Langmuir 1987, 3, 973-975 interaction plots l/$(h) are of the same order of magnitude. Accordingly, in all these cases the values of the effective decay length of the hydrophobic forces, or, in the other terms, of the attractive structural forces21responsible for the adhesion of particles in the primary minimum, are approximately the same (heffis about 1 nm). Thus the comparison of two sets of independent data on (y2Fh)c from the stability experiments and on l/$ = p l ( 2 r R ) from
973
adhesion force measurements leads to this typical value of the effective width of the primary minimum. This result is in line with the results of Derjaguin and co-workers, Israelachvili, Pashley and co-workers in Australia, Stenius and co-workers in Sweden, and our colleagues in several other research groups obtained by using other combinations of methods.8.21 Registry No. CPB, 140-72-7;silica, 7631-86-9.
Kinetic Analysis of Temperature-Programmed Desorption Curves: Application of the Freeman and Carroll Method J. M. Criado, P. Malet,* and G. Munuera Instituto de Ciencia de Materiales, C.S.I.C., Universidad de Sevilla, Sevilla, Spain Received December 5, 1986. In Final Form: April 13, 1987 The aim of this paper is to test the validity of the Freeman and Carroll method,’ widely used in the kinetic analysis of solid-state reactions under temperature program, when applied to temperature-programmed desorption curves. Conclusions achieved are confirmed by applying the method to simulated and experimental temperature-programmed desorption curves.
Introduction It is well-known that the Freeman and Carroll method has been widely used in the literatweld for determining simultaneously both the activation energy and the “order” of solid-state reactions fulfilling a “ n order” kinetic law. Moreover, it has been shown in a recent paper5 that the Freeman and Carroll equation is fitted by whatever would be the kinetic law obeyed by the reaction. The main advantage of this method is that the experimental data of the T P D profile are transformed into a straight line with slope -E/R, where E is the activation energy of the process, and a n intercept n characteristic of the reaction kinetics, resulting in a useful kinetic analysis method. Taking into account the similarity between the formal expressions of the kinetic equations of both solid-state reactions and desorption of gases from solid surfaces, this paper explores if the above method can be applied to the kinetic analysis of temperature-programmed desorption (TPD) curves and if it is able t o discern between pure desorption kinetics (no readsorption of previously desorbed species) and desorption under an equilibrium between the adsorbed and gas phases along the T P D run (free readsorption conditions).
Theoretical Section Desorption kinetics considered in the literature for the analysis of TPD profiles include first- and second-order desorption from energetically homogeneous surfaces, assuming two limiting cases: no readsorption of the previously desorbed species (desorption without readsorption) or equilibrium conditions between the adsorbed and the gas phases during the TPD run (desorption with (1)Freeman, E.S.;Carrol, B. J. Phys. Chem. 1958, 62, 394. (2)Wendlandt, W.W. Thermal Methods of Analysis; Wiley: New
York. - _._ -1974. - (3) Chen, D. T. Y.; Fong, P. H. I
J. Therm. Anal. 1976, 8, 295. (4)Johnson, D.W.; Gallagher, P. K. J. Phys. Chem. 1972, 76, 1474. (5) Criado, J. M.; Dollimore, D.; Heal, G. R. Thermochim. Acta 1982, 54, 159. 0743-7463/87/2403-0973$01.50/0
free readsorption). The desorption rate can be expressed by the general relationship (-dO/dt) = A exp(-E/RT) f ( 8 ) (1) where A is the Arrhenius preexponential factor, E the activation energy of the desorption process in the nonreadsorption case (or the adsorption enthalpy in the free readsorption limit), 8 the surface coverage, and f ( 8 ) a function depending on the desorption kinetics. f ( 8 ) expressions for different desorption kinetics are collected in Table I. Under nonreadsorption experimental conditions eq 1can be written in the form (-dO/dt) = A exp(-E/Rr)
On
(2)
By differentiating the logarithmic form of eq 2 with respect to In 8, we obtain
d In (-de/dt) d(l/T) = (-E/R)d In 8 d In 8 + n
(3)
A In (-dO/dt) A(1 / r ) = (-E/R)+n A In 8 A In 8
(4)
or
Therefore, the plots of the left-hand side of eq 3 and 4 vs. d(l/T)/d In 8 or A ( l / T ) / A In 8, respectively, should yield a straight line with a slope -E/R and an intercept equal to the desorption order n. This method, equivalent to that proposed by Freeman and Carrol’ in order to analyze nonisothermal solid decomposition traces, could be a useful and easy way ta perform line shape analysis of TPD curves in experimental conditions of nonreadsorption. However, experimental conditions in TPD make feasible the readsorption of previously desorbed species, and it is not evident that the former plots will fit straight lines with a characteristic apparent order n if these free readsorption kinetics apply. If we differentiate the logarithmic form of eq 1 with respect to In 8, we get d In (-de/dt) d(l/T) d le f ( 0 ) (5) = (-E/R)d In 8 dln8 dln8 +
0 1987 American Chemical Society
974 Langmuir, Vol. 3, No. 6, 1987
Criado et al.
