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Thermal desorption from heterogeneous surfaces; normalized curve treatment
Surface Science 137 (1984) 595-606 North-Holland, Amsterdam
595
THERMAL DESORPTION FROM HETEROGENEOUS NORMALIZED CURVE TREATMENT
SURFACES;
Pio FORZATTI, Massimo Enrico TRONCONI
and
BORGHESI,
Italo PASQUON
Dipartimento di Chimica Industriale ed Ingegneria Chimica de1 Politecnico, Piazza Leonardo da Vinci 32, I-20132 Milano, Italy Received
24 May 1983; accepted
for publication
28 October
1983
A new method for analyzing thermal desorption traces from heterogeneous surfaces is presented, based on the use of a single normalized desorption curve. The method provides the coverage dependence of the desorption activation energy and of the frequency factor A. It can also iterate to the correct kinetic order of the desorption reaction. Application of the procedure is demonstrated for a number of different normalized desorption energy profiles in the case of constant A, and by the analysis of published experimental data for a coverage dependent A.
1. Introduction The analysis of thermal desorption curves from a homogeneous surface is well established, and a variety of methods are available in the literature for deriving frequency factor, desorption activation energy and desorption kinetic order from experimental data [1,2]. Relatively little work has been done so far on the analysis of thermal desorption from heterogeneous surfaces. Heterogeneity has often been accounted for in terms of a coverage dependent desorption energy Ed(e) and of a constant frequency factor A. However, sometimes it has been found experimentally [2-41 that A varies in accordance with the desorption energy, so that both A and Ed are coverage dependent, displaying a compensation effect. In a first attempt to deal with the coverage dependence of the desorption rate parameters, Carter considered desorption from both discrete energy heterogeneous sites and a continuum of heterogeneous sites [5]. In his approach theoretical desorption transients are evaluated for desorption energies over the range of interest, which is easily estimated once the frequency factor A and the desorption order n are assumed. These desorption transients are weighted with parameters representing the initial site population distribution and the weighted theoretical transients are compared with the experimentally observed transients so that the parameters can be estimated. In general, the large number of parameters used makes questionable 0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
596
P. Foaatti
et 01. / Thermal
&sorptionfrom heterogeneoussurfaces
the conclusions resulting from this procedure. In fact, possible ambiguities resulting from such a treatment of poorly resolved spectra have been reported ]61. The same criticism seems to apply to another method of analysis of thermal desorption curves from heterogeneous surfaces due to Tokoro et al. 171, which makes use of a single desorption trace. Again, A is assumed to be constant, and the kinetic order must be known. The distribution pattern of Ed( 8) is empirically described by a polynomial form, E,(B)=
t k-0
a,(1
-B)“,
(1)
where the a,‘s are fitting parameters and N is an integer providing the best distribution function for Ed(B). The coefficients ak and the frequency factor A. representing the unknown parameters, are obtained by least-squares methods. Actually, the authors themselves observed ambiguities associated with different sets of values of A and ak’s representing equally well the same desorption curves. The most appealing methods presented so far to obtain the coverage dependence of the desorption rate parameters are due to King [8] and to Taylor and Weinberg [4]_ King’s method provides the coverage dependence of both Ed and A once the deso~tion order is assumed. A family of desorption traces corresponding to different initial coverages is required in order to construct suitable Arrhenius plots whose slopes yield Ed(d). If A is coverage independent, King’s method allows us to calculate the kinetic order too, as illustrated by Falconer and Madix [9] who report results of their analysis for desorption of a number of expe~mental systems. Taylor and Weinberg [4] have questioned the accuracy of this procedure, owing to the weak temperature variation corresponding to different surface coverages. They suggest instead to obtain a family of desorption traces by varying the heating rate j3 rather than the initial coverage 8,. As outlined by the authors, this results in a larger range of surface temperatures, so that the experimental accuracy is improved. However, the heating rate j3 must be changed by a factor of 102. For this reason the procedure seems useful only in the case of Flash Desorption. In fact, in the case of Temperature Programmed Desorption the usual hypothesis of isothermal bed conditions would not hold, as it is very hard to realize large variations of /3 without involving large differences in the temperature distribution inside the catalyst bed. The authors provide detailed results for the application of their approach to expe~mental data of CO desorption from Ir(ll0). In this paper a new method for analyzing thermal desorption curves from heterogeneous surfaces is proposed. The method is based on the use of a single normalized desorption curve, and is derived under the same physical hypotheses which apply for the methods discussed above; namely, a single adsorption state exists, precursor intermediates do not occur, and readsorption can be
591
P. Forzatti et al. / Thermal desorption from heterogeneous surfaces
neglected. a constant In section applied to
In section 2 the theory underlying the method is first developed for preexponential factor A, and some numerical applications are given. 3 the method is extended to the case of coverage dependent A, and the analysis of Taylor and Weinberg’s experimental data.
