Comprehensive Experimental and Theoretical Study of Chemical

Department of Physical Chemistry, UniVersity of Rostock, Hermannstrasse 14, 18051 Rostock, Germany. ReceiVed: December 6, 2006; In Final Form: April 1...
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J. Phys. Chem. B 2007, 111, 10975-10984

10975

Comprehensive Experimental and Theoretical Study of Chemical Equilibria in the Reacting System of the tert-Amyl Methyl Ether Synthesis Andreas Heintz,* Simon Kapteina, and Sergey P. Verevkin Department of Physical Chemistry, UniVersity of Rostock, Hermannstrasse 14, 18051 Rostock, Germany ReceiVed: December 6, 2006; In Final Form: April 16, 2007

The chemical equilibrium of the reactive system (methanol + isoamylenes / methyl tert-amyl ether) was studied in the temperature range 298-393 K in the liquid phase using the method of sealed ampoules as well as in the gaseous phase using a tubular flow reactor in the temperature range 355-378 K. In both cases, a cation exchanger Amberlist-15 was used as a heterogeneous catalyst. The reactive system of the methyl tertamyl ether synthesis exhibits a strong nonideal behavior of the mixture compounds in the liquid phase. The knowledge of the activity coefficients is required in order to obtain the thermodynamic equilibrium constants Ka. Two well-established procedures, UNIFAC and COSMO-RS, have been used to assess activity coefficients of the reaction participants in the liquid phase. Thermodynamic equilibrium constants KP measured in the gaseous phase together with the vapor pressures of the pure compounds have been used to obtain Ka in the liquid phase on a consistent experimental basis in order to check the results obtained from the UNIFAC and COSMO-RS methods. Enthalpies of reactions ∆rH° of the methyl tert-amyl ether synthesis reaction in the gaseous and in the liquid phase were obtained from temperature dependences of the corresponding thermodynamic equilibrium constants. Consistency of the experimental data of ∆rH° was verified with help of enthalpies of formation and enthalpies of vaporization of methyl tert-amyl ether, methanol, and methylbutenes, available from the literature. For the sake of comparison, high-level ab initio calculations of the reaction participants have been performed using the Gaussian-03 program package. Absolute electronic energy values, normal frequencies (harmonic approximation), and moments of inertia of the molecules have been obtained using G2(MP2), G3(MP2), and G3 levels. Using these results, calculated equilibrium constants and the enthalpy of reaction of the methyl tert-amyl ether synthesis in the gaseous phase based on the principles of statistical thermodynamics are found to be in acceptable agreement with the data obtained from the thermochemical measurements.

1. Introduction The synthesis of alkyl tert-alkyl ethers using the catalyzed reactions of iso-olefins with alcohols is an efficient process increasingly being used to produce ecologically clean additives for motor fuels. tert-Alkyl ethers, in particular, methyl tert-butyl ether (MTBE) and methyl tert-amyl ether (TAME), are produced on the industrial scale as nontoxic and high-octane gasoline additives. TAME is synthesized in an acid-catalyzed equilibrium reaction of a mixture of isoamylenes (2-methyl-1-butene and 2-methyl-2-butene) with methanol in the liquid phase.1-10 However, this reaction is also accompanied by the mutual isomerization of 2-methyl-1-butene (2MB1) and 2-methyl-2butene (2MB2). From the thermodynamic point of view, the reacting system consisting of four components is sufficiently described by the following two independent reactions:

methanol + 2-methyl-2-butene / TAME

(1)

2-methyl-1-butene / 2-methyl-2-butene

(2)

These reactions have been intensively studied in recent years. The main attention was paid to the investigation of the kinetics and catalysis of TAME synthesis in the liquid phase with emphasis on chemical engineering aspects.6-10 Reaction ac* To whom correspondence [email protected].

should

be

addressed.

