1918
Langmuir 2000, 16, 1918-1923
On the Structure of a Local Isotherm and Solution to the Adsorption Integral Equation J. P. Prates Ramalho* Departamento de Quimica, Universidade de E Ä vora, Apartado 94, 7000 E Ä vora, Portugal
Georgi V. Smirnov Centro de Matema´ tica Aplicada, Departamento de Matema´ tica Aplicada, Faculdade de Cieˆ ncias, Universidade do Porto, Rua das Taipas 135, 4050 Porto, Portugal Received June 10, 1999. In Final Form: October 20, 1999 The relationship between the measured adsorption isotherm and the unknown energy distribution function is described by the adsorption integral equation, a linear Fredholm integral equation of the first kind. We discuss the structure of its kernel. On the basis of general mechanical-statistical properties of the local isotherm and a simple and reasonable mathematical hypothesis, we derive a general analytical form for the local isotherm, which gives, as a particular case, the Langmuir isotherm. In this isotherm, all nonidealities, i.e., lateral interactions, both short and long range, are incorporated. Then we show that, under rather general assumptions, the adsorption integral equation with the general local isotherm as a kernel can be solved in an analytical form.
Introduction When a gas is in contact with a solid surface, it can be adsorbed by the solid, because of the existence of gassolid interactions. In many theoretical works on singlegas adsorption, the surface of the adsorbent is assumed to be homogeneous. This is a good model for the adsorption of simple gases on highly homogeneous surfaces, like some specially prepared samples of graphite. However, in general, the adsorption process is much more complicated, and surface heterogeneity plays an important role and must be taken into account. The surface heterogeneity appears because of the existence of local crystalline disorder, the presence of impurities strongly bonded with the surface, surface roughness, etc. In the theory of adsorption on energetically heterogeneous solid surfaces, it is assumed that the spectrum of adsorption energies in a given system is sufficiently dense to be represented by a continuous distribution of adsorption energies. Because this distribution function characterizes the adsorbent, there has been an enormous amount of work devoted to the development of experimental and theoretical techniques that allow this function to be obtained. The relationship between the measured adsorption isotherm and the unknown energy distribution function is described by the adsorption integral equation
θ(p) )
∫0∞θ(p,E) N(E) dE
(1)
where p is the equilibrium pressure, E is the adsorption energy, θ(p,E) is a local adsorption isotherm, θ(p) is the global adsorption isotherm which is experimentally measured, and N(E) is a relative number of adsorbing centers with the energy E. The function N is defined for nonnegative values of E, takes nonnegative values, and satisfies the condition
∫0∞N(E) dE ) 1
(2)
The local adsorption isotherm, θ(p,E), is the kernel of the integral equation and represents the accepted model
of adsorption. It describes adsorption on a homogeneous surface with adsorption energy E. From a mathematical point of view, to find the energy distribution function amounts to solving a Fredholm equation of the first kind. This problem is known as illposed. The difficulty in solving the equation arises from the fact that small variations in the total isotherm θ(p) may lead to significant changes in the distribution function. Because θ(p) is usually known only as a discrete set of experimental values (with inevitable experimental errors) in a limited range of pressures, the problem is much more involved. An extensive review of methods used to solve the adsorption integral equation was presented by Jaroniec and Madey.1 According to Re,2 three general classes of methods are used to solve (1): (1) Analytical methods first introduced by Sips3,4 and developed later by Misra5-7 based on integral transform theory. This approach assumes that the local isotherm and the global one are given in an explicit analytical form. (2) Numerical methods based on an elementary optimization of a parametrized form or on more accurate methods which take into account the ill-posed character of the Fredholm equation of the first kind. Examples of these methods are the singular value decomposition (SDV)8,9 regularization,10-15 and a combination of the (1) Jaroniec M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, The Netherlands, 1988. (2) Re, N. J. Colloid Interface Sci. 1994, 166, 191. (3) Sips, R. J. Chem. Phys. 1948, 16, 490. (4) Sips, R. J. Chem Phys. 1950, 18, 1024. (5) Misra, D. N. Surf. Sci. 1969, 18, 367. (6) Misra, D. N. J. Chem. Phys. 1970, 52, 5499. (7) Misra, D. N. Indian J. Pure Appl. Phys. 1971, 9, 358. (8) Vos, C. H. W.; Koopal, L. K. J. Colloid Interface Sci. 1985, 105, 183. (9) Koopal, L. K.; Vos, C. H. W. Colloids Surf. 1985, 14, 87. (10) Philips, D. L. J. Assoc. Comput. Mach. 1962, 9, 84. (11) Tikhonov, A. N. Sov. Math. 1963, 4, 1035, 1624. (12) Twomey, S. J. Assoc. Comput. Mach. 1963, 10, 97. (13) House, W. A. J. Colloid Interface Sci. 1978, 67, 166. (14) Merz, P. H. J. Comput. Phys. 1980, 38, 64. (15) Szombathely, M. v.; Bra¨uer, P.; Jaroniec, M. J. Comput. Chem. 1992, 13, 17.
