J. Phys. Chem. 1993, 97, 3314-3318
3314
Frequency Response Method for Study of Kinetic Details of a Heterogeneous Catalytic Reaction of Gases. 1. Theoretical Treatment Yusuke Yasuda Faculty of Science, Toyama University, Toyama 930, Japan Received: November 2, 1992
The frequency response (FR) method which has been developed to study kinetic details of a (nonreactive) gas/solid system is extended to a reactive system. A general expression to describe FR spectrum of the irreversible system is derived analytically, in which characteristic functions, {ri*],are introduced. According to the general expression, some simple reactions are discussed: (1) X Y,(2)X Y Z,and (3) X Y Z.In reaction 2, it is demonstrated that (i) the appearance rate of Y component (X Y) and that disappearance rate (Y Z)could be distinguished by means of the FR spectrum, because the two characteristic functions rl(+YX)* and yo(-YY)*describing both rates, respectively, are considerably different and (ii) the FR spectrum concerning the appearance rate of Z component could be divided into two parts: one is described by rl(+ZY)* which corresponds to the direct production of Z from Y in a single step and the other is described by yz(+ZY;+YX)* which corresponds to the indirect production from X via Y in a sequential process. Each characteristic function contains complex rate constants, in,/ - iwl,/], with respect to the transformation of I to m component at an elementary step. It seems of interest that the rate constant is similar in form to impedance of electric current so that "alternating reaction rate" has been introduced by the present method.
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1. Introduction Although catalysis is benefiting from surface science by means of new physical techniques, kinetic study represents one of the most accessible means of probing catalytic reactivity and one which has a direct appeal because the kinetics are the heart of any catalyst's function.' However, use of kinetic data to deduce reaction mechanism has been of limited value, because a reaction model usually involves too many parameters to be determined by traditional methods.* The frequency response (FR) method has been successfully applied to study kinetic details of a gas/solid system. Various characteristic functions to explain FR spectrum for a variety of dynamic systems have been reported as follows (X, Y,and Z will denote molecules in gas phase; A and B are adspecieson external surfaces; C is adspecies within micropores): (1) a,(@)and a s ( w ) available for a X F? A(adsorption4esorption) (2) sac(w), and aa,(w) for a X s A a B (( 1) via precursor A) (3) 6,,(w) and bnr(w), for a X F? C (diffusion in micropores) system$ (4) abn,(o)and abnr(w),for a X i= A e C ((3) via adspecies A) ~ y s t e m . ~ The FR method was extended to a reaction system of Hz+C3H6 C3H8 over Pt/Al,O, catalysts and a new kind of rate equation R(PH(t),kH(t))was found to be valid to explain the FR data, where PHdenotes the partial pressure of hydrogen and PH is the time derivative.6 In this work an analytical expression to describe FR spectrum obtained in a flow system (instead of a batch system previously reported) is derived on the basis of the unusual rate equation R(P,P). Actual proof of the fundamental assumption will be reported in the following paper.
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- --
-
+
The induced partial pressure variation of /-component PI(t ) may be expressed in general by
P l ( t ) = PI(1
+
1
p,P+~l)
(2)
where &denotes the pressure at the steady state;pr is the relative amplitude; 91is the phase difference between the volume and pressure variations. The short notation will be used: p,*
E pFiw/
(3) Dependence of p/ and 91on w would reflect kinetic details of the reactive system. 2.2. Material Balance. Material balance with respect to a source material m leads to
Jm
- u,'Pm(t) - R-m(t) + R+m(t) (4)
The total amounts of m molecule contained in a reactor (in the gas phase and on the external surface) are given in braces; J,,, denotes theconstant fluxof the injection; u,,,'P,(t) is theoverflow rate so that u,,,' would depend not only on the cross section of the outlet but also on the mass of molecule; R-,,,(t) and R+,(t) denote the disappearance and appearance rates, respectively. 2.3. Effects of Adsorption or Diffusion. A reaction is usually associated with adsorption on or diffusion in catalysts. At first, let us consider a case without reaction. Since R-,,, = R+, = 0, eq 4 becomes
