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CORRESPONDENCE Comment on “Effect of Local Composition Enhancements on the Esterification of Phthalic Anhydride with Methanol in Supercritical Carbon Dioxide” Nobumasa Toh, Hun Yong Shin, Shinichiro Mori, Yoshio Iwai,* and Yasuhiko Arai Department of Chemical Engineering, Faculty of Engineering, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan
Sir: A marked enhancement of the rate constants in the vicinity of critical point of carbon dioxide has been reported for the esterification of phthalic anhydride (1) with methanol (2) in supercritical carbon dioxide (3) by Ellington et al.1 They attributed the enhancement to the effects of local composition. It seems to be interesting to develop a rate-constant equation that can correlate the enhancement. In the present comment, a semiempirical equation is proposed based on transition-state (TS) theory including the local composition effect. According to Simmons and Mason,2 an expression for the rate constant kc (molar concentration basis) can be given as follows based on transition-state theory
kBT ‡φ1φ2 K kcTS ) κ ZRT h a φ‡
Table 1. Pure-Component Critical Parameters1 component
TC (K)
pC (bar)
ω
(1) phthalic anhydride (2) methanol (3) carbon dioxide (4) activated complexa
806.2 512.6 304.2 867.5
47.8 81.0 73.8 35.8
0.611 0.559 0.225 0.878
a
Approximated byproduct (methyl hydrogen phthalate).
(1)
4
where the symbols in the equation denote the transmission coefficient κ, the Bolzmann constant kB, Planck’s constant h, the equilibrium constant Ka‡ (activity basis) between the reactants (1 and 2) and the activated complex (4), the fugacity coefficient φ, the compressibility factor Z, the gas constant R, and the temperature T. When an experimental rate constant kc* at T* and p* is available, the following expression can be derived
kcTS(T,p) )
[
( ) T T*
2
( )
)]
‡ 1 (φ1φ2/φ4 )T,p ZT,p ∆H‡ 1 k *(T*,p*) exp R T T* (φ φ /φ ‡) ZT*,p* c 1 2 4 T*,p* (2)
(
Figure 1. Esterification of phthalic anhydride with methanol in supercritical CO2 at 40 °C versus pressure: (O) experimental points of Ellington et al.,1 (s) eq 3, (- - -) eq 2.
where the ratio of the equilibrium constants Ka‡(T,p)/ Ka‡(T*,p*) is approximated by the van’t Hoff relation. Further, according to Ellington et al.,1 kc(T,p) should be modified as follows by using the local properties when local compositions are much different from bulk compositions as a result of molecular clustering near the critical point
kc(T,p) ) kcTS(T,p)
( )( ) local Flocal y2 F y2
(3)
where F represents the molar density of the mixture and * Author to whom correspondence should be addressed. E-mail:
[email protected]. Fax: +81-92-642-3496.
y2 is the mole fraction of methanol, both of which are evaluated as bulk properties. Ellington et al. observed the local compositions of methanol about phenol blue (representative of phthalic anhydride) in supercritical CO2 by a solvatochromic technique.1 Their data suggest the following empirical expression for the local composition
[ (
)]
y2local p∞ ) exp -RZ 1 y2 p
(4)
where R and p∞ are empirical parameters. The parameter p∞ should be adjusted as y2local f y2 when p f p∞. In the present correlation, p∞ is assumed to be equal to p* for simplicity. Therefore, the experimental kc* datum in the higher-pressure region was adopted. To evaluate thermodynamic properties such as fugacity coefficients, the well-known Peng-Robinson cubic equation of state (PR EOS)3 was used. The physical properties of the pure substances involved in the study, such as the critical temperature TC, the critical pressure pC, and the acentric factor ω, are presented in Table 1.
10.1021/ie010273i CCC: $20.00 © 2001 American Chemical Society Published on Web 06/09/2001
Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4483
Figure 2. Esterification of phthalic anhydride with methanol in supercritical CO2 at 50 °C versus pressure: (O) experimental points of Ellington et al.,1 (s) eq 3, (- - -) eq 2.
The bulk mole fractions of phthalic anhydride (y1) and methanol (y2) used in this calculation were 2.9 × 10-5 and 1.8 × 10-2 (averaged value), respectively, which were obtained based on the experimental conditions of Ellington et al.1 The amount of reaction product (also activated complex) was ignored. This approximation has no significant effect because the amount of product is very small. To apply the PR EOS to mixtures, the conventional mixing rules were adopted with no interaction parameter. The summation of the local fractions should be unity, i.e.
y1local + y2local + y3local ) 1
(5)
The local mole fraction y2local was calculated using eq 4 for given y2 and p, where the compressibility factor Z was evaluated from the PR EOS with bulk mole fractions at T and p. y1local was approximated by y1 because the mole fraction of reactant 1 was quite small. y3local was obtained using eq 5. The local density Flocal was calculated using the PR EOS with the local mole fractions at T and p. In the present estimation, the two unknown parameters R and ∆Hq/R should be determined by using the experimental rate constant data. The rate constant datum kc* at 313.15 K and 14.6 MPa was selected. Then, R and ∆Hq were determined to be 9.5 and 150 kJ mol-1, respectively, by fitting eq 3 to the experimental rate constant data. As shown in Figures 1 and 2, good agreement is obtained. It is noted that the effect of the local mole fraction is important in representing the marked enhancement of the rate constant near the critical point. It would be interesting to study in the future the local mole fraction in the vicinity of critical point by performing a molecular simulation. Literature Cited (1) Ellington, J. B.; Park, K. M.; Brennecke, J. F. Effect of Local Composition Enhancements on the Esterification of Phthalic Anhydride with Methanol in Supercritical Carbon Dioxide. Ind. Eng. Chem. Res. 1994, 33, 965. (2) Simmons, G. M.; Mason, D. M. Pressure Dependency of Gas Phase Reaction Rate Coefficients. Chem. Eng. Sci. 1972, 27, 89. (3) Peng, D.-Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59.
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