2. Normalized thermal desorption curves with constant A 2. I. Theory Under the assumptions mentioned above, when diffusional limitations are negligible and differential bed conditions may be assumed, the material balance equation governing the desorption process yields [1,2] C= aB”A exp[ -&(B)/RT],
(2)
where cx= V&,/F, F is the carrier gas flow rate, C is the concentration of the sample in the carrier gas, V, is the volume of the solid phase and u, represents the amount of sample adsorbed per unit volume of the solid phase when the surface coverage is unity. When a linear heating schedule is chosen, the rate equation for the same process is conveniently written as [1,2] dB/dT=
-pm’@“A
exp[ -E,(B)/RT].
(3)
Normalization of eq. (2) with respect curve M yields
to the maximum
point
of the desorption
C’=(B/B,)“exp[c,-E(B)/T’],
(4)
where the following normalized variables have been introduced: T’= T/T,, ~(0) = fd(d)/RTM and cM = E,(B,)/RT,. Likewise, eq. (3) can be transformed into d@/dT’=
(de/dT’),
C’= y&I.
(5)
In the maximum point (T’ = 1, B = 0,) 0 yields the useful relationship eM + [n/e,
C’ = C/C,,
- (de/d8)M]
the necessary
condition
(dC’/dT’),
&, = 0.
=
(6)
Suppose now that a NTD curve, C’(T), is available. The corresponding profile 0( T’) can be immediately derived by graphical or numerical integration. In order to obtain the corresponding distribution of normalized desorption energies, c(e), eq. (4) can be solved to give
E(e)=7+M+nln(e/e,)-lnc~]. For
our purposes,
then,
estimates
(7) of Ed = r(B,)
and
n are necessary.
On
598
P. Forzatti et al. / Thermal desorption from heterogeneous surfaces
introducing one gets
a parabolic
E(e) = CM +(dr/dO)M
approximation
for ~(0) in the neighborhood
(M~,)-w,(B-&,,)~.
Substituting eq. (6) into eq. (8) followed after rearrangement In C’ = Ed [jl-+!$]+wy(~-~M)2
+rr[+[l If a value derived:
-&I
(8) by substituting
~(8) in eq. (7) yields
T
(9)
+ln[&)].
for the desorption
of 8,,
order
is assumed,
the following
linear
form is
where In
77=
cf-t~[h(e/e,)+(i/r)(l-e/e,)] (8-8,)*/T
+(l-l/T')-(e-eM)/~MT (e-e,)'p
.
(11) (12)
The points of the NTD curve close to the maximum can then be used to set up a linear regression according to eq. (10). cM is estimated as the slope of the resulting straight line. The applicability range of the approximation involved in eq. (8) is directly indicated by deviation of the plot 17 versus 5 from linearity. Once cM is available, the r(e) profile can be deduced via eq. [7], based on the assumed desorption order. Subsequently A is calculated from the C values. It is worth mentioning that this approach provides the energy distribution without requiring estimates of the frequency factor. The assumed desorption order is easily verified a posteriori. In fact, from eq. (6) n = eM[(dV’d@M
- %&Y~].
(13)
If n, calculated according to eq. (13) after approximately computing (dr/d@),, differs significantly from the assumed kinetic order, the whole analytical procedure should be iterated on the basis of the new estimate. It ought to be emphasized that physically acceptable values for n are just a few, typically including 0.5, 1 and 2. is most conveniently evaluated from the area The quantity yM = (de/dT’),
P. Forzatti et al. / Thermal desorption from heterogeneous surfaces
underlying temperature
the NTD curve, S,,. In fact, integration range results in
599
of eq. (5) over the whole
(14)
2.2. Numericui
applications
In order to test the analytical procedure for NTD curves reported in the previous section, seven normalized thermal desorption curves were simulated corresponding to the assigned normalized desorption energy profiles r(8) shown in fig. 1. First order desorption kinetics were assumed, and the simulation procedure was as follows. From eqs. (4), (5) and (6):
For each pair of values of the variables @ and T’, the RHS of the equation above depends on the value of the parameter 8, that must be determined to construct the corresponding NTD curve. Numerical integration of eq. (15) provides different profiles B(T’), depending on 8,. The correct value for this parameter is found by imposing 0 = 8, as T'-+0.Once 8, is found, the curves 8(r’) and C’(B, T’) are directly obtained by numerically inte~ating eq. (15) over the whole temperature range. A simple Euler procedure was employed for numerical integration after selecting a correct step length for each case. A very strong sensitivity to the value of 8, was observed while constructing the NTD curves corresponding to C(B) profiles 2 and 4 of fig. 1. The resulting NTD
E
Fig. 1. Distribution
profiles
of normalized
energies.