E-mail:

cording to eq 1 is thermodynamically limited concerning the product TAME, with an equilibrium conversion declining at higher temperatures. Several thermodynamic studies pertaining to TAME synthesis are available in the literature.1-5 The standard enthalpy of reaction ∆rH° in the liquid phase is usually obtained from the temperature dependence of the equilibrium constant of the reaction (see Table 1). The scattering of the derived values of ∆rH° is about 10 kJ·mol-1 for eq 1 and about 7 kJ·mol-1 for eq 2. Such relatively large scattering arises not only due to a very narrow temperature range (30-40 K) covered by most investigations but, more likely, because enthalpies of reactions have often been derived from the temperature dependence of the equilibrium ratio Kx, i.e. the ratio of mole fractions of the reaction participants TAME, 2-methyl-2-butene, and methanol rather than from the true thermodynamic equilibrium constant Ka. Ratio Kx strongly depends on the composition of the equilibrium mixture, especially when the mole fraction of methanol is low. In most of the experimental work published so far,2,3,5 the initial ratio of reactants methanol/isoamylene was close to unity. Thus, for the calculation of the thermodynamic equilibrium constants, Ka, activity coefficients γi of the reaction participants and their temperature dependence should be taken into account when ∆rH° is calculated from experimental values of Kx. The well-established UNIFAC method15 for calculating activity coefficients γi was applied in previous investigations.1-3,5 However, its parameters were insufficiently adjusted to the

10.1021/jp068388t CCC: $37.00 © 2007 American Chemical Society Published on Web 08/28/2007

10976 J. Phys. Chem. B, Vol. 111, No. 37, 2007

Heintz et al.

TABLE 1: Data Available from the Literature for the Reaction Enthalpy ∆rH °m of the TAME Synthesis Reaction (Equation 1) and Isomerization of Methyl-Butenes (Equation 2) in the Liquid Phasea ∆rH° (eq 1) (kJ·mol-1)

literature

(K)

method

Rihko1 Piccoli2 Serda3 Sola4 Syed5 this work

323-363 323-353 313-353 343 298-348 298.2-393.2

chem equilib chem equilib chem equilib calorimetry chem equilib chem equilib

a

∆rH° (eq 2) (kJ·mol-1)

-26.8 ( 2.3 -6.7 ( 3.0 -28.5 ( 2.5 -8.1 ( 0.5 -32.8 ( 1.4 -3.0 ( 0.3 -27.1 ( 2.5 -22.3 ( 3.3 -11.5 ( 1.9 -28.4 ( 1.5 -7.1 ( 0.3

The data is taken for comparison in Figures 3 and 6.

mixtures containing methanol, olefins, and tertiary alkyl ethers. Therefore, we decided to investigate the TAME synthesis reaction not only in the liquid phase but also in the gaseous phase. Chemical equilibrium study in the gaseous phase is much less affected by intermolecular interactions compared to the liquid phase, and corrections can be made for nonideality with high reliability in order to obtain the thermodynamic equilibrium constant KP in the gaseous phase. Several years ago, one of us already published18 the tentative data on the TAME synthesis reaction study in the gaseous phase at the three temperatures 373, 399, and 413 K in the presence of aluminosilicate catalyst. In this work, we have extended our study of reaction eqs 1 and 2 in the gaseous phase using a glass flow reactor and the cation exchanger Amberlist-15 as heterogeneous catalyst. Equilibrium constants for the TAME synthesis reaction in the gaseous phase KP,gas and in the liquid phase Ka,liquid depend on each other in the following way.

Ka,liquidΠ(fi0(T,P))νi ) KP,gas i

As a consequence Ka,liquid can be obtained from measured values of KP,gas and known values of the fugacities fi(T,Pi0) of the pure components according to eqs 3 and 7. KP,gas is defined as

(3)

KP,gas )

ln φi )

∑yiBii - ∑∑yiyjBij)P

(2

RT

ai ) xiγi is the activity of component i in the liquid phase, γi is the activity coefficient, and xi is the mole fraction in the liquid phase. KP,gas is the thermodynamic equilibrium constant in the ideal gas phase, νi are the general stoichiometric coefficients of the reactants, and fi0(T,P) is the fugacity of the pure compound i (index 0) at temperature T and pressure P of the reacting mixture. For the TAME synthesis reaction, the following holds:

Π(fi0(T,P)) ) νi

i

fTAME,0(T,P) fMeOH,0(T,P)‚f2MB2,0(T,P)

f2MB2,0(T,P) ) K′P,gas K′a,liquid f2MB1,0(T,P)

∫pp (Vi0/RT) dP]

(5)

