Statistical rates for reactions on a heterogeneous surface

CHEMICAL

Volume 97, number 1

STATISTICAL

PHYSICS

6 May 1983

LETTERS

RATES FOR REACTIONS ON A HETEROGENEOUS

SURFACE *

SD. PRASAD and L.K. DORAISWAMY National

Chemical

Laboratory.

Pune 411008.

India

Rcccived 9 February 1983; in final form 8 March 1983

Using the random patch model, kinetic expressions are derived for the pressure dependence of fist- and second-order surface reactions. Employing first- and second-order Stieltjes transforms and a Mellin transform. explicit expressions are derived for the rates ds functions of pressure and the parameters characterizing the site enera distribution. It is shown that the rates reflect the nature of the four distributions studied.

1. Introduction

is three-fold: (1) to relate the overall kinetics of a simple reac-

The random patch model of the surface has been extensively used to describe adsorption on heterogeneous surfaces [l-9] _in this model the surface is considered to be constituted of a number of randomly distributed homottatic patches, with adsorption equilibrium prevailing between each of the patches and the gas phase. Assumptions are then made about the nature of the local isotherm (which models adsorption on each of the patches) and the experimen-

tion scheme to the kinetics of the surface processes occurring on each of the patches with the assumption

tal data are fitted to a trial isotherm, which has the correct behavior at low and high pressures. The siteenergy distributions can then be found by the inversion of the adsorption integral equation using the

of adsorption equilibrium and surface reaction control; (2) to develop mathematical techniques to obtain analytical expressions for the rates as functions of pressure, assuming the Langmuir type local isotherm to be valid; and (3) to explain the genesis of some unusual pressure dependencies of many experimental catalytic rate laws. We choose the following catalytic reaction scheme:

---A,, %&

method of Stieltjes transforms [2,3] or Hilbert transforms [4] _ Four site-energy

IIA,-B

distribution

functions

4

(which are

kdb s -B

g’

11=1.2.

solutions of the adsorption integral equation) have been well studied: (I) the negative exponential distribution [2]. (2) the inverse exponential distribution [7], (3) the constant distribution [9], and (4) the positive exponential distribution [6,8] _The experi-

where g. s denote the gaseous and surface species respectively- The adsorption-desorption rate constants k,, k,,kdb are much larger than k,, so that the Grst- or second-order surface reaction would control the kinetics of the overall process. Since k&, is very large. the

mental dependence

surface concentration

of the differential

heat on surface

of B is negligibly

small.

coverage could indicate which of the above four functions would represent the site-energy distribution. We attempt to study the influence of the site-energy distribution on the overall kinetics for the four common distributions just mentioned. The purpose of the paper * NCL Communication

2. Theory We express the overall rate of a second-order reaction as

surface

No. 3113.

0 009-2614/83/0000-0000/603.000

1983 North-Holland

31

Volume 97, number

02 rS = &

1

CHEMICAL

,

6 (Q)

Oii(r-p)

s

dQ -

PHYSICS

(1)

isotherm B,i(T.p) and the function 6 (Q), which is norfirst-order surface reaction. eq. dependence on Bli_ The local as

11 +(bolp)e-Q/RT]-l

tf!:(T.~)=

_

Negative exponential: 6 (Q)= (l/Ri”)[sin(siv)/n]

(2)

6 (Q) = (eQ/R* + ff - 1)-l

-

I

S(Q)=

l/Q= -

h(Q)=O,

=h,/p+

QT
(5b)

esp(Q/Q,,>

Qp =GQ<-.

S
rp = cv(Qp/R7)

Stieltjes

trmsform

Second-order

--

Q_,f = 23 hc.ll nwlr -I ~I_

..

Stieltjes

Q

P

e\p (Q/R 7-j - I;

= Qr=

30 h-al mole -I

trmsform

$ IS’WI

-~r=ekp(Q~/RT)

bo = IO” Torr;

;

(6b)

V. (Y.Qnr are parameters appearing in the distributions (see also table 1). For the distributions represented by

(a - _I*‘)_v-l

x =

_ (4)

(Sal

(a

1:

[(cx - l)/ln (u](I/RT)

OGQ
RT

1

(3)

Positive exponential:

O
~-+a

.

