The adsorption integral equation: A new method of solution

Volume 60A, number 1

PHYSICS LETTERS

24 January 1977

THE ADSORPTION INTEGRAL EQUATION: A NEW METHOD OF SOLUTION* S.D. PRASAD and L.K. DORAISWAMY National Chemical Laboratory, Poona, India Received 18 October 1976 A classification of the adsorption integral equation in to finite and infinite limit problems is made. A method of solution based on the theory of finite Hilbert transforms is proposed.

The absorption integral equation representing an a priori heterogeneous surface is well known: O~P,T)=fO1~&(Q)dQ,

(1)

Considering integral equation (3) we find that its kernel has an infinite norm, i.e.

jj

dxd~

(4)

Qcharacterising is the heat ofthe adsorption, is the distribution abundance~(Q) of adsorption sites with a given heat of adsorption Q, O(P, T) is the overall surface coverage as a function of temperature and pressure, and ~ is the surface coverage on the ith patch. The surface is assumed to be a cluster of homotattic patches randomly distributed, with heat of adsorption and adsorption potential varying from patch to patch. The local isothetm ~ is of the Langmuir type:

compared to the2 finite limit case b b 01- 0r dx(x+y) dy
1 1+ tb / ~e_Qi/RT , (2) ‘ 0 ~ 1 where b0 is the entropy change factor. Conventionally the limits of the heat of adsorption are zero and infinity. By employing the substitutions

For the finite limit case solution of integral equation (3) is extremely difficult. Often solutions do not

0Ii

—

(I,0/p

1) y, exp (Q/RT) — 1 = ~(RTln(x + 1)). +

—

We transform integral [1] to the form RT

r ~5~(x)dx

=

~

(~,)

O(b / —1 T) = 0

~‘

(3)

g

g(y) is then inverted by means of Stieltjes transform, a method which has been adopted by a number of authors [2,3, 5,7]. Integral eq. (3) possesses a symetric kernel and the theory of such type of equations is known. *

NCL Communication No. 2073.

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a a

where k is a finite number; a, b are finite as the heats of adsorption are finite. Thus a classification into infinite and finite problems is possible based on the norm of the kernel of transformed equation (3).

exist for such an equation (denoted as the Fredholm equation of the first kind). However we give below conditions under which a solution can exist. It has been pointed out in the statistical mechanical studies of Hill [1] and Steele [6] that the assumption of [~oc] as the lower limit is questionable. On the other hand, in the theoretical study of the adsorption integral equation by Ross and Olivier [4] the assumption of a finite negative limit has been made. It would therefore be seen that an acceptable fmite limit problem can be formulated. We will illustrate a method of solving the Fredholm equation of the first kind with finite limits by reducing it to the well known Airfoil equation discussed by Tricomi [8]. We define the fmite Hilbert transform Fy as

f cb(X)dX

=

iy((P(X)),

—l
(6)

—1

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Volume 60A, number 1

PHYSICS LETTERS

We then employ the following substitutions in eq. (1): exp(—Q/RT)—1x,

Q2 > 1ORT, Q1 = —RTIn 2

0.7ORT.

=

Here the assumption Q1 = —RT1n 2 reduces the mathematics of the problem considerably. The upper limit Q2> lORTis physically meaningful because at a temperature around 500 K chemisorption heats are larger than 40 kJ/mole. We now write

r)=Rrf1* 6”(x)dx(y+l)

(7)

(y—x)~’

~

r Ø(y)dy 1*

1

=

0

> ~> ~

—

(11)

.

b0/p—1/~i+1),

together with the limits

~,

24 January 1977

where 6”(x) is given by

Since the homogeneous case is physically insignificant [cf. eq. (1)], as it implies 8 (F, T) = 0 always, we will not consider the solution further. If the function 0(x) belongs to the class L a —~0 [which is true as 6(Q) is normalized], it necessarily satisfies equation (9) which is the most general form of solution. A second form of solution is possible when 6 belongs to the class ~ e—~0. The solution is given by 2f(y)dy + c’ (12) —1 (\2) l_x2\” ~ (x)=— O’—x) (l—x2)1/2 ~

~f

Two additional forms are possible when the summability of the functions f(x)/(1 + x), f(x)/(l x) is obeyed. A function u(x) is summable if inequality —

6”(x) = 6(—RTln(x

+

1)).

With the definition of the Hubert transform given in [6] we get

(13)

holds:

X-~oo

(13).

fu~x)(1_x/x)dx
—8~y,T) =F

f(y)

irRT(y+l)

y(6”(x))

(8)

.

0

O

The integrals in eqs. (6), (7) and (8) are improper at the upper limit and hence are indicated by an asterisk sign. Eq. (8) is the well known Airfoil equation. The general solution of 6”(x) can be written as

(x) can then be written in two alternative forms: ‘1

6(x)

=

—1 1 1/2 1* 1/2 C” —,~-(~__~-) r (l—y) 1+j~ f(y)dy~ 3) —x (1 _x2)112

2,

1*

—1 r (1_Y2)1”2f(Y)dY+ 5 (x)=— I iT 1 —x2 (y —x) (1 _x2)l/ ,,

~‘

~

—1

2

‘

(9)

f

—i/i —x\ 1/2 1* (1+y 6 (x)— ‘~-i—-~---—) —1 \1y/ ,,

where Cis an arbitrary constant given by

(10)

ff(y)dy. —1

Here 6”(x) belongs to the class LP with 1

5 0(x)I~dx


1

k1 being an arbitrary constant.

The second term on the R.H.S. of equation (9) represents thee-÷0: solution of the homogeneous case of the 2e, class L 12

+

C”

Tj5;—i-~ (l_x2)u12’ (15)

C”, C” can

1

c

(14) 1/2 f(y)dy

‘~

be fixed by the normalizing condition on

6 “(x).

Thus the finite limit problem is solved by reducing

the adsorption integral equation to the well known Airfoil equation. We find the solutions are basically of two clases, one of LP with 0

Volume 60A, number 1

PHYSICS LETTERS

References

24 January 1977

[5] R. Sips, J. Chem. Phys. 18 (1950) 1024. [6] W.A. Steele, The solid-gas interface, vol. 1, ed. E.A. Flood

[1] T.L. Hill, J. Chem. Phys. 17 (1949) 762.

[2] M. Jaroniec, Surface Sci. 50 (1975) 553. [3] D.N. Misra, Surface Sci. 18 (1969) 367. [4] S. Ross and J.P. Olivier, On physical adsorption (WileyInterscience, New York, 1964) ch. 4.

(Marcel-Dekker, New York, 1967) ch. 10, p. 352.

[7] J. Toth et al., Acta Chim. Acad. Sci. Hung. 82 (1974) 11. [8) F.G. Tricomi, Quart. J. Math. (Oxford) Ser. (2), 2 (1951) 199.

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