b
Fig. 2. Normalized
thermal
desorption
curves for distribution
profiles
of fig. 1.
curves are presented in fig. 2, and the values of relevant parameters are given in tabie 1. The analytical procedure outlined above was then applied to the simulated curves, after superposition of a 5% random error. In all cases the plots of q
601
P. Forzatti et al. / Thermal desorption from heterogeneous surfaces Table 1 Parameters
for simulated
Curve
6%
1 2 3 4 5 6 7
0.656 0.631 0.256 0.388 0.279 0.409 0.917
NTD curves and estimates
of cM and P,,
YM
- 2.902 - 5.923 - 5.817 - 11.638 - 11.143 - 7.390 - 27.496
from slopes of 7-t
cr.4
CM
33.44 30.93 39.42 30.00 40.00 35.18 30.00
33.2 30.7 39.4 30.1 40.0 34.8 29.9
plots
versus t according to eq. (10) showed a linear portion, whose slope provided estimates iM in lzxcellent agreement with the correct values cM, as reported in table 1. Correlation indexes better than 0.99 were always obtained. A typical plot is presented in fig. 3 corresponding to curve 5. Fig. 4 shows a comparison
Fig. 3. Plot of n versus 6 for the NTD curve 5 of fig. 2. Fig. 4. Comparison between the theoretical calculated one for NTD curve 5 of fig. 2.
desorption
energy
distribution
profile
and
the
P. Forzatti et al. / Thermal desorption from heterogeneous surfaces
602
between the true ~(6) distribution, for curve 5, and the points of the normalized desorption energy profile resulting from the analysis of the corresponding NTD curve. Contrary to the case of simulation, no particular sensitivity to 8, values was observed, so that estimation of this parameter does not seem to be critical. Application of eq. (13) yielded n values very close to unity for all curves, as expected. For curves 2 to 5, the analysis was repeated assuming desorption orders 0.5 and 2. In all cases eq. (13) provided n values significantly different from the assumed ones, thus confirming that first order kinetics would be the correct assumption. Finally, a normalized thermal desorption curve was simulated corresponding to the normalized desorption energy profile 7 in fig. 1 and desorption order n = 2. Also, in this case application of eq. (13) confirmed that second order desorption kinetics were the correct assumption.
3. Normalized thermal desorption curves with coverage dependent A 3.1. Theory The treatment of this case follows the development already outlined for constant A, after adopting the expression proposed by Engel et al. [lo] for the desorption rate when the compensation effect occurs. Then, C=cuB%,exp[-E,(ti)/R(l/T-l/T,)], where Tb is an adjustable zation yields C’=(8/8,)“exp[e(8)
(16)
parameter
called “isokinetic
= (C&M
and a linear c(e)=cM Combining form 1’ = k,(’
(17)
results in
+ n/&,)/r.
approximation
+(e
- eM)(cM/YM
(18)
of ~(0) around
8,
gives
+ n/0,)/r.
(19)
eqs. (17) and (19) and rearranging + k
Normali-
(1-r-l/T’)+r6M],
where r = 1 - TM/Ts. Stationarity at the peak maximum (dc/dd)M
temperature”.
leads to the following
linear
(20)
23
where
-yM(l - l/T’)/e,] .$ = In c’-~~[h(e/e,) (e - e,)(i - l/~‘)
’
(21) (22)
P. Forzatti et al. /
Thermal
desorption from heterogeneous
The intercept and the slope of the straight related to the desired kinetic parameters by
line resulting
surfaces
603
from eq. (20) are
EM ==YM(k, - n/e,),
(23)
T = k/k,.
(24)
Once cM and r have been determined the energy profile is derived by solving eq. (17) for e(6). A check on the assumed value for n is again possible via eq. (18). Notice that in the limiting case of coverage independent A, c --, cc and T + 1. Correspondingly eq. (20) reduces to eq. (A.2) in the appendix, derived for a constant .A and a linear expression of ~(0). 3.2. Applications
to experimental
thermal desorption data
The experimental data of CO desorption from Ir(l0) published by Taylor and Weinberg [4] were analyzed according to the procedure presented in
.
2 .
4
6
8
t’
Fig. 5. Plot of 7’ versus t’ for curve a. in fig. 4 of ref. [4]; 0, K. Intercept = - 12.6; slope = - 8.0.
