CO M M U N ICATION S EFFECT OF PRESSURE ON GAS-PHASE REACTION RATE COEFFICIENTS Decomposition of Hydrogen Iodide Existing data on the effect of pressure on the reaction rate coefficient for HI decomposition is extended to higher pressures b y using transition state theory along with an equation of state. The rate coefficient defined in terms of molar concentration increases monotonically with pressure, displaying enhancement of rate over the simple effect of mass action. The rate coefficient defined in terms of activity goes through a broad maximum and subsequently decreases with increasing pressure. This plateau in the activity-based rate coefficient suggests, as found in ethylene polymerization, that in certain ranges of pressure, the rate coefficient is insensitive to pressure.
u
transition-state theory, expressions for reaction rate coefficients can be developed in terms of ai,the fugacity coefficients of the reactants and transition-state complex, and Z, the compressibility factor of the mixture. Consider the following elementary reaction involving two reactant species: SING
+
vAA vBB M + products (1) M is the transition-state complex, and v A and vB are the stoichiometric coefficients of the reactants, considered to be positive. The reaction rate, r, can be defined arbitrarily in terms of several mass-action expressions, two of which are :
r = kaaAvAaBvB r = kgAVAcavB
(2b)
where ai is the dimensionless activity and c i the molar concentration of species i. The resulting dimensional rate coefficients, k, and k,, are defined by Equations 2a and 2b, respectively. Using the relationship between ai and c I (with ft,the standard-state fugacity for gas-phase species usually being assigned a value of 1 atm.), it follows (Mason, 1965) that: (3) where k,, and k,, are the values of the respective rate coefficients as the total pressure of the system approaches zero. By application of transition-state theory (Eckert and Boudart, 1963), k,/k,, for any elementary reaction can be expressed in terms of the fugacity coefficient of the transition-state complex :
Combining Equation 4a with Equation 3, there results : k,
- =
kc,
(a A Z ) (QBZ) Q’MZ
The fugacity coefficient of the transition-state complex can be calculated either from a knowledge of its molecular structure (Eckert and Boudart, 1963) or from Equation 4b if experimental rate data for k,/k,, as a function of pressure are avail168
l&EC FUNDAMENTALS
able, assuming fugacity coefficients of reactants and the compressibility factor of the mixture are known. Experimental rate data leading to k,/k,, were obtained (Kistiakowsky, 1928) 12) as a function of for the decomposition of H I (2HI + Hz pressure. For this reaction, Equation 4b may be written:
+
(5) From Kistiakowsky’s data, along with a knowledge of @=I and Z,QAW can be calculated from Equation 5. Values of the virial coefficients for the transition-state complex were obtained assuming its structure (Eckert and Boudart, 1963). thus calculated agree closely with the values The values for aAW obtained by application of Equation 5 to the rate data. A relatively simple equation of state (Redlich and Kwong, 1949) will correlate rate data for the decomposition of HI when suitable values of the critical-property parameters of the transition-state complex are assumed. A modified equation of state (Redlich et al., 1965) has been used (Eckert, 1967) to predict solvent effects on the rate of decomposition of HI. The original Redlich-Kwong equation used herein is less complicated than the modified version and gives satisfactory results. From the fact that the mole fraction of the transition-state complex is extremely small, approaching zero, the RedlichKwong equation can be conveniently used to calculate by trial and error its critical temperature, T C Mand , pressure, PcM. Values of aYZ were calculated from Equation 5, based on A choice of T,, = 768’ K. experimental values of k,/k,,. and PcaU = 90 atm. correlated the experimental rate data for the HI decomposition well. These values of critical properties are in good agreement with the values based on the molecular structure discussed by Mills and Eckert (1 968) for the transition state complex. Using these values for the critical properties of the transitionstate complex, the Redlich-Kwong equation can be employed to extrapolate values of a M Z (and hence ka/k,,, Equation 4a, and k,/k,,, Equation 5) to pressures greater than employed by Kistiakowsky. The results of these calculations are shown in Figure 1, for the initial rate of decomposition of pure HI. Since the temperature of the reaction is well above the critical temperature of HI (424’ K.), the reaction takes place entirely
in the gas phase a t all pressures. The effect of other components on the reaction rate has been discussed by Mills and Eckert. k,/k,, increases monotonically with pressure (Figure l), indicating that a n enhancement of the reaction rate, over and above the mass-action effect due to increase in concentration, can be brought about by application of high pressure. O n the other hand, k,/k,, reaches a maximum. Neither k , nor k , is strictly constant over a wide pressure range. The behavior of k , is interesting in the light of the observation (Laird et al., 1956) that k, for the polymerization of ethylene remains fairly constant with pressure over the pressure range investigated (1200 to 2000 atm.). The polymerization of ethylene is represented by Laird et al. by the following sequence of reactions:
+ E 5 X (thermal initiation) x + E 2 x (propagation) X +X products (termination) E
01 0
1
I
400
800
I
1200
I
1600
I
2000
24 0
PRESSURE ( a t m )
Figure 1 . Reaction rate coefficient ratio for HI decomposition a t 300" C.
