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Selected Works of Yakov Borisovich Zeldovich
By J. P. Ostriker, G. I. Barenblatt, R. A. Sunyaev, A. Granik, E. Jackson PRINCETON UNIVERSITY PRESS
Copyright © 1992 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08594-4
CHAPTER 1
On the Theory of the Freundlich Adsorption Isotherm
§1. By far the majority of available experimental data on the adsorption of gases on the surface of a solid substance totally fails to agree with the simple laws derived by Langmuir. Moreover, even the linearity of the adsorption isotherm at very low surface coverage, which follows for adsorption on a uniform surface from the general laws of statistical mechanics, independently of any assumptions regarding the interaction between adsorbed particles, the dependence of the potential energy on distance, etc.—even this linearity of the initial portion of the isotherm represents not so much an unambiguous result of experiments as the assumptions of theorists.
Of the various empirical equations proposed for the adsorption isotherm, the most significant is undoubtedly the isotherm
q = Cp1/n. (1)
Its value for the description of experimental data is convincingly elucidated in the well-known monograph by McBain. This formula was introduced originally by analogy with the distribution of a substance between phases in which it exists in various states of association. However, experimental values very soon showed the impossibility of this explanation: it allows neither fractional values of n which vary smoothly with temperature, nor such values as n = 10 for the adsorption of iodine on starch.
Isotherm (1) is often obtained under conditions which physically exclude the dissociation of adsorbed molecules, for example, in the adsorption of O2 and Ar on SiO2 at low temperatures.
Let us note that since complete adsorption is limited by the size of the surface, at least under conditions which are far from condensation, for greater surface coverage deviations of experimental data from equation (1), where lim g = ∞, unavoidably occur. It would be incorrect, however, to consider that this equation satisfies the experimental data only insofar as it coincides in practice with the Langmuir isotherm in some middle portion. In many cases an isotherm which satisfies the Langmuir equation at high surface coverage (correctly describing the saturation of the surface) definitely diverges from it at low coverage, following the nonlinear law (1) (cf. 4 below).
§2. We shall consider several proposed explanations of isotherm (1). Chakravarti and Dhar derived isotherm (1) from the conception of multiple adsorption. Here, if one adsorbed particle occupies n empty locations on a surface, Chakravarti and Dhar assume that the probability of adsorption is proportional to (1 - vn)n, while the probability of desorption is vn.
However, such expressions may be obtained only by assuming that n empty sectors occupied by a particle being adsorbed may be located arbitrarily far from one another, and that any n occupied locations on the surface, if freed simultaneously, will release from the surface a molecule of the adsorbed substance.
In other words, the formulas which Chakravarti and Dhar write correspond to the assumption of dissociation of an adsorbed particle into n separate parts. The authors themselves reject this assumption at the beginning of the article.
Zeise, apparently independently of them, considered a special case of multiple adsorption: adsorption of a molecule at two empty locations of the surface. In addition, besides the proportionality of the probability of adsorption to (1 - θ2) (in our notation), he also makes quite strange assumptions about the mechanism of adsorption in his paper. Regardless of the fact that he is considering an equilibrium at which the temperature of the adsorbent does not differ from the temperature of the surrounding gas, Zeise writes: "Escape (of adsorbed molecules) from a unit of the surface must consist of those molecules which are knocked out by molecules (from the gas phase) which collide with them, since we cannot consider that molecules of gas, once adsorbed, may be freed solely by the effect of the insignificant impulses of the thermal motion of molecules of the underlying solid body."
But most extraordinary is the fact that, if we accept Zeise's assertion, it turns out that both adsorption and desorption are proportional to the pressure in the gas phase. The equilibrium adsorbed amount turns out not to depend on the pressure. Zeise manages to produce an isotherm only because, several lines later, he forgets about the proportionality of the probability of desorption, v = xµ, to the number of collisions with the surface, µ, and subsequently treats v as a constant.
We mentioned above the general agreement on the linearity of the initial sector of the adsorption isotherm on a uniform surface. The isotherms derived by Chakravarti and Dhar, q = Cp1/n/(b + p1/n), and by Zeise, q = [ap/(p + b)]1/2, are interesting in that, together with the nonlinearity of the isotherm at the beginning (q ~ p1/n; q ~ [square root of p]), they also reflect the tendency of adsorption towards saturation for p -> ∞.
A number of derivations of isotherm (1) are based on the well-known formula of Gibbs,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where equation (1) is obtained with the help of some incomplete assumption as to the dependence of the surface tension σ on q: πΩ = nRT where π = σ0 - σ is the surface pressure, Ω is the surface occupied by one gram molecule of adsorbed substance, the reciprocal of q (Rideal); or σ = σ0 (1 q/q∞) (Chakravarti and Dhar); or, finally, the ad hoc (σ0 - σ)/σ = Cp1/n (Freundlich).
Meanwhile, it is evident that with sufficient dilution any adsorbed substance behaves, independently of its two-dimensional mobility, as a two-dimensional ideal gas with the equation of state πΩ = RT. Together with the Gibbs equation, this is one proof (if such is in fact necessary) of the linearity of the initial part of the isotherm on a uniform surface.