Table I. f ( B ) and a(@)Functions for Different DesorDtion Kinetics kinetics f(8) AB) first order without readsorption (1W) 8 -In (8/80) first order with first-order readsorption (1R1) (F/VeVm)(8/(1 - 8))' (V,V,/F)(-ln (8/80) + 6' - 80) 82 118 - l / S O second order without readsorption (2W) second order with second-order readsorption (2R2) (V,Vm/F)(1/8 - 8 + 2 In (8/80) - l/8, (F/V,V,)(82/(1 -
+ 80)
"F,carrier gas flow rate; V,, volume of the solid phase in the catalytic bed; V,, amount of gas adsorbed per unit volume of the solid phase at 8 = 1.
kinetics 1R1 2R2 a
Table 11. Results of the Linear Regression Analysis According to Equation 11 (0.1 Q 0 Q 0.9) d In p(8)ld In 8 d In f(8)ld In 8 ba ab (8 - l)/(-ln ( 8 / 8 0 ) + 8 - 80) 0.738 -0.495 1/(1- 0) 2/(1 - 8) 1.124 -0.661 (2 - 8 - 1/8)/(2 In ( 8 / 8 0 ) - 8 + 80 + l / 8 - 1/80)
rc 1.0000 1.0000
Apparent desorption order. *Apparent activation energy E' = E(l + 1.05~).'Linear regression coefficient.
If the temperature of the sample is increased at a constant rate
P = dT/dt (linear temperature program), eq 1 can be integrated as
where g(8) is a function depending on the desorption kinetics and
I is the integral of the Arrhenius equation (7) g(8) expressions for first- and second-order desorption kinetics
with and without readsorption are collected in Table I. Consider Doyle's approximation for I:6 In I = In ( E / R ) - 5.34 - 1.05(E/RT)
(8)
Table 111. Kinetic Analysis Parameters for Different Desorption Kinetics kinetics apparent order E'/E 1w 1.000 1.000 1R1 0.738 0.480 2w 2.000 1.000 2R2 1.124 0.306 Table IV. Kinetic Analysis of Simulated TPD Curvesa kinetics n E', kcal/mol r E, kcal/mol 1w 1.00 i 0.01 1.0000 24.0 i 0.1 24.0 f 0.1 1R1 0.75 i 0.02 11.02 k 0.08 0.9999 23.4 i 0.4 2w 1.97 i 0.01 0.9999 24.0 i 0.1 24.0 f 0.1 2R2 1.16 i 0.04 7.18 f 0.14 0.9998 23.5 i 0.5 Confidence limits have been calculated within a confidence interval of 95%.
This is an expression that according to the literature7 leads to
E values with an accuracy better than 5% for (EIRT)3 15, when used to determine kinetic parameters of solid-state reactions. Taking logarithims in eq 6, and taking into account eq 8, one gets In g(%) = In
AE - - 5.34 - 1.05(E/RV PR
(9)
which can be transformed into
If the linear relationship d In f ( 0 ) -d In 8
- a-
d In g(0) d In %
+
(11)
could be established from eq 5 , 10, and 11, we could obtain d In (-d8/dt) = -(I d In 8
,A
.
d(l/T) + 1A ,05a)(E/R)-
d In 0
+b
(12)
Comparison of eq 3 and 12 shows that if the linear relationship 11were fulfilled, Freeman and Carroll's plot would yield a straight line with a n apparent desorption order n = b and an apparent activation energy E' = (1 + 1.05a)E.
Table II summarizes the mathematical expreasions of d In f(8)/d In 8 and d In g(8)/d In 8 for first- and second-order desorption kinetics with free readsorption and the a, b, and linear correlation coefficient values obtained from the linear regression analysis according eq 11. In all cases a n initial surface coverage 0, = 1 has been assumed. As shown in Table 11, eq 11 is fulfilled for free readsorption desorption kinetics, and it is possible to conclude from this fact, together with the above considerations, that Freeman and Carroll's plots will yield straight lines with an (6) Doyle, C. D. J. Appl. Polym. Sci. 1962, 6, 639. (7) Criado, J. M.; Ortega, A. Int. J. Chem. Kinet. 1985, 17, 1365.