-- Experimental data (Kistiakowsky, 1 9 2 8 ) - - - Extrapolated data using Equations 4 a and
5, with values of
@iZ
calculated from Redlich-Kwong
(6)
kt +
where E represents the rthylene molecule and X a free radical of any chain length. By using the steady-state approximation for species X, the rate of the reaction defined in terms of activities can be written as:
(7) where the composite rate coefficient, k,, is given by:
Since this k, is composed of a ratio of individual rate coefficients, some compensation of pressure effects may take place. Also, the individual rate coefficients may themselves be sluggish functions of pressure over wide ranges of pressure, as demonstrated by k, for H I decomposition from about 300 to 800 atm. The calculation of @MZfor reactions in general from structural data for the transition-state complex is more direct than the use of pressure dependence of rate coefficients mentioned here. However, unambiguous structural data are almost always lacking; thus the measurement of
rates in a kinetically simple reaction over a wide range of temperature and pressure is desired to test the applicability of a n equation of state such as the Redlich-Kwong equation and the transition-state theory in correlating and interpreting rate data a t high pressures. I n the course of the study the fugacity coefficients of reactants and 2 for the mixture can be determined from P-V-T measurements, leading to direct experimental determination of k,/k,, (Equation 3) and @.+fZ(Equation 4a). literature Cited
Eckert, C. A., 2nd. Eng. Chem. 59, No. 11, 20 (1967). Eckert, C. A , , Boudart, M., Chem. Eng. Sci. 18, 144 (1963). Kistiakowsky, G. B., J . Am. Chem. Soc. 50, 2315 (1928). Laird, R. K., Morrell, A. G., Seed, L., Discussions Faraday SOC., No. 22,126 (1956). Mason, D. M., Chem. Eng. Sci. 20,1143 (1965). Mills, T. R., Eckert, C. A., IND.ENG.CHEM.FUNDAMENTALS 7, 327 (1968). Redlich, O., Ackerman, F. J., Gunn, R. D., Jacobson, M., Lau, S., IND.END.CHEM. FUNDAMENTALS 4, 369 (1965). Rediich, O., Kwong, J. N. S., Chem. Rev. 44,233 (1949). GEORGE M . SIMMONS DAVID M. MASON Stanford University Stanford,Calif. 94305
RECEIVED for review January 5, 1968 ACCEPTED August 2, 1968
T H R E E EXAMPLES OF TRIDIAGONAL MATRICES IN T H E DESCRIPTION OF CASCADES A discussion is presented on tridiagonal matrices arising in calculations of transients in countercurrent systems, transients in stirred tanks with backmixing, and consecutive first-order reversible reactions. For identical consecutive steps, only the matrix for the backmixing case has a simple inverse, which is why this problem could b e solved b y algebraic methods. N A
previous communication (Klinkenberg, 1966) an equa-
I tion in closed form was derived for moments of residence
time distributions for a series of stirred tanks with main forward flow and with backmixing between them. More recently (Klinkenberg, 1968) the analysis of the consequences of the general equation (for arbitrary order of moment and for arbitrary places of injection and observation of tracer) has been better streamlined and carried further. This left the
question as to why the analysis remained so simple that the result could be given in closed form. Accordingly, some recent chemical engineering textbooks, dealing with matrix methods, have been further consulted (Amundson, 1966 ; Lapidus, 1962). For a linear series of states with first-order transitions between the adjacent ones, one meets a "tridiagonal matrix" of the coefficients of the equations. T h e textbooks mention VOL 8
NO. 1
FEBRUARY 1 9 6 9
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