J. Phys. Chem. 1989, 93, 7185-7190
7185
Frequency Response Method for the Study of Kinetics of a Heterogeneous Catalytic Reactlon of Gases Yusuke Yasuda Faculty of Science, Toyama University, Toyama 930, Japan (Received: January 17, 1989; In Final Form: April 18, 1989) A new frequency response method is proposed on the basis of actual data on C3H6+ H2 3 C3Hs over Pt/A1203 at 273 K observed under each partial pressure of ca. 10 Pa: the gas space of a continuous-flow reactor was varied sinusoidally, and every partial pressure variation induced was followed by a mass spectrometer. Both amplitude and phase difference of AR observed in the angular frequency region from 4 to 60 rad/min were described well by AR(PH.PH)= (R,/PH@)')(nM'H(t) - K o H ( f ) ) , where R, and pH(') denote the overall reaction rate and the partial pressure of H2 at the steady state before the oscillation and APH is the time derivative of the pressure variation, dAPH/dt. The rate "constant" n and K were 0.15 and 7 X lo-' min, respectively. The unordinary rate equation involving PHwas interpreted by R = q,&, in terms of the driving force or the free energy drop, C(d, and the frequency factor, ud, at the rate-limiting step; Aud/ud = nAPH/P$) and AC(d/C(d = -K&/PH('). The newly derived rate constant K seemed to decrease with increasing temperature. The turnover frequency could be given by n / K .
1. Introduction
During the life time of a good catalyst, the reaction turns over a million times, while the free energy of the system irreversibly keeps on decreasing. The essence of catalysis would be in the catalytic cycle. The turnover frequency v defined as R / *N is usually used to characterize the catalytic cycle, where R denotes product molecules in unit time and *N is the number of active sites. However, precise determination of *N is usually difficult; because every exposed surface site is not necessarily active,l *N could depend on the surface coverage of the reactant,2 and further kinetic coupling and coupled cycles are e ~ p e c t e d . ~ In spite of remarkable progress in the surface science in the past decade, new kinds of techniques for the study of kinetics of a heterogeneous catalytic reaction have been limited. Most of the previous methods are based on the data of time response. After the pioneering work by Naphtali and P ~ l i n s k i the , ~ frequency response (FR) method has been developed by the author and co-workers to study mass transfer kinetics of gas/surface s y ~ t e m s , ~ vapor s liquid systems,6 and gas-zeolite systems.' Since a resonance of the system to artificial oscillation is observed in the FR method, a fine difference in the time-response data can clearly be detected and the kinetic constants derived are necessarily independent of the amount of adsorbent. The FR method is first applied in the present work to a heterogeneous catalytic reaction system. It was expected that the rate constants derived would be independent of the amount of catalyst and then characteristic of the catalytic cycle. As shown below, a new kind of rate constant K being unexpected was derived, of which the nature is discussed.
2. Experimental Section 2.1. Materials. The catalyst of powders was prepared by impregnation of AI203 (of which the mean diameter was 60 bm and surface area was 285 m2/g; JRC-ALO-2 supplied by Catalysis (1) Somorjai, G. A. Catalytic Design; Wiley: New York, 1987; p 11. (2) Ertl, G. In Catalysis; Anderson, J. R., Boudart, M., Eds.; SpringerVerlag: Berlin, 1983; Vol. 4, p 209. (3) Boudart, M. Catalyst Design; Wiley: New York, 1987; p 141. (4) Naphtali, L. M.; Polinski, L. M. J . Phys. Chem. 1963, 67, 369. (5) (a) Yasuda, Y. J . Phys. Chem. 1976, 80, 1867. (b) Yasuda, Y. J. Phys. Chem. 1976,80, 1870. (c) Yasuda, Y.; Saeki, M. J. Phys. Chem. 1978, 82, 14. (6) (a) Yasuda, Y.; Sugasawa, G. J. Phys. Chem. 1982, 86, 4786. (b) Sugasawa, G.; Yasuda, Y. Nippon Kagaku Kaishi 1983, 1241. (7) (a) Yasuda, Y. J. Phys. Chem. 1982, 86, 1913. (b) Yasuda, Y.; Sugasawa, G.J . Catal. 1984, 88, 530. (c) Yasuda, Y.; Yamamoto, A. J. Catal. 1985, 93, 176. (d) Yasuda, Y.; Yamada, Y.; Matsuura, I. In Proceedings of the 7th International Zeolire Conference, Tokyo, Japan, 1986; Murakami, Y., Iijima, A,, Ward, J. W., Eds.; Kodansha-Elsevier: Tokyo, 1986; p 587. (e) Yasuda, Y.; Shinbo, S . Bull. Chem. Soc. Jpn. 1988, 61, 745. ( f ) Yasuda, Y.; Matsumoto, K. J. Phys. Chem., in press.