2. FR Method in a Flow System 2.1. FR Spectrum. In the FR method the gas space of a continuous-flow reactor V is varied sinusoidally after a steady state is attained (complex notation will be used):
v(r)= P(I - ue'"')
The change in A,,, caused by the pressure variation may be described in general by3s4
(1)
where Pdenotes the mean value of the oscillation; v is the small ( S I O - ' ) relative amplitude; w is the angular frequency. 0022-3654/93/2097-33 14$04.00/0
where K is the constant proportionalto the gradient of adsorption 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 13, 1993 3315
Heterogeneous Catalytic Reaction of Gases. 1
In a case with porous catalysts, eq 5 should be altered to
-iy
0.51 According to similar consideration analogous to the adsorption, we have4
-
Pm*- 1 = bnF(W)*
where the characteristic function for a diffusion process 6"~*is given by
'i
0% 06
-0.5 Figure 1. Characteristic function for adsorption, ( Y F ( w ) * (=x + iy). The number indicates the value of log w; w w / w C where wE = ( u K . A ) ~ / Q: *. (-K/{1 + K + ( U / K ,+,))O),where it is assumed that K = 1 and U / K A = 0.25. The circle asymptotes (0, 0) as w 0 and w -,
-
-
isotherm:
K=-(-) RT, aA,
(7) V 'pm s The subscript s indicates the derivative at the steady state. Substituting V(t), Pm(t),and A,,,(?) in eqs 1, 2, and 6 into eq 5 , we have
P, P
i d , , , *
RTO
(18)
6&)*
= - (u
+
wK(8, + itin,) wK6,) iw(1 + Kb,,)
+
(19)
The characteristic functions in a batch system, a, and asfor adsorption and ,6 and 6, for diffusion, are explicitly given in ref 4. 2.4. Characteristic FMC~~OIIS for Reaction. ( I ) Fundamental Assumption. The term stemming from the adspecies in eq 4 is omitted in this section for simplicity:
The fundamental assumptionin this workis that the rate equation is a function of PI as well as PI, so that we have
- u + K(a,- ias)p, * )eiwf J , - u / P , - u,'P,,,p,*eiwr
(8)
where the term of v X p was neglected because of the small u (Sl O-I). Since eq 8 must be satisfied at any time, comparing the time-independentor time-dependentterms on both sides, we have two equations:
-
J , = um'Pm
(9)
Since the perturbation by means of the volume change is small in this method, we may have after Taylor series expansion to the first order of p
where the short notation is introduced:
To remove apparent changes in the FR data due to apparatus, it would be valid to add a rare gas to the starting material as a reference standard. Since the rare gas (indicated by 0) neither adsorbs nor reacts, eq 10 leads to (u,
+ io)po* = iou
where the following short notation is introduced:
(12) nrm/
It is assumed in this work that molecular weight of the rare gas is almost identical with that of m molecule so that we have u,
+ u,(=u)
(13)
Let us introduce Pm and Acp,, defined by
P,
Av, E ~m - ~ 0 ~ 0P;m Pm/Po Dividing eq 10 by eq 12, we have
E
E
?*,
(14)
- 1 = (YF(w)*
(15) where the characteristic function for an adsorption system a F * is given by (YF(w)* e
-
+ ia,) ( u + wKa,) + iw(1 + Ka,) wK(a,
The function for a choice of the parameters is shown in Figure 1.
*)s
The coefficient n corresponds to the reaction order, but I is a new kind of rate constant discussed previously.6 Both of them are evidently independent of the amounts of catalysts and therefore characteristic of the reaction. It is worth noting that the causal sequence requires the same sign for n and I, because the phase lag between the reaction rate and the pressure variations must be negative. Substituting V(t),P,,,(t), and R,,(t) in eqs 1, 2, and 22 into eq 20, we have
P, P (16)
(
jw-@,*
- u)eiwl
J,
I
- u,'P,(
1 + pm*eiwf)- R-,
R Tn
R+, - R-,,Fk-,,,I*pI*eiwf + R,,~k+,,'pl*eiUt
+ (26)
3316 The Journal of Physical Chemistry, Vol. 97, No. 13, 1993
Yasuda
where the complex rate coefficient &* is introduced:
k,,,*
= nrml- iwlT,,,/
-iy (27)
Comparingtime-independentor time-dependentterms on both sides of eq 26, we may have two equations:
r-, - r+, = (RT,/F,,,V~J, - a,
(28)
Figure 2. Characteristic function y,o(7mm)* (=x + iy) for reaction. The number indicates the value of log w ; w E w / w , ( m ) , where it is assumed + N",) = 0.5, L.n,nl/(I - Lnl) = 0.5, and w,(m) 5 ( u + that N.nlnl/(u Nnl)/( I - Ln,)= IO rad/min. The semicircle asymptotes (0.5, 0) as w 0 and (-0.5, 0) as w =.
