rectiliuear diameter plots €or a variety of simple liquid-vapor systems. Moreover, the usefulness of this lattice model extends well into the gaseous region, and in the limit of low particle density, the perfect gas law is predicted. The versatility of this approach suggests that it could be usefully adapted to a description of monolayer adsorption processes. In general, one deals with an adsorbate which behaves like a perfect two-dimensional gas at low coverage, and like a liquid or solid as monolayer coverage is approached (0 3 1). Furthermore, close packing of the adsorbate may not be realized if the spacing between potential minima on the surface differs from the diameter of the adsorbate atoms or molecular units; in that event, fractional holes remain on the surface even when 8 = 1. We test the utility of the fractional-sized vacancy model by deriving both the isotherm equation and the partial configurational entropy for a surface phase consisting of particles and fractionalsize holes. As a prerequisite, an expression for the free energy of such an assembly will be set up; the required expressions for the entropy and energy of the system are discussed in the next section. Thermodynamic Properties for a Binary Mixture in which the Components Differ in Size.Consider the configurational entropy of a system of N r molecules of an r-mer, built up from r monomer units,, and NI molecules of a monomer, distributed on a lattice of L, sites in such a maiinw that each monomer or r-mer segment occupies one lattice site. Then Lr
N1+
rNr
(1)
(1) Operated with support from the U. S. Army, Navy, and Air Force. (2) C. R. Mueller and D. C. Stupegia, J . Chew. Phys., 26, 1522 (1957).
(3) A more detailed set of derivations and discussion of the remainder of this paper has been given b y J. M. Honig in M.I.T.Lincoln Laboratory Technical Report No. 252. Copies available on request.
Moreover, Flory4 and Guggenheim6 have shown that the configurational entropy of mixing for the components in the solution is given by AS, = -kL, { (1 - 4 ) 111 (1 - 4)
+
(@/TI
In cbf
(2)
where the fraction of lattice sites occupied by rmer and monomer is # = rNr/Lr and 1 - 4 = N1!Lr, respectively. Equation 2 holds in the limit where the coordination number of the lattice is very large. In the present situation we consider the rmer to be analogous to an adsorbed atom or niolecule and the monomer analogous to a hole. Since an adsorbed atom is far more symmetric than an r-mer, it is physically more palatable to consider the particle as occupying one surface site of area a, and the hole as occupying an area ffh = (l/r)aa. To see the implication in this shift of viewpoint, we can write the area A of the surface as A = Nags N h c r h , where N , and N h are the total number of each species. Then
+
N,
+
Nh/T
=
A/as
E La
(3)
which leads t o the following expressions for the fractional surrace coverage: 0 = Na/La and 1 0 = N h / r L a . It is seen that the substitution of rL, for Lr and of e for 4 will convert eq. 2 to the form So = -JGLa(e In 6
+ r(1 - e) In (1 - e)}
(4)
where AS, = So in the present case, since a lattice devoid of atoms or completely filled with atoms has zero configurational entropy. To S, we must later adjoin the contribution of the thermal degrees of freedom to the entropy, ST,of the adsorbed particles. The configurational energy of the system is found by noting that, neglecting edge effects, the maximum number of nearest neighbor pairs in a lattice of La sites is ZLa/2, where Z is the coordination number. In a purely random distribution the probability of encountering an occupied pair of nearest neighbor sites is e2. If now the lateral energy of interaction associated with (4) P. Florg, J . Chem. Phys., 10, 51 (1942). ( 5 ) E. A. Guggenheim, “Mixtures,” Clarendon Press, Oxford ,1952.