= 0.503, yM = - 2.39, n = 1, TM = 528
604
P. Forzatti et al. / Thermal desorption
from heterogeneous
surfaces
\ \ I_
0
0.2
04
Fractional
06
0.8
Coverage
Fig. 6. Ed( 0) and A( 8) profiles obtained from analysis of curves a, b and c shown in fig. 4 of ref. [4]. Full lines: our results. Broken lines: Taylor and Weinberg’s results [4].
section 3.1. A linear heating schedule was assumed in the analysis, with /? values corresponding to the average heating rates reported by the authors. The ~‘-5’ plot (eq. (20)) corresponding to our analysis of curve a in fig. 4 of Taylor and Weinberg’s paper is shown in fig. 5. The estimates tM = 23.8 and i = 0.63 result in the Ed( 0) and A( 0) profiles given in fig. 6, which are in reasonable agreement with Taylor and Weinberg’s own results, also shown in the same figure. The preliminary assumption of first order desorption kinetics was then validated by application of eq. (18) (n found = 1.03). Our profiles were eventually confirmed by the analysis of curves b and c, also taken from fig. 4 of the paper by Taylor and Weinberg.
4. Conclusions
An original treatment of desorption traces from heterogeneous surfaces has been presented which can provide a complete description of the desorption reaction kinetics. It determines the coverage dependence of the desorption energy and of the frequency factor, if any. It can also iterate to the correct
P. Forzatti et al. / Thermal desorption from heterogeneous surfaces
605
order of the desorption reaction. The method of analysis proposed makes use in principle of data points from a single desorption trace, thus utilizing the experimental data information at best. It is also worth mentioning that the present approach circumvents the experimental difficulties of achieving large temperature changes at constant coverage, as required by previous methods [4,8,9]. This method of analysis suffers from three limitations. First, an a priori determined form is supplied to describe the coverage dependence of the frequency factor. However, this form is in agreement with experimental evidence for the occurrence of the compensation effect. Second, the mathematical structure of the n and 6 variables (eqs. (11) and (12) or (21) and (22)) makes the estimates of the parameters sensitive to errors in 8 and T values. In order to improve the accuracy of such estimates, along with that of the A(B) and E,(B) profiles, more than one desorption trace, obtained for instance by replicated runs, could be conveniently considered. In any case, the straightforward two-variable linear regressions technique used allows the identification of unreliable data by direct visual check of the linearity of the 77-t plot. Finally, the linear approximation of c(6) in the neighborhood of the peak maximum might result in higher-order components of C(O), if relevant, being transferred to the coverage dependence of the frequency factor through the estimate of 7. The extension to a parabolic approximation of ~(0) was found impractical because of strong correlations between the fitting parameters. In any case, the linear approximation is expected to provide A(8) and E,(O) profiles sufficiently approximate in most practical cases. The feasibility of the treatment proposed has been demonstrated in the case of constant A by the numerical examples given in section 2.2, and in the case of coverage dependent A by its application to actual, experimentally obtained desorption spectra.
Acknowledgements The authors wish to thank Minister0 Publica Istruzione (Rome) for financial support, and one of the referees for critical suggestions which helped to improve the original manuscript.
Appendix Homogeneous surfaces
As a particular case the analytical treatment proposed in section 2.1 can be simplified to deal with thermal desorption from homogeneous surfaces. Since
606
P. Forzatti et al. / Thermal desorptron from heterogeneous surfaces
e( 0) = cM = c, eq. (10) can be solved for In C’ e= (l-l/T’)-(8,/y,)
Surfaces
1*(8,&J
= -nyM’8M.
with linear energy distribution
If the desorption energy is a linear function of the surface coverage, the approximate method of analysis for a constant A can be made rigorous, and the kinetic order is estimated directly. The linear coverage dependence of E( 13) is expressed by eq. (8) with a,,,, = 0, from which, as already discussed in section 2.1, the following linear form can be easily derived 9” = ~~5’~ + n,
64.2)
where $’ = (ln C’)/[ln(
e/6$,,,) + (I - &&,,,)/~],
E”=
+-
[(I - l/T’)
~M)/(YMTI)]/[in(e/eM)
(A.3) +(l
-e/eM)/T’].
(A.4)
All the points of the NTD curve can be used to construct a +E” plot, whose slope yields eM and whose intercept yields n. Then, the desorption energy distribution is obtained via eq. (7).
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo]
R.J. Cvetanovic and Y. Amenoniya, Catalysis Rev. Sci. Eng. 6 (1972) 21. M. Smutek, S. Cerny and F. Buzek, Advan. Catalysis 24 (1975) 343. H. Pfntir, P. Feuleur, H.A. Engelhardt and D. Menzel, Chem. Phys. Letters 59 (1978) 481 J.L. Taylor and W.H. Weinberg, Surface Sci. 78 (1978) 259. G. Carter, Vacuum 12 (1962) 245. C. Pisani, G. Rabin0 and F. Ricca, Surface Sci. 41 (1974) 277. Y. Tokoro, T. Uchijima and Y. Yoneda, J. Catalysis 56 (1979) 110. D.A. King, Surface Sci. 47 (1975) 384. J.L. Falconer and R.J. Madix, J. Catalysis 48 (1977) 262. T. Engel, H. Niehus and E. Bauer, Surface Sci. 52 (1975) 237.