10.1021/la990746e CCC: $19.00 © 2000 American Chemical Society Published on Web 12/16/1999
Structure of a Local Isotherm
regularization principle with the B-spline representation of the distribution function.16 (3) Approximate methods based on approximations imposed on the local isotherm.17-25 The most popular and commonly used approximation method is the condensation approximation.17 There are also hybrid analytical-approximate methods like the method proposed by Villie´ras et al.26 The methods can also be classified as local and global.27 The global methods give the distribution function in the whole domain of energies and require knowledge of the experimental global isotherm in the complete range of pressures, which means, for p ∈ (0, +∞). The local methods give the distribution function in a limited range of adsorption energies and require knowledge of the global isotherm in a bounded interval of pressures. Usually the approximate methods are of local character. The method presented here can be classified as an analytical one and, in some sense, can be seen as an extension of the Sips method. Motivated by general mechanical-statistical properties of the local isotherm and using some simple and reasonable mathematical hypotheses, we derive a general analytical form for the local isotherm. This form contains as a particular case the Langmuir isotherm. In this isotherm all nonidealities, that is, lateral interactions, both short and long range, are incorporated. We show also that under rather general assumptions the adsorption integral equation with the general local isotherm as a kernel can be reduced to the Stieltjes integral equation3,28 with a modified global isotherm, and, therefore, solved in an analytical form. The paper is organized as follows. In the second section we discuss the structure of the local isotherm. The class of kernels considered in the sequel is introduced in the third section. In the fourth section we recall the wellknown inversion formula for the Langmuir integral equation. The fifth section contains an inversion formula for the adsorption integral equation with a special (polynomial) kernel, while the following section is devoted to the general case. Finally in the last section we present concluding remarks. 2. Structure of the Local Adsorption Isotherm
Langmuir, Vol. 16, No. 4, 2000 1919
This equation means that, provided that the inverse function of ξ exists, the local isotherm θ(p,E) has the form
θ(p,E) ) Θ(K(E) p) where K(E) ) K0 exp(E/KBT). In this section we present a simple proof of this conjecture. We assume that the solid surface consists of a lattice of M sites. Any site can adsorb one molecule. The surface is homogeneous, and the sites have adsorption energy E. As a general case, the adsorbed molecules can interact with each other, and we assume that this interaction does not change the internal levels of the molecule. The canonical ensemble partition function of a set of N adsorbed molecules can be written as
QN,M ) [qa(T)]N exp(NE/KBT)Q ˜ N,M where Q ˜ N,M is the canonical configurational partition function of N structureless particles interacting with each other, in a M site bidimensional lattice, and qa(T) is the partition function of one adsorbed molecule. The partition function qa(T) for an adsorbed polyatomic molecule contains terms connected with its vibration with respect to the surface and its rotations and internal vibrations. For atoms qa(T) is a vibrational partition function. It describes vibrations of the atom with respect to the surface. This partition function also contains terms describing internal electronic degrees of freedom. The exact form of the partition function Q ˜ N,M depends on the details of the interaction between the particles and on the structure of the lattice, but it does not depend on the interaction between the molecules and the surface. The chemical potential µs is given by