Constant:

s;. [6’(X)]

1 n-1 -RT lna

[eQ/RT - l]-”

Inverse exponential:

In these eyuamns Q denotes the heat of adsorption, Q, . Q2 represent the lower and upper limits of the hr~t of adsorption.p denotes the partial pressure of A in Tom. and b, is the entropy change factor (the standxd state of the adsorbed molecules being that of half ~o\cr~ige). 1-q. (2) tmplies negligible interaction between adsorbed molecules and immobile adsorption (in rhemlsorption. interaction forces add to. or subtrxt from. the adsorption heat in insignificant amounts). The vartdtion of the partition function of the adsorbed mokxules with adsorption heat is ignored in the present N ork. an assumption which has been justified [IO] _ z, IS the mean surface rate constant which is found b> mAinp the assumption that the surface rate constmt hers an Arrhenius type relationship and the distnburion characterizing the activation energy is statistlcJl> independent of 6 (Q). Then 311 average is calcu-

First-order

6 May 1983

lated over the entire surface assuming a uniform distribution of activation energies. Since we are only interested in the pressure dependence of rs, we can consider any arbitrary distribution of activation energies_ zr is thus considered as a temperature-dependent constant_ The four distributions considered may be expressed as [3,5-91:

(21 I.e. III terms of the local site energ>’ distribution malrzed IO unity. For a (1 ) H 111he first power isotherm can be written

LETTERS

n=l

-

- In (a/_~) - y)’

xT

1:

x 109:

_ , --

o - 1 In a

CHEMICAL

Volume 97, number 1

PHYSICS LETTERS

eqs. (3) and (4) the assumption Q, = 0, QZ = 00 is made. For the distributions defined by eqs. (5) and (6), since they are not defined for an infinite limit, finite cut-off heats QTY Qp are assumed_ We employ the substitutions exp(Q/RT) - 1 =x; ho/p + 1 =y [2];6’(~)=6(RTln(x + 1));and the following definitions:

s’

[6’(x)]

= RT f

=RTj

rs = &C(RT[&(Z+,/p)

- [11/(/J +

‘)lxy

‘%&!?,

{F(2, +..I; 1 -p; (2+p)F(2./I

x/sin@r)}

--x2) XT@

+ 2;/.i + 3; -Xi-l)

.

(ll4

(7) ‘s = k,C(RTl~)(b~/p)~

(x +y)’

pressure range, we have

and for a first-order reaction:

6’0 0

bution. For the most common for a second-order reaction:

- dj.t + 1)

0 S$‘(x)]

6 May 1983

@I

- W.P

+ l)lXi(

{F(l.

-P; 1 -P;

-x2)xjl’

“ll’F(l,~+l;il+2;-Xi’)

’

where S$ , S$ denote respectively the first- and secondorder Stieltjes transforms with respect toy of the function S’(x)_ Table 1 contains a list of first- and second-order Stieltjes transforms for the three distributions represented by eqs. (3)-(5). When we assume a first-order surface reaction, the overall rate can be direct!y written in terms of the

_

(1 lb)

3. Results and discussion In fig. 1 the rs calculated for the different distributions are plotted against pressure for a seconderder surface reaction_ The behavior of the first-order reaction is qualitatively the same and hence is not shown separately_ Clearly, the divergence between the predic-

first-order Stieltjes transform (the mean coverage)_ For second-order kinetic behavior on each of the patches, we have, after combining eqs. (7) and (S) with eq. (1): I-S= Kr cs; [6’(x)] - 0 - 1) s; [S’(x)])

- pn/sin (jn7)) _

tions is good enough to discriminate between the different distributions for the parameter values chosen_ (9)

Sincey is related to pressure. the variation of the statistical rafe rs can be very easily predicted by consulting table 1_ Since the first-order Stieltjes transform 5’: is an analytic function ofy for a contour cut along the negative real axis, the second-order Stieltjes transform S: can easily be found by differentiating S: with respect toy in eq. (7) (see table 1): S; [6’(x)] = -dS;

[6’(x)]ldy

_

(10)

That this is actually the case is illustrated in the appendix for the case of the negative exponential distribution defined by eq. (3), where comparison is made with another method. Eq. (10) thus provides a simple method to calculate the second-order Stieltjes transform_ For the positive exponential function defined by eqs. (6a) and (6b) we suggest a Mellin transform method [6,1 I] _The details are given in the appendix_ The solution can be written in a hypergeometric series in pressure and the parameters characterizing the distri-

Fig. 1. Plot of statistical rate for a second-order surface reaction (arbitrary units) rs against partial pressure. QM = 23 kcaljmole, QT= QP = 30 kcal/mole.