based on the assumption of a random distribution, which can only occur if w = 0. However, it call be showns that eq. 4 and 5 are the limiting cases for Zw/2 1, per < p(1/2). (2) The effect of lateral interactions is offset to some degree by increasing the value of r. Thus, it is seen from Fig. 2 that the isotherm corresponding to C = 3 and r = 2 exhibits loops, whereas those for which r > 2 do not. Similarly, in Fig. 3, the curves for C = 4 and r = 2,3 indicate the onset of condensation, whereas the curve for C =: 4 and r = 4 does not. Also, the various diagrams show that it requires lxogressively higher pressures to reach a given value of 8 as r is increased. All of the above is tl reflection of the fact that E, is independent of r while S, changes so as to make it more difficult to fill the surface completely when r becomes large. (3) One of the most interesting findings is- that the configurational partial molal entropy ,So is no longer anlisymmetric about the point 8 = l/2 when r > 1. Recently, it has been shown by Kleiner7 that when the lateral interaction energies are assumed to be pairwiee additive and localized adsorption prevails, Sc will always exhibit the inversion symmetry about 8 = 1/2, so long as r = 1. Figure 4, in which S1= - log L8/(l = &/2.303k - (r - 1)/ 2.303 is plotted us. 8, shows the progressive departure from this situation; the partial molal entropies obtained for a given 8 become more negative as r is increased. (4) Figures 1-3 are very similar to those obtained from isotherms based on the two-dimensional van der Waals equation of state.* Mere examination of adsorption data thus does not suffice to distinguish between the two theories pertaining to localized and mobile adsorption. The model discussed here should be used primarily in the interpretation of data taken at very low temperature, and is competitive with the van der Waals isotherm equation in the intermediate region between 77 and 90°K. Where the present theory is applicable, deviations from the symmetry properties discussed in (1)-(3) may be considered to arise because of the mismatch between the extent of an adsorption site (minimum in the adsorption potential) and the dimensions of the adsorbate. We will conclude with a brief discussion comparing theory and experiment. One of the important aspects of prior t h e o r i e ~ ~ dealing ,~ with monolayer adsorption of gases on energetically homogeneous surfaces and with the situation where r = 1 is that the point of inflection in the sigmoidal isotherms or in the loop occurs a t e = l/2. Moreover, letting 8' and 8" represent the two fractions of occupied sites at which two-dimensional condensation sets in and terminates, it can be shown6 that 8" - 1/2 = 1/2 - 8'; ie., that 8' and 8" are equidistant from the inflection point when T = 1. It is not a simple matter to find experimental data against which the theory can be checked; for, to briing out the features discussed earlier, it is necessary to employ homogeneous adsorbents, adsorbates that display a high degree of molecular symmetry, and systems where the formation of (7) W. H. Xleiner, J . Chem. Phys., 3 6 , 76 (1962) 18) J . H. de Boer, "The Dynamical Character of .4dsorpti0n,'~ Oxford, 1953, Chap. VITI. (9) R. R u m b l ~and J. M. Nonig. J , Chem. P l r y ~ . 58, , 424 (1960).
.5
Ic
$0
-I.OL
-1.5
1
-2,ot
.I
Fig. 4.-Plot
.2
.3
.4
.5
.6
.7
.8
.9
1.0
e. of the quantity [&/2.303/c - ( T - 1)/2.303]
s1US. e.
higher layers does not begin until after a monolayer has beerr deposited. Severt'heless, there exist data that conform to the requirements of the present theory. As an example, one may cit'e the measurements by Prenzlow and I-IalseylO; t,heir adsorption isotherms for argon on graphitized carbon black covered by a preadsorbed layer of xenon are sigmoidal in character, and the infleetmionpoint' occurs at 8 w, 0.4. Moreover, by extrapolat'ing t'he results to t'emperat'ures slight'ly lower than the range covered in the measurements, one can see that 8' and 8" mill not be equidistant from the inflection point. The parameters C = 3 and r = 3 are roughly in accord with the data by Prenzlow and Halsey for temperat'ures near 650K.11 A similar situation is encountered in the adsorpt'ion of argon at 77OK. on graphitized carbon black,12 argon on one layer of xenon preadsorbed on graphit'ized black (77OK.) , I 3 krypton on graphitized carbon black, (77.8°K.),14and xenon, CH4, CzHa on {100) surfa'ces of NaC115 (103-113°K., 77-93'K., 123-140OK.). The list' is not, exhaustive, but sufficient to show t,hat there are cases where a lattice theory in which r = 1 fails t'o apply to experimental data. The second important aspect of the theory developed here, concerning t'he asymmetry of 3, about' the point 8 = 1/2, is not checked as readily, since there is a real paucity of suitable experi(10) C. F. Prenzlow and G. D. Halsey, Jr., J. Phus. Chem., 61, 1188 (1957). (11) W. Steele (personal communication) has attempted to reinterpret these data b y lowering the numerical value for the number of atoms required for monolayer coverage of the surface. This does bring the inflection point of the sigmoidal isotherms closer t o B = ' / a ,
b u t the thermodynamio functions nevertheless do not exhibit the symmetry associated with the value T = 1. (12) J. H. Singleton and G. D. Halsey, Jr., J . Phys. Chem., 58, 1101 (1954). (13) J. H. Singleton and G. D . Halsey, Jr., i b i d . , 58, 330 (1954). (14) S Ross a n d W. Winkler, J. Colloid Sci., 10, 330 (1955). (15) 9. Ross and H. Clark, J . Am. Chem. Soc., '76, 4281 11954).