(
where E is the adsorption energy, T is the temperature, KB is the Bolzmann constant, and K0 is a function of the temperature. (16) Jagiello, J. Langmuir 1994, 10, 2778. (17) Harris, L. B. Surf. Sci. 1968, 10, 128. (18) Cerofolini, G. F. Thin Solid Films 1974, 23, 129. (19) Jaroniec, M.; Rudzinski, W.; Sokolowski, S.; Smarzewski, R. Colloid Polym. Sci. 1975, 253, 164. (20) Rudzinski, W.; Jagiello, J.; Grillet, Y. J. Colloid Interface Sci. 1982, 87, 478. (21) Hsu, C. C.; Wojciechowski, B. W.; Rudzinski, W.; Narkiewicz, J. J. Colloid Interface Sci. 1978, 67, 292. (22) Rudzinski, W.; Jagiello, J. J. Low Temp. Phys. 1981, 45, 1. (23) Rudzinski, W.; Narkiewicz, J.; Patrykiejew, A. Z. Phys. Chem. (Leipzig) 1979, 260, 1097. (24) Nederlof, M. M.; Van Riemsdijk, W. H.; Koopal, L. K. J. Colloid Interface Sci. 1990, 135, 410. (25) Jagiello, J.; Ligner, G.; Papirer, E. J. Colloid Interface Sci. 1990, 135, 410. (26) Villie´ras, F.; Michot, L. J.; Bardot, F.; Cases, J. M.; Francois, M.; Rudzinski, W. Langmuir 1997, 13, 1104.
M,T
(see ref 30). If the adsorbed phase is in equilibrium with the gas phase at pressure p, the phases have the same chemical potential. If the gas phase behaves as an ideal gas, its chemical potential µg is
[
Without proof, but giving several examples, Rudzinski and Everett29 showed that the detailed form of any theoretical isotherm is determined by the form of function ξ(θ,T) satisfying
ln(pK0) ) -E/KBT + ln ξ(θ,T)
)
∂ ln QN,M µs )KBT ∂N
]
µg Λ3 ) ln + ln p KBT KBTq(T) where Λ ) h/(2πmKBT)1/2 and q(T) is the internal partition function of one molecule in the gas phase and contains terms of rotational and internal vibrations and electronic degrees of freedom. Combining this two equations, we obtain
[
exp -
]
∂ ln Q ˜ N,M ) K(E,T) p ∂N
(3)
where
K(E,T) )
Λ3 qa(T) exp(E/KBT) KBT q(T)
(27) Cerofolini, G. F.; Re, N. Riv. Nuovo Cimento 1993, 16, 1. (28) Widder, D. V. An Introduction to Transform Theory; Academic Press: New York, 1971. (29) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (30) Hill, T. L. An Introduction to Statistical Thermodynamics; Addison-Wesley Pub. Co.: Reading, MA, 1960.
1920
Langmuir, Vol. 16, No. 4, 2000
Prates Ramalho and Smirnov
Another general property of isotherms that can be easily proved is the function θ monotonicity. Using the chain rule, one obtains
In the thermodynamic limit one can write
Q ˜ N,M ) exp[-βMf(θ)]
∂θ ∂θ ∂µs ) s ∂ ln p ∂µ ∂ ln p
where f(θ) is the free energy per site defined by
f(θ) )
-
lim
Mf∞,N/M)θ
1 ln Q ˜ N,M βM
and θ ) N/M stands for the fraction of occupied sites. Now eq 3 can be written as
[
exp -β
]
It is easy to prove29 in the grand-canonical ensemble that the derivative of ∂θ/∂µs can never be negative. In equilibrium, the chemical potential µs of the adsorbed phase must be equal to the chemical potential of the gas phase. Therefore, considering the gas as ideal, we have
∂f(θ) ) K(E,T) p ∂θ
∂θ ∂θ ) K T ∂ ln p ∂µs B
Setting
[
Because the right side of this equality is positive, the isotherm is a monotonic nondecreasing function of the pressure.