Society of Japan) with an aqueous solution of H2PtC16(Nakarai Chemicals; extra pure reagent). The Pt content was 3.9 wt %. It was dried at 383 K and calcined at 773 K. It was contacted before each run with 250 Torr of H2 (1 Torr = 133.32 Pa) at 623 K for a t least 10 h and evacuated at that temperature for 1 h. Both H2 and Ar (in commercial cylinders; 99.99% pure) were used without any purification. Both C3H6 and C3Hs (in commercial cylinders; 99% pure) were purified by a freeze-thaw cycle in vacuo with cold traps at 195 and 77 K. 2.2. Apparatus. The FR apparatus is schematically shown in Figure la. In a standard run, a ternary mixture of H2 and C3H6involving Ar as a probe gas was prepared in the gas holder G I (ca. 600 Torr) and injected into the reactor containing catalyst C (1.26 g) through the needle valve N,. The gas mixture in the reactor involving the product C3Hs was evacuated through a variable-leak valve L (Granville-Phillips, type 203). A small amount of the mixture was leaked to a quadrupole mass spectrometer M F (Anelva, Model AGA-100) through a needle valve N2. The total pressure was monitored by a diaphragm pressure gauge P (MKS Baratron, type 223). The gas space of the reactor was 0.815 dm3, of which 10.0% was changed sinusoidally with the metal bellows B,; two identical bellows Bl and B2 were connected as shown in Figure 1b in order to compensate for forces on both sides due to atmospheric pressure and make the harmonic oscillation easy. They were driven by a wave generator D, of which a stroke was A8 mm in a standard run. The volume maximum was monitored by an on-off switch V. The flux of the injection was determined by the pressure change in the source G M by means of a mercury manometer. 2.3. Procedure. After a steady state of the reaction was attained, the volume V was varied, with use of complex notation, as
+
V ( t ) = Ve(l
- ve'")
(1) where V, denotes the mean volume; u is the small relative amplitude; and w is the angular frequency. The induced partial pressure variation of m-component P, was expressed well by
where pm denotes the relative amplitude and p(, is the phase lag. The following short notation is introduced. (3) The value at the steady state will be indicated by the subscript S.
The subscript m distinguishes each component: m = H for H2, m = E for C3Hs, m = A for C3H8,and m = Ar for argon. Each partial pressure was determined from the intensity of each parent
0022-365418912093-7 185$01.50/0 0 1989 American Chemical Societv
7186
The Journal of Physical Chemistry, Vol. 93, No. 20, 1989
Yasuda
TABLE I: Experimental Conditions in Each Run expt
no.
V/%
1
10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 14.9 14.9 4.9 4.9 10.0 10.0
Ib 2
2b 3 3b 4
4b 5’ S’b 5”
5“b 6 6b
Jr’l~mol min-I 116
PTb/Torr 0.4 1 0.42 0.72 0.72 0.84 0.78 0.72 0.72 0.40 0.40 0.44 0.38 0.31 0.31
287 79s 217 122 146 86
ak/min-l 12.6 15.* 7.1 14., 12.6 12.6 ll.,
xnc
XE
XAf
0.60 0.61 0.60 0.61 0.41 0.41 0.41 0.40 0.6 1 0.6 1 0.61 0.6 1 0.40 0.45
0.31
0.09 0.09 0.09 0.09 0.05 0.06 0.39 0.40 0.09 0.09 0.09 0.09 0.10 0.09
0.3 1 0.20 0.20 0.30 0.30
0.50
XA
Xnc
0.30 0.30 0.34 0.32
0.21 0.20 0.30 0.30 0.45
‘Total flux of injection. bTotal pressure. CMolefraction of each component in the source.