Figure 1. Simulated TPD curves for different desorption kinetics (labelsas in Table I): E = 24 kcal/mol, A = 108/s-', P = 20 K/min. apparent order b and an apparent activation energy related to the true E value through a characteristic parameter a. In summary, a comparison of eq 3 and 12 allows one to conclude that the plot of d In (-d8/dt)/d In 8 values, as calculated from a T P D curve, vs. d ( l / T ) / d In 8 leads to a straight line whose intercept n = b is characteristic of the kinetic model fitted by the desorption process, and the slope yields the apparent activation energy E'. Once the kinetic model has been established from the value of n, the actual E value can be calculated from the apparent one through the relationship E'/E = 1 1.05a. The values of n and E'/E are summarized in Table 111.
+
Applications In order to test the validity of the above theoretical considerations, TPD profiles corresponding to desorption
Langmuir, Vol. 3, No. 6, 1987 975
Kinetic Analysis of Desorption Curves kinetics lW, 1R1, 2W, and 2R2 have been simulated, by assuming in all cases E = 24 kcal/mol, A = los s-l, 0, = 1,and a linear heating rate /3 = 20 K/min (Figure 1). For these simulations the integral of the Arrhenius equation has been evaluated by using a third-order degree rational approximations with an accuracy better than The results of the kinetic analysis of these TPD curves through eq 4 for 0.1 < 8 < 0.9 are given in Table IV, showing that activation energy and desorption order are obtained with accuracy (error less than 2%) in all cases. Therefore, the introduction of Doyle’s approximation in the deduction of eq 12 is valid, as could be expected since the lower value of E / R T is 15 in all the curves considered, within the range of validity of eq 8’ for kinetic analysis of temperatureprogrammed curves. The suitability of the method for the kinetic analysis of experimental TPD curves has also been tested by using a thermal desorption profile of C 0 2 (Figure 2a) reported by Ying and Madixg for the catalytic decomposition of formic acid on a (110)-oriented copper single crystal. This example was chosen since the C 0 2 T P D peak, obtained under vacuum experimental conditions, has been well characterized in the original paper as a first-order desorption peak from a homogeneous surface, without readsorption of the desorbed gas. Ying and Madixgobtain an activation energy of desorption of 31.9 kcal/mol from the change in maxima position with the heating rate or an E value of 33.3 kcal/mol by applying a line shape kinetic analysis method. It should be noted that, in both analysis techniques, no assumptions of the reaction order were made in order to determine the activation energy of the process. The analysis of this T P D curve through eq 4 (Figure 2b) leads to a n value of 0.97 h 0.05 and an E’value of 35 f 1.5 kcal/mol (within a confidence interval of 95%). The n value corresponds to a first-order desorption process without readsorption (see Table 111),and therefore the activation energy of the process is directly determined as E = E’ = 35 f 1.5 kcal/mol. These results are in accordance, within the experimental error, with the 33.3 kcal/mol value determined by Ying and Madix from the line shape kinetic analysis of their T P D curve and, therefore, demonstrate the validity of the proposed method to determine the true desorption order and activation (8)Senum, G.I.; Yang, R. T.J. Therm. Anal. 1977,11,445. (9) Ying, D.H.S.;Madix, R. J. J. Cotal. 1980, 61, 48.
a
200
100
1
1
I
I
2
3 m
x
300
T/ C
,
[
\
4 0
4
bine
Figure 2. (a) TPD curve of COPdesorption from Cu(llO), ref 9. (b) Kinetic analysis by the Freeman and Carroll method.
energy of the process, since, in spite of the experimental errors in the T and -d0/dt values taken from an experimental T P D curve, the kinetic analysis provides the n value with enough accuracy to distinguish among the theoretically determined n values for the different kinetic models considered. Consequently, this method could be used for line shape analysis of T P D profiles as an alternative to other methods proposed in the literature. It must be pointed out that no well-described desorption processes with freely occurring readsorption from a homogeneous surface have been found from a perusal of the literature. Therefore, it has not been possible, unfortunately, to apply the above method of calculation to analyze this type of desorption reaction in order to obtain additional tests to prove the validity of the above theoretical considerations.