]
∂f(θ) ξ(θ,T) ) exp -β ∂θ
3. Analytical Form of the Local Adsorption Isotherm
we obtain
ξ(θ,T) ) K(E,T) p
(4)
This equality is very general in the sense that it is valid for interacting particles for both short- and long-range lateral interactions. It is also valid for any geometry of the lattice. Considering independent particles, ξ(θ,T) can be easily evaluated, and one obtains the Langmuir isotherm. In the case of lateral interactions, ξ(θ,T) can be estimated approximately by introducing approximations like, for example, the Bragg-Williams31 or the quasi-chemical approximation.32,33 As we have seen before, provided that the function ξ has an inverse, equality (4) also means that the local isotherm θ(p,E) has the form
θ(p,E) ) Θ(K(E) p) where K(E) ) K(E,T). Most of the isotherms considered in the literature, both theoretical and empirical, are of this type. As is well-known, the function θ obtained in some theories can exhibit a loop as a function of p for temperatures lower than a critical temperature, while it is a single-valued function for temperatures above the critical temperature. The appearance of these loops makes the function θ multivalued and is a limitation of the theory. In an exact theory, θ has to be a single-valued function. According to Hill,30 a loop always appears in a theory developed in a canonical ensemble. A theory developed in the grand-canonical ensemble can never lead to a loop.30 Also important is the behavior of the isotherm at p ) 0 and p ) ∞. All isotherms must satisfy the conditions
lim θ ) 0 pf0
and
lim θ ) 1 pf∞
The second condition means that the monolayer coverage is achieved at infinitely high equilibrium pressure. (31) Bragg, W. L.; Williams, E. J. Proc. R. Soc. London A 1934, 145, 699. (32) Bethe, H. A. Proc. R. Soc. London A 1935, 150, 552. (33) Peierls, R. Proc. Cambridge Philos. Soc. 1936, 32, 471.
As we have seen, the local isotherm depends on the product K(E) p:
θ(p,E) ) Θ(K(E) p)
(5)
Introducing the variable K ) K(E) and setting
{
K ∈ [0,K0] 0, N (K) ) KBT N(E(K)), K ∈ (K0, +∞) K
(6)
from eqs 1 and 2 we get
θ(p) )
∫0∞Θ(Kp) N (K) dK
(7)
and
∫0∞N (K) dK ) 1
(8)
respectively. We assume that the function Θ ) Θ(s) satisfies the following conditions: 1. The function Θ is monotonic nondecreasing in the real positive ray: Θ′(s) g 0, whenever s g 0. 2. Θ(0) ) 0 and limsf+∞ Θ(s) ) 1. 3. The function Θ is analytical in the half-plane Re(z) > -sa/2, sa > 0. The function Θ satisfies the first two conditions because of physical reasons1,3,4,29 as we have seen in the previous section. Condition 3 is essential for the approach presented below. Without loss of generality, sa ) 1. (The case sa * 1 can be reduced to this by a change of variables.) Consider the transformation q ) s/(1 + s). It maps the semiplane Re(s) > -1/2 onto the unit disk D ) {q| |q| < 1}. The function Q(q) ) Θ(s(q)) is analytical in D, and we have
Q(q) ) Q1q + Q2q2 + ... + Qnqn + ..., |q| < 1 (9) From this we obtain ∞
Θ(Kp) )
(
Kp
)
Qj ∑ 1 + Kp j)1
j
, Re(Kp) > - 1/2
(10)
From these hypotheses it follows that the global isotherm θ(p) is analytical and monotonic nondecreasing
Structure of a Local Isotherm
Langmuir, Vol. 16, No. 4, 2000 1921
in the real positive ray, and there exist limits limpf0 θ(p) ) 0 and limpf+∞ θ(p) ) 1. For more mathematical details, the reader is referred to ref 34. This new form of the isotherm was derived using only general statistical physical ideas and simple mathematical assumptions. We did not use any particular model of lateral interactions or any approximation of the molecules distribution. Therefore, we think that it can be of interest. Note that setting Q1 ) 1 and Qj ) 0, j > 1, we get the Langmuir local isotherm