TABLE 11: Experimental Cunditions and Parameters Concluded expt no. 1 2 3 4 5’ 5” 6 aa I
R,/pmol min-l 18.7, 37.5, 16.67 18.31 17.5, 20.6, 22.7;
a ‘ 2.22 1.78 2.1, 1.7, 2.14 2.12 0.5;
Yb 1.09
0.73 1.4, 0.7;
::::} .
n~
K A / ~ O - min ~
PEW
PE(~)
n~
KE/~O-’min
0.12 0.10 0.02 0.14
0.78 1.20 0.60 1.oo
0.15 0.12 0.02 0.17
6.0 7.5 8.5 7.1
0.10 0.05 0.17 0.08
0.42 0.56 0.3 1 0.36
0.11 0.08 0.15 0.13
3.5 3.5 4.4 2.6
0.24
0.81
0.29
6.2
0.14
0.38
0.18
2.9
0.13
0.91
0.16
8.3
0.11
0.92
0.12
8.4
P A W
PA(-)
I
1.1,
pn(0I P (I) E . b Y -= pA (a) / pE (a),
m r a i Figure 1. Frequency response apparatus: (a) MF, quadrupole mass spectrometer; N l and N2, needle valves; G I and G2, gas holders; P, diaphragm pressure gauge; T, thermocouple; C, catalyst; S,stopcock; L, variable-leak valve; BI, metal bellows; V, on-off switch. (b) D, sinusoidal-wave generator; B, and B2,identical metal bellows. 4,
ion. Contribution of a dauther ion to the other parent ion was negligible under the present conditions in Tables I and 11. I n order to remove apparent changes in p m and (pm due to adsorption-desorption phenomena as well as delay of the partial pressure reading, a blank experiment (indicated by b in Table I) was carried out immediately after each run under almost the same conditions whereby the flux from the gas holder GIwas switched to the flux from Gzwhere a similar ternary mixture, involving C3H8instead of C3H6, had been prepared. 3. Results 3.1. Choice of Temperature. In a preliminary experiment, the angular frequency was kept at 4 X 10 rad/min, of which the period was about 10 s, and the temperature of the catalyst was gradually increased. Changes in the phase lags, pAand ‘ p ~ and , also the amplitudes, pA and pE,are shown against the temperature in parts a and b, respectively, of Figure 2. Steep changes in (pA and pA were observed around 273 K, while ‘ p and ~ pE were almost unaltered except at the lowest temperature. It is worth noting that
0
-LO 0 Lo T/’C Figure 2. Change in partial pressure oscillations in the course of elevating temperature at a fixed angular frequency: (a) phase lag p,,, versus temperature; (b) relative amplitude p,,, versus temperature. Key: 0,the results of product, C3H8;0, those of reactant, C3H6.
the positive ‘ p was ~ in the same direction as that of an adsorption system,5b while the negative cpA was in the opposite direction. On the basis of the results, the reaction temperature was kept at 273 K in the following runs and w was scanned over a range between 4 and 6 X 10 rad/min. The composition of the mixture in the source and other experimental conditions are summarized in Tables I and 11; experiment 1 was the standard run, and the others were carried out in order to confirm conclusions derived from the standard run. 3.2. p m and (pm versus w . Results of pm and (pm are plotted as functions of w in Figure 3, where those of the blank experiment are shown together for comparison. It is worth noting that the sequence of pA > pAo> p~ in the lower w-region seems reasonable, because the reaction was accelerated with the increasing pressure
The Journal of Physical Chemistry, Vol. 93, No. 20, 1989 7187
K‘inetics of a Heterogeneous Catalytic Reaction of Gases a
10
,
I
A
AA1
Substituting eq 1, 2, and 6 into eq 5 and neglecting tarms of higher order than u (- lO-I), we have
s
R , = UA’PA‘’)
\ aE
(7)
and also 5
(UA
+
+ kd)pA*
SA*
=
iWU
+ RsAr*
(8)
where the following short notation is introduced. U,
”
0.5
1
s,*
1.5 log ( w/rad min-’1
R,,
I
(RTO/V,)U,’
I
(RTo/Ve)s,’*
I
(RTo/P,(S)Ve)R,
(9) (10) (1 1 )
The target is r* in eq 8. In the blank experiment with propane, eq 8 becomes (UA
+ SA* + iw)pAO* = iwv
(12)
because R , or R,, is evidently zero and sA* may be regarded as identical with that in eq 8. Consequently, we have r* = iwu@Ae-iAqA- l)/uA ( z r A * )
(13)
where
PA log ( d r a d min-‘ 1 Figure 3. FR data of the standard run (experiment 1): (a) pmversus w ; (b) rp, versus w. Key: 0,the results of C,H,; 0,the results of C,H,;
the results of Ar probe gas; 0 , those results of C3Hs obtained in the blank experiment. A,
PA/PAO
A‘PA E
‘PA0
- ‘PA
(15)
and the relation in eq 7 is used. On the other hand, R may be determined from the consumption rate of propene. After the treatment similar to that of propane, we have r* = iov(1
- PEdA-)/(y~A) (=rE*)
(16)
PE
(17)
where PE/PAO
‘PE - ‘PA0
A‘&
I
y
p,(s)/p,(s)
(18)
and Figure 4. Polar plots of the FR data shown in Figure 3. Semicircles are theoretical ones, of which the parameters are given in Table 11.