33

Volume

97. number

CHEhfICXL

1

PHYSICS

6 hlay 1983

LETI-ERS

Table 2 E~presslons for rs in the limit p - 0 Dismbution 6 ‘(X) 1. [sin(_;v)/?;fx--“/RT

otxc-

2_[1a - I)/ln(n)RT](x+a)3. l&7-

o<.v<_rT

0

sT< x < -

1 c_$

0 -c Q =zQ,

0

First-order surface reaction

Second-order surface reaction

Fr@/bOY

~#P/boYU

EJ(cY- 1)/h a] @/ba)(l

O<_Y<-

z/Z(RT/p +

1)(x?

-

l)(p/bo)

X;C(RT/B -t2,(x?

The inverse exponential distribution predicts the least, and the positive exponential distribution the largest.

values of catalytic fates. II is interesting to analyze the limiting forms of zs when p - 0. For the negative exponential distribution defined by eq. (3). we have for a second-order reaction - 17) .

(13

Asp/b,, IS of the order lo-lo at 0.01 Torr and Y = 0.05. we find power-law behmlior at low pressures.

I-urther, WCgel the sme pressure dependence for both first- and second-order surface reactions_ This is m contrast :o the behavior of the uniform surface wherein we would have obtained Hem-y’s law behavior md second-power dependence on pressure respectnel? _The results and conclusions for all the distribu-

trons arc summarized in table 2. For rhe constmt distribution and the positive exponential distribution. second-order and first-order surface reactions show second- and first-power dependence on pressure respectively. Compared to this. the invrrsc r~ponentral distribution shows first-power dependenz on pressure for a second-order surface reackm. no simple expressmn for rs is possible for a firstorder surtkr

+ In Q)

&CR T/Q,> +dbd

&(RT/&)XT@/bo)

Qp
J-, = Z,@/b”)l’(l

- 4

efficient of the hypergeometric kinetic order terms.

- l)@/bo)*

series by appropriate

4. Conclusions Rigorous methods are proposed to relate the overall kinetics of a catalytic reaction to the kinetics of surface processes for first- and second-order reactions. The kinetics of a first-order reaction can be expressed as a first-order Stieltjes transform whereas that of a second-order reaction can be written as a difference of two terms involving a first- and second-order Stieltjes transform for three distributions (negative exponential, inverse exponential and constant)_ For the positive exponential distribution, expressions can be derived both for first- and second-order reactions on the basis of the Mellin transform. The pressure dependence of the rates reflect the nature of the site energy distribution and as such can be used to confirm the nature of site energy distribution. This can supplement the information obtained from studies of differential heat variation as a function of surface coverage_

reaction.

The above analysis c.m be extended to fractional order lmerics for the first three distributions by readmg from a table of fractional Stieltjes transforms or rrl~tmg them to fractional integrals ( I?]_ Corresponding expressions for the positive exponential distributron can be obtained simply by replacing the first co-

Appendix (1) It has been shown by employing able methods [ 121 that

compiex

vari-

6 May 1983

CHEMICAL PHYSICS LETTERS

Volume 97, number 1

,

Re(3>0_

(Al)

M,=j xfl-W(x)

(A84

dx ,

0 We employ

the substitution

x=eQ/RT-1

;

MP =jrlx$‘F(n,~;~

Then the distribution comes

function

defined by eq. (3) be-

6’(x) = (l/RT)[sin(xv)ls]x-”

_

&2)

We also make use of the identities r(v) r(l

-v)

‘; =

= Ti/sin(Irv) ,

(A4)

S; [S’(x)] = z+‘-l

(A2) with eqs. (Al),

k,(CRT/~)(b~/p)fi

= - clS$ [S’(x)]jdy

;

[F(z

[F(2, -fl;

_

p1 = b. ;

exp(-Q@3

PZ = b.