Kp 1 + Kp
Θ(Kp) )
The other terms on the expansion will reflect the effects of interactions and all nonidealities. 4. The Langmuir Kernel and the Stieltjes Integral Equation As we have already mentioned in the case Q1 ) 1 and Qj ) 0, j > 1, we get the Langmuir local isotherm
Because limsf∞ Θ(s) ) 1, we have J
Qj ) 1 ∑ j)1 With this polynomial local isotherm, the adsorption integral equation (7) takes the form J
θ(p) )
Qj∫0 ∑ j)1
(
Kp
∞
)
j
1 + Kp
N (K) dK
(14)
Integral equation (14) can be solved by applying the Mellin transform, or after some change of variables, it can be reduced to a convolution form and solved using the Fourier transform (see ref 28). However, all of these general methods work only under rather restrictive assumptions on the functions θ and Θ that are not satisfied in the problems arising in the adsorption theory. Therefore, some special methods are needed. Define the function
L (N )(p) )
Kp Θ(Kp) ) 1 + Kp
∫0∞1 +KpKpN (K) dK
(15)
It satisfies the conditions In this case, eq 7 can be reduced to the Stieltjes integral equation.28 Indeed, make t ) K, ξ ) 1/p, φ(t) ) tN(t), and Φ(ξ) ) θ(1/ξ). Then from eq 7, we have
lim L (N )(p) ) 0
lim L (N )(p) ) 1
and
pf0
pf∞
(16)
Make
Φ(ξ) )
∫0
∞
φ(t) dt t+ξ
(11)
where Φ(ξ), ξ g 1, is a known function and the problem is to find φ(t) g 0, t ∈ [0,∞). Moreover, φ satisfies
∫0
∞
φ(t) dt )1 t
(12)
For the sake of simplicity, we shall consider this problem in the class of continuous functions φ. From eq 11 it follows that Φ(ξ) is analytical in the complex plane cut along the ray L ) {ξ| Re(ξ) e 0, Im(ξ) ) 0} and
φ(x) ) lim yV0
Φ(-x - iy) - Φ(-x + iy) , x>0 2πi
(13)
5. Polynomial Case Now consider the case when Q(q) is a polynomial: J
Qjqj ∑ j)1
In this case the local isotherm has the form J
Θ(Kp) )
(
Kp
Qj ∑ 1 + Kp j)1
)
j
(34) Smirnov, G. V. Preprint of CMA/6/99; University of Porto: Porto, Portugal, 1999; submitted.
(17)
It is easy to see that its derivatives are given by
G(j)(p) )
∂jG(p) j
∫0∞(1 +KKp)
) (-1)jj!
∂p
j+1
N (K) dK
From this and eq 14 we see that the function G is a solution to the Euler equation
θ(p) p
(see ref 28). This solution to eq 11 was first used by Sips3,4 to find the energy distribution function.
Q(q) )
1 G(p) ) L (N )(p) p
)
J - 1(-1)jQ j+1
∑ j)0
j!
pjG(j)(p)
(18)
and satisfies the boundary conditions
lim pG(p) ) 0 pf0
and
lim pG(p) ) 1 pf∞
(see eq 16). As is proved in ref 34, G(p) is a unique solution of eq 18, satisfying the above boundary conditions. The substitution t ) ln p transforms the Euler equation (18) in a linear differential equation with constant coefficients that can be solved by standard methods.35 Solving the differential equation, we find the function equation G(p). (The mathematical details can be found in ref 34.) After that the function N(p) can be obtained from eq 17. From eq 15 we see that L(N)(p) plays a role of the global isotherm for a simpler problem with the Langmuir local isotherm. Thus, the general adsorption integral equation with a polynomial kernel can be reduced to the adsorption integral equation with the Langmuir kernel and solved using the Sips method. (35) Hirshe, M. W.; Smail, S. Differential equations, dynamical systems, and linear algebra; Academic Press: New York, 1974.