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-
series: Kmm*
+ iw Km/* +-E-+ e/* 0
pm* - 1 = -
em*
em*
a
/(#m)
+ iw
em*
4 Zm)
where the short notation is introduced:
K/k*
Km/*
7 -+ ... (38) e/ k(ir/) ek*
It should be emphasized that eq 38 has been derived on the basis of the material balance only, although the fundamentalassumption in eq 21 is essential. Now, let us introduce characteristic functions defined by
yo(rmm)* = For the component without the feed, Le., J , = 0, eq 28 becomes
r+, - r-,
-- bm
pm* - 1 = [-(K-,,* +
K-,,*p,*)
+ N,) + iw(1-
L,)
(39)
y , ( r m l ) *=
(32)
(2) General Expression. To remove apparent changes in the FR data due to apparatus, it would be valid to introduce Pm* defined in eq 14. Dividing eq 30 by eq 12 and considering the approximation in eq 13, we have
N,mm - iwLTmm
(a
a ( a + N,)
y,(rml;*:lk)* =
+ iw + iw(1 - L,)
(u
N T m / - iwL,m/ (40) ( a + N I ) iw(1 - LI)
a + iw + N,) + io(1 - L,)
+
X
+
I(+,)
(K+mm*
K+,/*PI*HI~,* (33)
+
where
&+m)
where the short notation is introduced:
KTml*5 N,,) - iwL,,,
Since eq 33 is available for every component,pI* can be described by
The general expression in eq 38 may be rewritten in terms of the characteristic functions by
C
(y2(-ml;+lk)*- y2(-ml;-lk)*)
+ ...I +
(y2(+ml;+lk)*- y2(+mk-lk)*)+ ...I (43) Substituting PI*in eq 33 by eq 35, we have Kmm*
+---E-+e/*
pm* - 1 = em*
a + iw em*
Km/*
/(+m)
I( # m ) k( #/)
These characteristic functions are shown (for a choice of the parameters) in Figures 2-4. It is worth noting that (i) yo* gives a semicircle, (ii) y I is distinguishable from yo* by the different w-dependence as well as the shape, and (iii) the angle of y2* composed of two steps increases from 0 to 2 r with increasing w.
3. Application to Some Simple Reactions
where
K,,,/* I -K -m/*+ K+m?
(37) Repeating the substitution for PI.*in eq 36, we have the infinite
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3.1 X Y. The reaction rate is presumed to depend on PX and Px,that is, R(Px,&). According to the general expression in eq 43, we have in this case
The Journal of Physical Chemistry, Vol. 97, No. 13, 1993 3317
Heterogeneous Catalytic Reaction of Gases. 1
-iy
-iy 0
Figure 3. Characteristic function yl(+ml)* (=x + iy) for reaction. The number indicates the value of log w ; w w / w c . where it is assumed that u N/)= 0.5, J ~ + ~ ~ / / -( I U/(U + Nnl) 0.5, 1/(1 - LI) 1.5, N + ~ / / (+ L/) = 0.5, and wc(l) ( u + N/)/(1 - 151) = wc(m) = wc = 10 rad/min. The function asymptotes (0.25, 0) as w 0 and (-0.75, 0) as w a.
-
-
0.5--
1
-asp
-
Figure 5. Frequency response curves of X-and Y-components propused in the reaction X Y. The _number indicate! the value of log w; w u / w C . It is assumed that (PY/Px)n.xx = 0.5, (Py/Px)ul..xx = 0.3, ~ + Y X = 1, ul+yx = 0.5, and wc = 2.1 rad/min.