I
(19)
under the approximation of
of the reactant (see Table 11), but the sequence reversed in the higher w-region. On the other hand, vA and c p ~were on the
SE* = SA*
opposite side of pA0over the whole w-region. 3.3. Polar Plots of FR Data. The results in Figure 3a,b can be displayed together by the polar plot as shown in Figure 4. The plots seem to be along the semicircles.
Now, only if uAhas been determined in eq 13 and 16 can we find r* values from the actual data in Figure 4. 4.2. Evaluation of uA. The material balance concerned with Ar is simpler, because neither reaction nor adsorption occurs
4. Data Analysis
4.1. Material Balance. The variation of R caused by the pressure change may be expressed in general by R ( t ) = R,(1
+ ,*eiut)
(4)
where R, denotes the reaction rate at the steady state and r is the relative amplitude in complex notation. The material balance of propane molecules, for example, can be described as
wherejArdenotes the rate of injection. After consideration similar to that for eq 5, we have PA1
=
iwV/(uAr
+ iw)
(22)
Comparing the real and imaginary parts on both sides, we find PAr
= wu/(bA?
+ w2)1/2
(23)
and The left-hand side is the change in the gas phase, where RTo is the conversion factor: ASA denotes the change due to adsorption-desorption processes; the last term is the leak rate of the A component from the drain, which was assumed to be proportional caused by APA may be to the partial pressure. The change MA expressed in general bySa h S A ( t ) = -SA’*
ApA(t)
(6)
‘PAr
= tan-’
(‘Ar/w)
(24)
Actual data on pArand pArobtained in the standard run are plotted in Figure 5 against w. The solid curves represent the results calculated from eqs 23 and 24, leading to the parameters of u = 0.10
uAr=
12.6 min-’
(25)
Although the deviation of vArfrom the theoretical curve became
7188
The Journal of Physical Chemistry, Vol. 93, No. 20, 1989
Yasuda
Figure 6. Theoreticallyderived semicircles for (a) C,H8 and (b) C3H6. 90
b
4.4. Theoretical Curves. Combining eq 36 with eq 13, we can derive theoretical formula for P A ( w ) and A(PA(w) with the parameters n and K: (UH
0,5
1
1.5 log lo/radmin-l)
considerable in the higher w-region, the fair agreement of both qArand pArsupports the basic equation of eq 5 as well as eq 21 applied for the present apparatus. The approximation =
bE
=
(26)
bA
is allowed because of almost the same molar masses for each species: 40, 42, and 44, respectively. Now we can derive empirically rA*and rE*,according to the relations of eq 13 and 16. The next problem is what rate equation can derive theoretically the results corresponding to semicircles (see Figure 4). 4.3. Rate E q u a t i o n . It is assumed that R is described by a function of two variables R(t) = R(PH,PH)
- 1) / UA = n A - iWKA
(37)
Since the values of n and K evaluated from pA*and pE*were different, the subscript A (or E) will be added to them. Comparison of the real and imaginary parts on both sides leads to
Figure 5. FR data of Ar probe. Solid curves are theoretical ones.
bAr
+ iw)($Ae-'A'A
(27)
where PHdenotes the time derivative, dPH/dt. The pressure is varied in this method as
(gA(w)=) { O P A sin ACP, - OH( 1 - P A cos A q A ) ) / u A =
nA
(38)
and (hA(w)=)
{ u ~ sin A
+ w(l -PA
Ac~A
COS A ~ A ) ) / u A
=
KAW
(39)
The other variables, pA and 6A, shown in Figure 4 would be profitable to describe r*, because the dashed lines indicate the deviation from the results obtained in the blank experiment, (1 ,O); eq 38 and 39 can be transformed into simpler forms:
= uA{(nAZ + KA2w2)/(uH2
pA(W)
+ w2))1/2
(40)
and
+ KAUH)/(KAd
8A(w) = tan-' {w(nA
-
UHnA))
O-