C= CQ,+~[w(QJQ~~) rs = Fr

exP(-QplR;r3

x2 = (bolp)

-

s”’ eyi(7’,P)

1 -

P:

-x,)x~~

-X*)XiP]

lll-l;

6 @I dQ G

_

(A%

For low pressures. since the argument of the hypergeometric series may be greater than unity, we use the techniques of analytical continuation [ 131:

WI

FG, b, c, ~1

=pJRT;

Qi’

1 -_I.r;

=B1(-z)-uF(a,

1 -c+a;

1 -b+aa;z-I)

+B2(-z)-bF(b,

1 -c+b;

1 -a+b;zml),

;

B

=

1

r(c)

r@- a)

r(b) ryc-~)'

n=1.2,

(A6)

B

=

2

r(c) r(a - b) r(a)r(c - b)

-

(Al 1)

By combining eqs. (A9), (AlO), (A3), (A4) and (Al 1) we get an expression for ‘; in the most useful range, p-,

-

[/J/h + 2)] xi(a+P’)F(2,

-

@+

;

-Q = exp(QJR73;

(AIO)

with

rs = ~,(CRT/P)@~/P)~ x1 = bolp ;

-P;

(A3),

Thus. once the first-order Stieltjes transform is known, the second-order transform can easily be calculated by differentiation. (2) We now propose a Mellin transform method for deriving expressions for the overall rates rs for the positive exponential distribution, eqs. (6a) and (6b). This transform method was originally proposed [6] for deriving adsorption isotherms and we extend it to the analysis of first- and second-order surface reactions. We define (bo/p)e-QIRT=x

C-b)

WI

+ 1) = Yr(V) -

By employing definition (A4). we get (see table 1)

_

In the high-pressure region p > bo, eqs. (ASa) and (A8b) can readily be used. For a second-order reaction, by employing formulae (A7a), (A7b) and (ASa), (ASb) we have

-

I+

+ 1; -cr.xl)

cc--1=-v_

1) p7r/sin &r)]

_

-P; 1 -P;

-x2)x2

&I+ 3; p + 3;

-P

-Xi’)

(Al?)

Similar expressions can be derived for any kinetic order by employing the procedure suggested above, after using the appropriate value of n in eq. (ASb).

Q,

where n denotes the order of the surface reaction_

References

We recall [ 111 that for 6’(x) defined as 6’(x) =(I

+QX)-”

= 0, the Mellin transform

)

OXl, MN is

,

[l] G-D_ Halsey and HXTaylor.

(A74 W’b)

J_ Chem. Phys. 15 (1947) 624. [Z] R. Sips, J. Chem. Phys. 16 (1950) 1024. 13) Xl. Jaroniec. Surface Sci. 50 (1975) 553_ [4] S.D.Prasad and LX. Doraiswamy, Phys. Letters 60A (1977) 11.

35

Volume

97. number

1

CHEMICAL

[5 ] S.D. Prasad and L.K. Doraisuamy, Phys. Letters 94A (1983) 219. [6] LA_ Rudnitsay and A.M. Alexeycv. J. Catal. 37 (1975) ‘32. ]7] D.N. Misra. J. Chem. Phys. 52 (1970) 5499_ (81 hl.1. Trmkin. Kinet. Catal. USSR 5 (1967) 1005. 19) hl.1. Temkin and V. Lrvich. J. Phys. Chem. USSR 20 (1946)

36

1441.

PHYSICS LETTERS

6 May 1983

[lo]

SE. Hoary and J-M. Prausnib. Surface Sci. 6 (1967) 377. [ 111 Tables of inte_gal transforms, Vol. 1 (McGraw-Hill, New York, 1954) pp_ 308,310. (1’1

Tables of integral transforms, Vol. 2 (McGraw-Hill,

New York, 1954) p_ 233. [ 13 ] Higher transcendental functions, Hill, New York, 1953) p_ 108.

Vol. 1 (McGraw-