1308
JAESHI CHOI AXD
%‘ALTER
J. MOORE
Vol. 66
mental data. In a review by Everett and Young,16 involving holes of fractional size, as presented in So was computed from experimental data for a this paper, is a convenient extension of prior theolarge number of different adsorbent-adsorbate r i e ~to~ which . ~ it reduces in the limit r = 1. Subject systems. In almost all cases calculated, So us. 0 to the assumptions inherent in the theoretical curves deviated from the “ideal” curve (eq. 11 development, the theory specifies adsorption isowith r = l),but for 0> 1/2 the experimentalvalues therms and other thermodynamic quantities for fcll above the ideal curve rather than below, as in the adsorbed phase in terms of an additional paFig. 4. This is attributed to two effects. In the rameter r. For reasonable numerical values of the particular cases cited by Everett and Young, the latter, the theory is in better accord with a variety surfaces were energetically heterogeneous, and of experimental data than the earlier formulations multilayer formation set in long before the first for which r = 1. layer was completed. Both of these phenomena To obtain further checks on the theory discussed contribute to the above mentioned e f f e ~ t . l ~above ~ ~ ~it is highly desirable to have data which are The only instanct: that has come to the writers’ taken with the specific objective of testing the attention where the deviation of the experimental results cited here. In particular, the data should from the ideal So us. e curves is as indicated in be amenable to the determination of So. If systemFig. 4 is the work by Hill, Emmett, and Joyner18 atic deviations from the above theory are noted, f_or the adsorption of nitrogen on graphon. The it may be necessary to resort to a more sophistiS, us. 0 ciirve cited in the above reference roughly cated derivation in which the self-contradictory matches the curve in Fig. 4 for which r = 3. assumptions concerning the random distribution Summarizing, it may be said that the theory coupled with a non-zero configurational energy are eliminated. (16) D. H. Everett and D. M. Young, Trans. Faraday Soc., 48, 1164 (1952). Acknowledgment.-The authors are very greatly (17) L. E. Drain and J. A. Morrison, ibid., 48, 316 (1952); 49, 654 indebted to Dr. Walter H. Kleiner for his un(1953). stinting assistance in the preparation of this (18) T. L Hill, P. H. Emmett, and L. G. Joyner, J . Am. Chem. manuscript and for many fruitful discussions. Soc., 73, 5102 (1951).
DIFFUSIOX OF NICKEL I N SINGLE CRYSTALS OF NICKEL OXIDE’ BY JAESHI CHOIAND WALTERJ. MOORE Chemical Laboratory, lndiana University, Bloonaington, Indiana Received January 18, 1988
The diffusion of 63Ni in single crystals of NiO has been measured by a sectioning method. From 1000 to 1470’ for NiO in air, D(Ni) = 1.83 x 10-3 exp( -45.6 kcal,/RT). The D was almost the same in crystals containing 4 X lo-* atom fraction trivalent impurities as in those containing 60 x 10-4 cobalt. A mechanism based on singly ionized cationic vacancies gives a quantitative interpretation of the AH* and AX* for the diffusion.
Previous studies of the diffusion of 83Niin NiO have yielded discordant results. Shim and Moore2 reported D N ~ = 4.4 X e x p ( 4 4 . 2 kcal./RT) ems2set.-?, whereas Lindner and Akerstrom,ausing similar techniques and in some cases crystals from the same boule, found D N ~= 1.72 X exp(-56.0 kcal./RT). We have undertaken the further measurements described in this paper in an effort to resolve this discrepancy. Both earlier studies were made by the method of decline in surface activity. The isotope BaNiis not well suited to this method since it emits a soft (63 kv.) p, and consequently very thin layers of the nickel oxide suffice to absorb most of the radiation. We therefore used in the present measurement of D a different method, which is more suitable for this soft tracer radiation. This is the method of surface activity after ~ectioning.~We also made some special tests of the activation energy of D by the method of Zhoukho~itzky,~ which is indepen(1) Work assisted by the U. S. Atomio Energy Commission, Contract
AT .(11-1)-250. (2) M. T. Shim and W. J. Moore, J . Cham. Phgs., 26, 802 (1957). (3) R. Lindner and A. Akerstrom Dzscussions Faraday SOC.,23, 133
(1957). (4) R. H. Condit and C. E. Birchenall. J . Metals, 8 , 1341 11956). (5) A. A. Zhoukhoyitzky. J . A p p l . Rad. and Isotopes, 6 , 159 (1959).
dent of the value of the absorption coefficient. A further improvement in technique was the annealing of the crystals before the diffusion run. Recent work6 has emphasized the necessity of such preannealing of crystals made by the Verneuil method. All our new results support the lower value of the activation energy. From 1000 to 1400” for monocrystalline NiO in air, D = 1.83 X exp(45.9 i 2.0 kcal./RT). The activation energy by the Russian method was E = 41 kcal. for polycrystalline specimens. Experimental Methods The single crystals of nickel oxide were cut as 4 to 5 mm. squares, 2 to 2.5 mm. thick. Crystals were used from two sources: Tochigi Chemical Industry Company, Osaka, Japan, and General Electric Company, Schenectady, New York. Spectrographic analyses were made of these materials. The Japanese NiO contained 0.6’% cobalt and 0.1% Mg, with traces of other elements. The G.E. crybtals contained less than 10 p.p.m. cobalt.’ Both square faces of a specimen were polished flat on a precision grinder, using silicon carbide paper of grades 1/0 to 4/0. The specimens then were pre-annealed in a stream of dry tank (6) Y. Oishi and W. D. Kingery, J . Chem. Phys., 33, 480 (1960). (7) Analysis %: A1 0.003, Co