595
THERMAL DESORPTION FROM HETEROGENEOUS NORMALIZED CURVE TREATMENT
SURFACES;
Pio FORZATTI, Massimo Enrico TRONCONI
and
BORGHESI,
Italo PASQUON
Dipartimento di Chimica Industriale ed Ingegneria Chimica de1 Politecnico, Piazza Leonardo da Vinci 32, I-20132 Milano, Italy Received
24 May 1983; accepted
for publication
28 October
1983
A new method for analyzing thermal desorption traces from heterogeneous surfaces is presented, based on the use of a single normalized desorption curve. The method provides the coverage dependence of the desorption activation energy and of the frequency factor A. It can also iterate to the correct kinetic order of the desorption reaction. Application of the procedure is demonstrated for a number of different normalized desorption energy profiles in the case of constant A, and by the analysis of published experimental data for a coverage dependent A.
1. Introduction The analysis of thermal desorption curves from a homogeneous surface is well established, and a variety of methods are available in the literature for deriving frequency factor, desorption activation energy and desorption kinetic order from experimental data [1,2]. Relatively little work has been done so far on the analysis of thermal desorption from heterogeneous surfaces. Heterogeneity has often been accounted for in terms of a coverage dependent desorption energy Ed(e) and of a constant frequency factor A. However, sometimes it has been found experimentally [2-41 that A varies in accordance with the desorption energy, so that both A and Ed are coverage dependent, displaying a compensation effect. In a first attempt to deal with the coverage dependence of the desorption rate parameters, Carter considered desorption from both discrete energy heterogeneous sites and a continuum of heterogeneous sites [5]. In his approach theoretical desorption transients are evaluated for desorption energies over the range of interest, which is easily estimated once the frequency factor A and the desorption order n are assumed. These desorption transients are weighted with parameters representing the initial site population distribution and the weighted theoretical transients are compared with the experimentally observed transients so that the parameters can be estimated. In general, the large number of parameters used makes questionable 0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
596
P. Foaatti
et 01. / Thermal
&sorptionfrom heterogeneoussurfaces
the conclusions resulting from this procedure. In fact, possible ambiguities resulting from such a treatment of poorly resolved spectra have been reported ]61. The same criticism seems to apply to another method of analysis of thermal desorption curves from heterogeneous surfaces due to Tokoro et al. 171, which makes use of a single desorption trace. Again, A is assumed to be constant, and the kinetic order must be known. The distribution pattern of Ed( 8) is empirically described by a polynomial form, E,(B)=
t k-0
a,(1
-B)“,
(1)
where the a,‘s are fitting parameters and N is an integer providing the best distribution function for Ed(B). The coefficients ak and the frequency factor A. representing the unknown parameters, are obtained by least-squares methods. Actually, the authors themselves observed ambiguities associated with different sets of values of A and ak’s representing equally well the same desorption curves. The most appealing methods presented so far to obtain the coverage dependence of the desorption rate parameters are due to King [8] and to Taylor and Weinberg [4]_ King’s method provides the coverage dependence of both Ed and A once the deso~tion order is assumed. A family of desorption traces corresponding to different initial coverages is required in order to construct suitable Arrhenius plots whose slopes yield Ed(d). If A is coverage independent, King’s method allows us to calculate the kinetic order too, as illustrated by Falconer and Madix [9] who report results of their analysis for desorption of a number of expe~mental systems. Taylor and Weinberg [4] have questioned the accuracy of this procedure, owing to the weak temperature variation corresponding to different surface coverages. They suggest instead to obtain a family of desorption traces by varying the heating rate j3 rather than the initial coverage 8,. As outlined by the authors, this results in a larger range of surface temperatures, so that the experimental accuracy is improved. However, the heating rate j3 must be changed by a factor of 102. For this reason the procedure seems useful only in the case of Flash Desorption. In fact, in the case of Temperature Programmed Desorption the usual hypothesis of isothermal bed conditions would not hold, as it is very hard to realize large variations of /3 without involving large differences in the temperature distribution inside the catalyst bed. The authors provide detailed results for the application of their approach to expe~mental data of CO desorption from Ir(ll0). In this paper a new method for analyzing thermal desorption curves from heterogeneous surfaces is proposed. The method is based on the use of a single normalized desorption curve, and is derived under the same physical hypotheses which apply for the methods discussed above; namely, a single adsorption state exists, precursor intermediates do not occur, and readsorption can be
591
P. Forzatti et al. / Thermal desorption from heterogeneous surfaces
neglected. a constant In section applied to
In section 2 the theory underlying the method is first developed for preexponential factor A, and some numerical applications are given. 3 the method is extended to the case of coverage dependent A, and the analysis of Taylor and Weinberg’s experimental data.