(6)

i0

The exponential term is called the Poynting correction factor with Vi0 being the liquid molar volume of compound i. If the pressure difference is small (a few bar), the Poynting correction factor is close to unity and can be neglected. This is the case for the TAME synthesis reaction which means that in eqs 3 and 5 it can be written:

fi0(T,P) = fi0(T,Pi0)

(10)

with

K′a,liquid )

x2MB2 γ2MB2 x2MB2 ≈ x2MB1 γ2MB1 x2MB1

(11)

and

y2MB2 φ2MB2 y2MB1 φ2MB1

(12)

The activity coefficients γ2MB1 and γ2MB2 are certainly different from unity in the complex mixture, but due to the high similarity of the two isomers, it can be assumed that γ2MB1 ≈ γ2MB2 in the real liquid mixture (K γ′ = 1). The same holds for the fugacity coefficients, i.e., φ2MB2 = φ2MB1. Standard reaction enthalpies are obtained from the temperature dependence of the equilibrium constants

(

)

(13)

(

)

(14)

∆rH°id.gas ) RT2

∂ ln KP ∂T

P

and

The term fi0(T,P) is related to the fugacity fi0(T,Pi0) of the pure compound i at its saturation pressure Pi0 by the following expression:

fi0(T,P) ) fi0(T,Pi0) exp[

(9)

where Bii are the second virial coefficients of the pure components and Bij (i * j) are for the cross virial coefficients. Equation 2 describes the isomerization reaction of methylbutenes occurring simultaneously with the TAME synthesis reaction eq 1. For this reaction, it holds in analogy to eq 1:

' ) KP,gas

aTAME xTAME γTAME ) ) Kx‚Kγ aMeOH‚a2MB2 xMeOH‚x2MB2 γMeOH‚γ2MB2 (4)

(8)

where yTAME, yMeOH, and y2MB2 are the mole fractions in the gaseous phase of the reacting mixture in the thermodynamic equilibrium, Ptot is the total pressure, and P0 ) 101325 Pa. The terms φi (i ) TAME, MeOH, 2MB2) are the corresponding fugacity coefficients in the gaseous phase given by

where

Ka,liquid )

yTAME φTAME Ptot yMeOH‚y2MB2 φMeOH‚φ2MB2 P0

(7)

∆rH°liquid ) RT2

∂ ln Ka ∂T

P

where the standard reaction enthalpy in the ideal gaseous phase ∆rH°id.gas is related to the standard reaction enthalpy ∆rH°liquid by

∆rH°liquid ) ∆rH°id.gas -

∑i νi∆gl Hmi

(15)

where ∆gl Hmi is the molar enthalpy of vaporization of the pure compound i at the reference temperature. As a consequence, from eq 4 to 13, KP values measured in the gaseous phase could be used for calculation of the true thermodynamic constant Ka and K′a, respectively, in the liquid phase, provided that the saturated vapor pressures of the pure reaction participants are known.

Equilibria of tert-Amyl Methyl Ether Synthesis

J. Phys. Chem. B, Vol. 111, No. 37, 2007 10977

TABLE 2: Results of Chemical Equilibrium in the Liquid Phase of the TAME Synthesis Reaction (Equation 1) and Isomerization of Methyl-Butenes (Equation 2)a n

xMeOH

x2MB1

x2MB2

xTAME

γMeOHb

γ2MB1b

γ2MB2b

γTAMEb

Kγ(eq 1)b

Kx(eq 1)

Ka (eq 1)c

Kx (eq 2) ≈ Ka (eq 2)

8 298.2 6 7 8 10 12 313.2 8 7 9 323.2 9 8 7 333.2 6 9 9 353.2 6 10 373.2 10 10 393.2 10 10

0.017 0.157 0.561 0.141 0.202 0.314 0.485 0.508 0.737 0.519 0.804 0.055 0.305 0.565 0.232 0.570 0.755 0.311 0.599 0.310 0.591

0.033 0.006 0.002 0.011 0.010 0.007 0.005 0.005 0.002 0.007 0.002 0.039 0.016 0.008 0.031 0.014 0.006 0.039 0.020 0.054 0.029

0.432 0.083 0.029 0.129 0.110 0.082 0.060 0.060 0.024 0.079 0.021 0.382 0.160 0.082 0.264 0.118 0.054 0.303 0.152 0.367 0.198