Comparing experimental results on the left-hand sides with the calculated results on the right-hand sides, we could easily determine the three parameters, n, I-XX,and I+Yx;the value of a could be determined from the data of a rare gas on the basis of eq 12. 3.2. X Y Z. Each elementary reaction rate is presumed to be given by
--
R+x = 0
R-, = R - x ( P x , P x )
- 0.5-Figure 4. Characteristic function yz(+_ml;+lk)* ( = x + iy) for reaction. The number indicates the value of log w ; w w / w C . where it is assumed thatu/(u+N,,) =0.5, 1/(1 - L n J = 1 . 5 , N + , , ~ / ( a + N ~ ) = 0 . 5 , L + , , ~ / ( l According to the general expression in eq 43, we may have in this - L,) = 0.5, N + / r / ( u+ N A )= 0.5, L + / A / -( ~Lk) = 0.5. and wc(k) I ((I case + N A ) / ( -~ L A ) = w,(m) = wc(l) = wc = 10 rad/min. The function 0 and (0.375,O) as w a. asymptotes (0.125,O) as w
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-
px* - 1 = -yo(-XX)* py* - 1 = Yl(+YX)*
(44)
Since K+xx* = K*YY* = 0, eq 44 can be explicitly given by pxe'Aa - 1 = -
N-xx - i w L X x ( a + N - x x )+ i 4 1 - L x x )
r+y = a
(46) Considering this and the obvious relation of & = R + y in eq 3 1 ,
we may have (47)
Inserting eqs 46 and 47 into eq 45 and considering the obvious relation of n-xx = n + y x ( = n ) , we have the final expressions: pxe'Aa - 1
a ( P y / P x ) ( n- iwl-xx)
a( 1
(50)
py* - 1 = -yo(-YY)*+ YI(+YX)*
(51)
&* - 1 = yl(+ZY)*+ Y~(+ZY;+YX)*
(52)
The first and second terms in eq 5 1 correspond to the disappearance and appearance rates of Y component, respectively. The two terms in eq 52 give the appearance rate of Z; one is from Y in a single step and the other is from X via Y in a sequential process. Since these characteristic functions are considerably different, they could be separated in the data analysis. Equations 50-52 can be explicitly given as follows:
On the other hand, eq 32 leads to
r-x = ( P y / P x ) a
px* - 1 = -y0(-XX)*
+ ( P y / P x ) n )+ iw(1- (PY/Px)aI-,,)
These semicircles are plotted (for a choice of the parameters) in Figure 5 .
Yasuda
3318 The Journal of Physical Chemistry. Vol. 97,No. 13, 1993
it would not be difficult to determine them by a computer simulation;e.g., if the FR data on the left-hand sides are observed at five different w's, eqs 53-55 lead to 30 (=3 X 2 X 5) simultaneous equations, because each equation leads to two equations concerning the real and imaginary parts on both sides. 3.3. X Y Z. Each elementary reaction rate is presumed to be given by
Pz,N..xx and L-xxmay be neglected so that we have Pxe""
-1 0
-
+
R+x = 0
R + z = R+z(Px,Px$'y,Py)
R-, = 0
(56)
4. Conclusion The most general expression to describe FR spectrum obtained in a flow system can be given in terms of various characteristic functions by
Px* - 1 = -y0(-XX)*- yl(-XY)*
Pm* - 1 = aF*(or
py* - 1 = -yo(-YY)* - YI(--YX)*
r-x = (P,/Px)u
r-y = (pz/py)u ( 5 8 )
In a simpler case where every rate in eq 56 does not depend on Py and PY, eq 57 may be expressed by
fixe''"
-1
=-
+
- [-yo(-mm)*
(57)
Considering eq 32 with respect to Z and also & = ft-y = R+z in eq 3 1, we have
r+, = u
(60)
These equations correspond to those discussed previously in a reaction of Hz + C3H6 C,Hs over Pt/Al,Oj at 273 K;6 n - i w was used in the previous paper instead of n - io1 in this work.
The general expression in eq 43 leads to
pz* - 1 = yl(+ZX)* + yl(+ZY)*
+
t)zeisCz - 1 = u(n+zx - i o f + z x ) / ( u iw)
-
R-x(Px,Px;Py,Py)
N-xx - i w L X x (a + NTXx)+ iw(1 - L x x )
(y2(+ml;+fk)*- y2(+ml;-fk)*]+ ...I ( 6 1 ) I(#m) & ( # I )
The term of yi* gives the effect of a pressure variation of some component to pm* via i's steps. Therefore, the contribution of yi* ( i L 3 ) could usually be neglected. Although thefinite series still contain many parameters, it would be possible to determine them by a computer simulation provided that the FR data onpm* were observed over a wide range of w, as demonstrated by the simple reactions in section 3.
References and Notes
If the reaction was carried out under excess X or Px >> Py
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( I ) Joyner, R. W. J . Chem. Soc., Faraday Trans. 1990.86, 2675. ( 2 ) Froment, G. F.; Hosten, L. H . Catalysis; Anderson, J. R., Boudart, M., Eds.; Springer-Verlag: New York, 1981; Vol. 2, p 97. (3) Yasuda, Y. J . Phys. Chem. 1976, 80, 1867. (4) Yasuda, Y. Bull. Chem. Soc. Jpn. 1991,64,954. (5) Yasuda, Y . J . Phys. Chem. 1982, 86, 1913. (6) Yasuda, Y . J . Phys. Chem. 1989, 93, 7185.