2. Normalized thermal desorption curves with constant A 2. I. Theory Under the assumptions mentioned above, when diffusional limitations are negligible and differential bed conditions may be assumed, the material balance equation governing the desorption process yields [1,2] C= aB”A exp[ -&(B)/RT],
(2)
where cx= V&,/F, F is the carrier gas flow rate, C is the concentration of the sample in the carrier gas, V, is the volume of the solid phase and u, represents the amount of sample adsorbed per unit volume of the solid phase when the surface coverage is unity. When a linear heating schedule is chosen, the rate equation for the same process is conveniently written as [1,2] dB/dT=
-pm’@“A
exp[ -E,(B)/RT].
(3)
Normalization of eq. (2) with respect curve M yields
to the maximum
point
of the desorption
C’=(B/B,)“exp[c,-E(B)/T’],
(4)
where the following normalized variables have been introduced: T’= T/T,, ~(0) = fd(d)/RTM and cM = E,(B,)/RT,. Likewise, eq. (3) can be transformed into d@/dT’=
(de/dT’),
C’= y&I.
(5)
In the maximum point (T’ = 1, B = 0,) 0 yields the useful relationship eM + [n/e,
C’ = C/C,,
- (de/d8)M]
the necessary
condition
(dC’/dT’),
&, = 0.
=
(6)
Suppose now that a NTD curve, C’(T), is available. The corresponding profile 0( T’) can be immediately derived by graphical or numerical integration. In order to obtain the corresponding distribution of normalized desorption energies, c(e), eq. (4) can be solved to give
E(e)=7+M+nln(e/e,)-lnc~]. For
our purposes,
then,
estimates
(7) of Ed = r(B,)
and
n are necessary.
On
598
P. Forzatti et al. / Thermal desorption from heterogeneous surfaces
introducing one gets
a parabolic
E(e) = CM +(dr/dO)M
approximation
for ~(0) in the neighborhood
(M~,)-w,(B-&,,)~.
Substituting eq. (6) into eq. (8) followed after rearrangement In C’ = Ed [jl-+!$]+wy(~-~M)2
+rr[+[l If a value derived:
-&I
(8) by substituting
~(8) in eq. (7) yields
T
(9)
+ln[&)].
for the desorption
of 8,,
order
is assumed,
the following
linear
form is
where In
77=
cf-t~[h(e/e,)+(i/r)(l-e/e,)] (8-8,)*/T
+(l-l/T')-(e-eM)/~MT (e-e,)'p
.
(11) (12)
The points of the NTD curve close to the maximum can then be used to set up a linear regression according to eq. (10). cM is estimated as the slope of the resulting straight line. The applicability range of the approximation involved in eq. (8) is directly indicated by deviation of the plot 17 versus 5 from linearity. Once cM is available, the r(e) profile can be deduced via eq. [7], based on the assumed desorption order. Subsequently A is calculated from the C values. It is worth mentioning that this approach provides the energy distribution without requiring estimates of the frequency factor. The assumed desorption order is easily verified a posteriori. In fact, from eq. (6) n = eM[(dV’d@M
- %&Y~].
(13)
If n, calculated according to eq. (13) after approximately computing (dr/d@),, differs significantly from the assumed kinetic order, the whole analytical procedure should be iterated on the basis of the new estimate. It ought to be emphasized that physically acceptable values for n are just a few, typically including 0.5, 1 and 2. is most conveniently evaluated from the area The quantity yM = (de/dT’),
P. Forzatti et al. / Thermal desorption from heterogeneous surfaces
underlying temperature
the NTD curve, S,,. In fact, integration range results in
599
of eq. (5) over the whole
(14)
2.2. Numericui
applications
In order to test the analytical procedure for NTD curves reported in the previous section, seven normalized thermal desorption curves were simulated corresponding to the assigned normalized desorption energy profiles r(8) shown in fig. 1. First order desorption kinetics were assumed, and the simulation procedure was as follows. From eqs. (4), (5) and (6):
For each pair of values of the variables @ and T’, the RHS of the equation above depends on the value of the parameter 8, that must be determined to construct the corresponding NTD curve. Numerical integration of eq. (15) provides different profiles B(T’), depending on 8,. The correct value for this parameter is found by imposing 0 = 8, as T'-+0.Once 8, is found, the curves 8(r’) and C’(B, T’) are directly obtained by numerically inte~ating eq. (15) over the whole temperature range. A simple Euler procedure was employed for numerical integration after selecting a correct step length for each case. A very strong sensitivity to the value of 8, was observed while constructing the NTD curves corresponding to C(B) profiles 2 and 4 of fig. 1. The resulting NTD
E
Fig. 1. Distribution
profiles
of normalized
energies.
b
Fig. 2. Normalized
thermal
desorption
curves for distribution
profiles
of fig. 1.