1922
Langmuir, Vol. 16, No. 4, 2000
Prates Ramalho and Smirnov
Below we derive an explicit formula for the function L (N )(p). Consider the characteristic polynomial of eq 18
eist, from eqs 16 and 20, we get
blm ) P (R) )
J-1(-1)
Qj+1 R(R - 1)...(R - j + 1) j!
∑ j)0
(-1)k-m+1
J-1 Qj+1 ) 1, and therefore R ) -1 Observe that P (-1) ) ∑j)0 is not a root of the characteristic polynomial. Let Rl, l ) 1,L, be different roots of P(R) with multiplicities nl, l ) 1,L, respectively. Then a general solution to eq 18 is given by34
G(p) )
∑ ∑ (clkIk(Rl,p) + blkSk(Rl,p)) l)1 k)1
(-1)
clk
(
)
R)Rl
∫1p(ln pr)
1 (k - 1)!
k-1
Sk(R,p) ) (ln p)
p Rθ(r) dr r r2
()
L (N )(p) )
[
k-1 R
p
nl
rRl+2
l
pRl+1
∑ ∑ p(clkIk(R,p) + blkSk(Rl,p)) l)1 k)1
nl
[
clk
l∈J+
l
1
∑ P ′(R )(R
l∈J-
-θ(p) +
[
() ∫()
l l Rl+1
p
∫0p r ∞
p
(
)
∫
]
rRl+2
×
] ]
1
∑ P ′(R )(R
dθ(r) -
Rl+1
p
+ 1)
r
l∈J+
×
l + 1) δp (R)
l
∑ resR)R P (R) -
dθ(r) )
l
l∈J-
δ+ p (R)
∑ resR)R P (R)
(19)
k-1 × m-1 l)1 m)1 k)m (k - 1)! θ(r) dr p k-m (-ln r) + blm (ln p)m-1pRl+1 (20) 1 Rl+1 r
∑∑ ∑
θ(r) dr
∞ ∑ P ′(R )∫p
l
l∈J+
where
where the constants blk, k ) 1,nl, l ) 1,L, are determined by condition (16). If Re(Rl) * -1, l ) 1,L, then the constants blk, k ) 1,nl, l ) 1,L, can be easily found. Indeed, observe that L
-
nl
L
L (N )(p) )
dr
pθ(r)
∑ P ′(R )∫0
θ(p) +
and blk, k ) 1,nl, l ) 1,L, are arbitrary constants. Therefore, we have
L (N )(p) )
θ(r) dr , rRl+1 Re(R) < -1 (22)
1
where J- ) {l|Re(Rl) < -1} and J + ) {l|Re(Rl) > -1}. Integrating by parts, we get
, k ) 1,nl, l ) 1,L
Ik(R,p) )
k-1
pRl+1
l∈J-
nl 1 dml-k (R - Rl) (nl - k)! dRnl-k P(R)
θ(r) dr , rRl+1 Re(R) > -1 (21)
∫0 (-ln r)k-m ∑ ( ) m 1 k)m(k - 1)! nl
k-m
L (N )(p) )
where
clk )
blm )
clk
Assume also that all roots of the characteristic polynomial are simple. Then, substituting eqs 21 and 22 for blk in eq 19, we obtain
nl
L
k-1 ∫1∞(-ln r)k-m ∑ ( ) m 1 k)m(k - 1)! nl
j
δp (R) )
1 -θ(p) + R+1
δ+ p (R) )
[
1 θ(p) + R+1
[
R
∫0p(pr)
]
dθ(r)
R
∫p∞(pr)
]
dθ(r)
The last equality can also be written in the following form:
L (N )(p) )
∫Γ
1 2πi
-
δp (R) F
P (R)
dR -
1 2πi
∫Γ
+
δ+ p (R) F
P (R)
dR
where the contours Γ( F are given by
Because the integrals
θ(r) dr , Re(R) > -1 rR+2
∫1∞(ln r)β
ΓF ) {R| |R + 1| ) 1/F, Re(R) e -1} ∪ {R|Re(R) ) -1, Im(R) ∈ [-F, -1/F] ∪ [1/F, F]} ∪ {R| |R + 1| ) F, Re(R) e - 1} and
and
θ(r) dr 1 (ln r)β R+2 , Re(R) < - 1 0 r
∫
Γ+ F ) {R| |R + 1| ) 1/F, Re(R) g -1}| ∪ {R|Re(R) ) -1, Im(R) ∈ [-F, -1/F] ∪ [1/F, F]} ∪ {R| |R + 1| ) F, Re(R) g -1}
Structure of a Local Isotherm
Langmuir, Vol. 16, No. 4, 2000 1923
and F > 0 is big enough. Passing to the limit as F f ∞, we obtain
∫
1 1 L (N )(p) ) + 2 2πi
∫0
1 -∞ γP (-1 + iγ) +∞
+∞
p r
()
ιγ
dθ(r) dγ
where the integral is in the sense of principal value. If the roots of the characteristic polynomial are not simple, the polynomial can be approximated by polynomials with simple roots, and taking the limit, we get the same formula.