§3. For some number of elementary sectors with given properties (heat of adsorption, vibrational volume) arbitrarily distributed on a surface, the adsorption isotherm of Langmuir,
q = ap/[p+b], (2)
should strictly obtain (under the usual assumptions that not more than one molecule can be adsorbed in each sector and that mutual interactions may be neglected). In this formula, b depends upon the properties of the sectors, while a is proportional to their number.
The isotherm is linear for p of the same order as b the increase of q begins to lag strongly behind that of p. Therefore, the nonlinearity of the experimental adsorption isotherm in some region of pressures should be considered (again, in the absence of dissociation or strong interaction between adsorbed particles) as proof of the presence on the surface of points with values of b from the Langmuir formula which are (for a given gas) of the order of magnitude of the pressures under consideration.
A rational explanation of equation (1) will be obtained if we find a distribution of points of the surface according to their adsorptive capacity such that the superposition of the Langmuir isotherm on each "kind" of points leads to q = Cp1/n. Mathematically, the problem is formulated as an integral equation of the first kind,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)
where q(p) is the equation of the observed adsorption isotherm, and a(b) is the function sought. Convergence of the integral [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to q∞ corresponds to the finite limit of adsorption for p -> ∞. In Langmuir's original interpretation of adsorption on a non-uniform surface, we encounter two kinds of formulas: for adsorption on a crystal surface, Langmuir writes
q = [summation over i] [aip/p + bi], (4)
while for adsorption on an amorphous surface, where continuous variation of b is possible, he gives the following formula for q:
q = ∫ ap/[p+b] dS. (4a)
The integral is taken over the entire geometric surface, without separating out or collecting together points with equal values of b or introducing the function a(b).
We shall now proceed to the solution of integral equation (3). Mathematics gives no general method for the solution of equations of this type (the equation is singular). This situation more than justifies the method we apply, one of guessing rather than finding the solution, based on the special properties of the "nucleus" of the equation of the function, p/(p + b). Therefore the emphasis shifts to verification of the solution obtained. We shall check the correctness of the "guessed" solution by the irreproachable method of substituting it into the exact integral equation. In this same way we shall also find the numerical coefficients.
We schematize the Langmuir isotherm with b given by replacing it with two straight lines,
q = ap/b, 0 < p ≤ b; q = a, b ≤ p, (5)
which qualitatively describe the initial and saturation portions of the isotherm. After this substitution, the solution is obtained very simply:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)
Differentiating equation (6) twice with respect to p, we easily find
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and
q" (p) = -a(p)/p, a(p) = -pq"(p), (7)
whereby equation (3) is (approximately) solved. It is easy to see that this solution correctly reflects the basic properties of the true a(p); the requirement that a(p) > 0 gives, for a superposition of the Langmuir isotherms, the obvious q"(p)< 0.
It is also easy to verify that, automatically,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
Formula (7) is a mathematical expression of the fact that the nonlinearity of the isotherm for some p1 is related to the presence of sectors of the surface for which b = p1.
§4. Let us return to equation (3); from it we easily find the corresponding expression
a(b) = Ab1/n-1. (9)
In order to make [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] convergent, it is sufficient to break off a(b) at some b0. The infinite value of b1/n-1 at b = 0 is of no consequence since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges.
After substitution into formula (3), the expression found for a(b) gives
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)
In finite form the integral cannot be taken for an arbitrary value of n. Let us separately consider its asymptotic behavior at p < b0 and p > b0. At low pressure
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)
and, taking the limit,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
since, as we know,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)
Expressing A in terms of the saturation, which we determine by formula (8), we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)
Formulas (12) and (14) prove that the distribution (9) which we have found actually leads to isotherm (3) at low pressure.
For p > b0, we expand and p/(p + b) = 1/(1+ b/p) = 1 –b/p + ..., and integrate by terms, after which we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)
With accuracy to within higher order terms, expression (15) coincides with the expansion of separate Langmuir isotherms in 1/p at high pressure:
ap/p + b' = a(1 - [b'/p] + ...); b' = bo/n + 1.
Thus, distribution (9) gives us an adsorption isotherm which not only coincides with the given Preundlich equation (1) for small p, but also describes the tendency of q to saturation for large p which occurs in experiment (cf. §1). Precisely this character, incidentally, is exhibited by the experimental data of Zeise [7], Chakravarti and Dhar [6], and Roginskii and the author [10].
§5. The quantities b with which we characterized the adsorptive properties of points on the surface can be written in the form g exp(—Q/RT), where g varies only slightly with the temperature compared to the exponent. Assuming that the value of g is the same for all points on the surface, we may find a distribution function of points on the surface according to the heat of adsorption of the gas, A(Q):
A(Q)dQ = a(b)db = D'e- Q/nRT dQ, Q > Q0, [16]
(Continues...)
Excerpted from Selected Works of Yakov Borisovich Zeldovich by J. P. Ostriker, G. I. Barenblatt, R. A. Sunyaev, A. Granik, E. Jackson. Copyright © 1992 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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