curves are presented in fig. 2, and the values of relevant parameters are given in tabie 1. The analytical procedure outlined above was then applied to the simulated curves, after superposition of a 5% random error. In all cases the plots of q
601
P. Forzatti et al. / Thermal desorption from heterogeneous surfaces Table 1 Parameters
for simulated
Curve
6%
1 2 3 4 5 6 7
0.656 0.631 0.256 0.388 0.279 0.409 0.917
NTD curves and estimates
of cM and P,,
YM
- 2.902 - 5.923 - 5.817 - 11.638 - 11.143 - 7.390 - 27.496
from slopes of 7-t
cr.4
CM
33.44 30.93 39.42 30.00 40.00 35.18 30.00
33.2 30.7 39.4 30.1 40.0 34.8 29.9
plots
versus t according to eq. (10) showed a linear portion, whose slope provided estimates iM in lzxcellent agreement with the correct values cM, as reported in table 1. Correlation indexes better than 0.99 were always obtained. A typical plot is presented in fig. 3 corresponding to curve 5. Fig. 4 shows a comparison
Fig. 3. Plot of n versus 6 for the NTD curve 5 of fig. 2. Fig. 4. Comparison between the theoretical calculated one for NTD curve 5 of fig. 2.
desorption
energy
distribution
profile
and
the
P. Forzatti et al. / Thermal desorption from heterogeneous surfaces
602
between the true ~(6) distribution, for curve 5, and the points of the normalized desorption energy profile resulting from the analysis of the corresponding NTD curve. Contrary to the case of simulation, no particular sensitivity to 8, values was observed, so that estimation of this parameter does not seem to be critical. Application of eq. (13) yielded n values very close to unity for all curves, as expected. For curves 2 to 5, the analysis was repeated assuming desorption orders 0.5 and 2. In all cases eq. (13) provided n values significantly different from the assumed ones, thus confirming that first order kinetics would be the correct assumption. Finally, a normalized thermal desorption curve was simulated corresponding to the normalized desorption energy profile 7 in fig. 1 and desorption order n = 2. Also, in this case application of eq. (13) confirmed that second order desorption kinetics were the correct assumption.
3. Normalized thermal desorption curves with coverage dependent A 3.1. Theory The treatment of this case follows the development already outlined for constant A, after adopting the expression proposed by Engel et al. [lo] for the desorption rate when the compensation effect occurs. Then, C=cuB%,exp[-E,(ti)/R(l/T-l/T,)], where Tb is an adjustable zation yields C’=(8/8,)“exp[e(8)
(16)
parameter
called “isokinetic
= (C&M
and a linear c(e)=cM Combining form 1’ = k,(’
(17)
results in
+ n/&,)/r.
approximation
+(e
- eM)(cM/YM
(18)
of ~(0) around
8,
gives
+ n/0,)/r.
(19)
eqs. (17) and (19) and rearranging + k
Normali-
(1-r-l/T’)+r6M],
where r = 1 - TM/Ts. Stationarity at the peak maximum (dc/dd)M
temperature”.
leads to the following
linear
(20)
23
where
-yM(l - l/T’)/e,] .$ = In c’-~~[h(e/e,) (e - e,)(i - l/~‘)
’
(21) (22)
P. Forzatti et al. /
Thermal
desorption from heterogeneous
The intercept and the slope of the straight related to the desired kinetic parameters by
line resulting
surfaces
603
from eq. (20) are
EM ==YM(k, - n/e,),
(23)
T = k/k,.
(24)
Once cM and r have been determined the energy profile is derived by solving eq. (17) for e(6). A check on the assumed value for n is again possible via eq. (18). Notice that in the limiting case of coverage independent A, c --, cc and T + 1. Correspondingly eq. (20) reduces to eq. (A.2) in the appendix, derived for a constant .A and a linear expression of ~(0). 3.2. Applications
to experimental
thermal desorption data
The experimental data of CO desorption from Ir(l0) published by Taylor and Weinberg [4] were analyzed according to the procedure presented in
.
2 .
4
6
8
t’
Fig. 5. Plot of 7’ versus t’ for curve a. in fig. 4 of ref. [4]; 0, K. Intercept = - 12.6; slope = - 8.0.