The inversion formula can also be written as
L (N )(p) ) 1 1 + 2 2πi
(
(p/r)iγ
∫-∞+∞∫0+∞γ1 P (-1 + iγ) - 1
)
dθ(r) dγ
Here all integrals are usual improper integrals. The rigorous formulation of this result and the proof can be found in ref 34. 7. Conclusions
6. General Local Isotherm As we have seen, any local isotherm satisfying the conditions of section 3 can be written as an infinite series
(
∞
Θ(Kp) )
Kp
)
Qj ∑ 1 + Kp j)1
j
, Re(Kp) > -1/2
(23)
For this more general case, the formula
L (N )(p) )
∫-∞+∞γP(-11 + iγ)∫0+∞(pr)
1 1 + 2 2πi
iγ
dθ(r) dγ (24)
can also be used. In this situation, P(R) is an infinite series
(-1)jQj+1 R(R - 1)...(R - j + 1) j! j)0 ∞
P (R) )
∑
The series of this type are known as the Newton series. As a generalization of the characteristic polynomial, we shall call P a characteristic function. If the characteristic function is well-defined, that is, if the Newton series converges and has no zeros with the imaginary part equal to -1, then G(p) ) L(N)(p)/p, where L(N)(p) is given by eq 24, satisfies the infinite order Euler equation
(-1)jQj+1 ) pjG(j)(p) p j! j)0
θ(p)
∞
∑
We have shown that for the monolayer adsorption on a surface, including arbitrary range lateral interactions, the isotherm can be written as a power series of the Langmuir isotherm. If a polynomial isotherm or a complete series isotherm is used as the kernel for the adsorption integral equation, it can be reduced to an integral equation with the Langmuir kernel and a modified global isotherm. Because the integral adsorption equation with the Langmuir kernel can be solved in an analytical form (see eq 13), the integral equation with the polynomial or complete series kernel can also be solved in an analytical form. The fact that there exists a formula for the solution to the integral equation does not mean that the problem is completely solved. Because the global isotherm is usually known as a discrete set of experimental values in a limited range of pressures, the use of numerical methods is inevitable. The knowledge of the local isotherm structure can be used to develop adequate numerical methods. The proposed techniques combined with the method of complex approximation with constraints (a method developed to solve the adsorption integral equation with the Langmuir kernel36) were tested in a series of numerical experiments. It was confirmed that a finite series with three to five terms is enough to get good approximations of many known kernels, and the use of these approximations allows the reconstruction of the energy distribution. The results of computations based on the methods developed here will appear in a forthcoming paper.37 LA990746E
and the boundary conditions
lim pG(p) ) 0 pf0
and
lim pG(p) ) 1 pf∞
Combining eqs 13 and 24, we find the distribution N.
(36) Bushenkov, V. A.; Ramalho, J. P. P.; Smirnov, G. V. Preprint of CMA/8/99; University of Porto: Porto, Portugal, 1999; to appear in J. Comput. Chem. (37) Bushenkov, V. A.; Ramalho, J. P. P.; Smirnov, G. V., in preparation.