= 0.503, yM = - 2.39, n = 1, TM = 528
604
P. Forzatti et al. / Thermal desorption
from heterogeneous
surfaces
\ \ I_
0
0.2
04
Fractional
06
0.8
Coverage
Fig. 6. Ed( 0) and A( 8) profiles obtained from analysis of curves a, b and c shown in fig. 4 of ref. [4]. Full lines: our results. Broken lines: Taylor and Weinberg’s results [4].
section 3.1. A linear heating schedule was assumed in the analysis, with /? values corresponding to the average heating rates reported by the authors. The ~‘-5’ plot (eq. (20)) corresponding to our analysis of curve a in fig. 4 of Taylor and Weinberg’s paper is shown in fig. 5. The estimates tM = 23.8 and i = 0.63 result in the Ed( 0) and A( 0) profiles given in fig. 6, which are in reasonable agreement with Taylor and Weinberg’s own results, also shown in the same figure. The preliminary assumption of first order desorption kinetics was then validated by application of eq. (18) (n found = 1.03). Our profiles were eventually confirmed by the analysis of curves b and c, also taken from fig. 4 of the paper by Taylor and Weinberg.
4. Conclusions
An original treatment of desorption traces from heterogeneous surfaces has been presented which can provide a complete description of the desorption reaction kinetics. It determines the coverage dependence of the desorption energy and of the frequency factor, if any. It can also iterate to the correct
P. Forzatti et al. / Thermal desorption from heterogeneous surfaces
605
order of the desorption reaction. The method of analysis proposed makes use in principle of data points from a single desorption trace, thus utilizing the experimental data information at best. It is also worth mentioning that the present approach circumvents the experimental difficulties of achieving large temperature changes at constant coverage, as required by previous methods [4,8,9]. This method of analysis suffers from three limitations. First, an a priori determined form is supplied to describe the coverage dependence of the frequency factor. However, this form is in agreement with experimental evidence for the occurrence of the compensation effect. Second, the mathematical structure of the n and 6 variables (eqs. (11) and (12) or (21) and (22)) makes the estimates of the parameters sensitive to errors in 8 and T values. In order to improve the accuracy of such estimates, along with that of the A(B) and E,(B) profiles, more than one desorption trace, obtained for instance by replicated runs, could be conveniently considered. In any case, the straightforward two-variable linear regressions technique used allows the identification of unreliable data by direct visual check of the linearity of the 77-t plot. Finally, the linear approximation of c(6) in the neighborhood of the peak maximum might result in higher-order components of C(O), if relevant, being transferred to the coverage dependence of the frequency factor through the estimate of 7. The extension to a parabolic approximation of ~(0) was found impractical because of strong correlations between the fitting parameters. In any case, the linear approximation is expected to provide A(8) and E,(O) profiles sufficiently approximate in most practical cases. The feasibility of the treatment proposed has been demonstrated in the case of constant A by the numerical examples given in section 2.2, and in the case of coverage dependent A by its application to actual, experimentally obtained desorption spectra.
Acknowledgements The authors wish to thank Minister0 Publica Istruzione (Rome) for financial support, and one of the referees for critical suggestions which helped to improve the original manuscript.
Appendix Homogeneous surfaces
As a particular case the analytical treatment proposed in section 2.1 can be simplified to deal with thermal desorption from homogeneous surfaces. Since
606
P. Forzatti et al. / Thermal desorptron from heterogeneous surfaces
e( 0) = cM = c, eq. (10) can be solved for In C’ e= (l-l/T’)-(8,/y,)
Surfaces
1*(8,&J
= -nyM’8M.
with linear energy distribution
If the desorption energy is a linear function of the surface coverage, the approximate method of analysis for a constant A can be made rigorous, and the kinetic order is estimated directly. The linear coverage dependence of E( 13) is expressed by eq. (8) with a,,,, = 0, from which, as already discussed in section 2.1, the following linear form can be easily derived 9” = ~~5’~ + n,
64.2)
where $’ = (ln C’)/[ln(
e/6$,,,) + (I - &&,,,)/~],
E”=
+-
[(I - l/T’)
~M)/(YMTI)]/[in(e/eM)
(A.3) +(l
-e/eM)/T’].
(A.4)
All the points of the NTD curve can be used to construct a +E” plot, whose slope yields eM and whose intercept yields n. Then, the desorption energy distribution is obtained via eq. (7).
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo]
R.J. Cvetanovic and Y. Amenoniya, Catalysis Rev. Sci. Eng. 6 (1972) 21. M. Smutek, S. Cerny and F. Buzek, Advan. Catalysis 24 (1975) 343. H. Pfntir, P. Feuleur, H.A. Engelhardt and D. Menzel, Chem. Phys. Letters 59 (1978) 481 J.L. Taylor and W.H. Weinberg, Surface Sci. 78 (1978) 259. G. Carter, Vacuum 12 (1962) 245. C. Pisani, G. Rabin0 and F. Ricca, Surface Sci. 41 (1974) 277. Y. Tokoro, T. Uchijima and Y. Yoneda, J. Catalysis 56 (1979) 110. D.A. King, Surface Sci. 47 (1975) 384. J.L. Falconer and R.J. Madix, J. Catalysis 48 (1977) 262. T. Engel, H. Niehus and E. Bauer, Surface Sci. 52 (1975) 237.