0.518 0.754 0.408 0.719 0.678 0.597 0.450 0.427 0.237 0.395 0.173 0.524 0.519 0.345 0.473 0.298 0.185 0.347 0.229 0.269 0.182

6.40 (7.17) 2.66 (2.97) 1.30 (1.42) 2.82 (3.19) 2.42 (2.71) 1.90 (2.12) 1.45 (1.58) 1.41 (1.53) 1.11 (1.15) 1.40 (1.50) 1.06 (1.09) 4.51 (4.92) 2.02 (2.16) 1.33 (1.40) 2.47 (2.52) 1.34 (1.38) 1.11 (1.13) 2.16 (2.08) 1.32 (1.32) 2.24 (2.02) 1.36 (1.32)

1.10 (1.01) 1.47 (1.18) 3.10 (2.16) 1.38 (1.14) 1.51 (1.22) 1.81 (1.40) 2.49 (1.81) 2.60 (1.89) 4.73 (3.19) 2.57 (1.89) 5.68 (3.78) 1.13 (1.04) 1.65 (1.34) 2.77 (2.01) 1.39 (1.20) 2.61 (1.93) 4.38 (2.92) 1.47 (1.27) 2.59 (1.92) 1.39 (1.23) 2.37 (1.78)

1.10 (1.02) 1.47 (1.21) 3.12 (2.23) 1.38 (1.16) 1.51 (1.24) 1.81 (1.43) 2.49 (1.86) 2.62 (1.94) 4.78 (3.32) 2.58 (1.94) 5.76 (3.94) 1.13 (1.05) 1.65 (1.37) 2.78 (2.07) 1.39 (1.22) 2.62 (1.98) 4.42 (3.04) 1.47 (1.29) 2.60 (1.97) 1.39 (1.25) 2.37 (1.83)

1.04 (0.99) 1.02 (1.03) 1.46 (1.53) 1.01 (1.02) 1.03 (1.04) 1.10 (1.12) 1.29 (1.35) 1.33 (1.39) 1.91 (2.19) 1.32 (1.40) 2.17 (2.62) 1.01 (0.98) 1.07 (1.10) 1.38 (1.49) 1.01 (1.03) 1.34 (1.46) 1.85 (2.15) 1.02 (1.06) 1.33 (1.48) 1.00 (1.04) 1.26 (1.41)

0.15 (0.14) 0.26 (0.29) 0.36 (0.48) 0.26 (0.27) 0.28 (0.31) 0.32 (0.37) 0.36 (0.46) 0.36 (0.47) 0.36 (0.57) 0.37 (0.48) 0.36 (0.61) 0.20 (0.19) 0.32 (0.37) 0.37 (0.51) 0.29 (0.33) 0.38 (0.53) 0.38 (0.63) 0.32 (0.40) 0.39 (0.57) 0.32 (0.41) 0.39 (0.58)

70.5 53.7 25.1 39.5 30.5 23.2 15.5 14.0 13.4 9.6 10.3 24.9 10.6 7.5 7.7 4.4 4.5 3.7 2.5 2.4 1.6

10.4 (9.6) 14.0 (15.5) 9.0 (12.1) 10.3 (10.9) 8.6 (9.5) 7.4 (8.6) 5.5 (7.1) 5.0 (6.6) 4.8 (7.7) 3.5 (4.6) 3.6 (6.3) 4.9 (4.8) 3.4 (3.9) 2.8 (3.8) 2.3 (2.6) 1.7 (2.4) 1.7 (2.8) 1.2 (1.5) 1.0 (1.4) 0.8 (1.0) 0.6 (0.9)

13.09 13.83 14.50 11.73 11.00 11.71 12.00 12.00 12.00 11.29 10.50 9.79 10.00 10.25 8.52 8.43 9.00 7.77 7.60 6.80 6.83

T (K)

a Experimentally determined mole fractions xi of equilibrium mixtures (n is the number of measured points at a given composition). b The terms γi and Kγ are calculated by COSMO-RS or UNIFAC (in parentheses). c Ka ) Kx‚Kγ with Kx from experiment and Kγ from either COSMO-RS or UNIFAC (in parentheses).

The true thermodynamic equilibrium constant Ka of the TAME-synthesis reaction (eq 1) calculated in such way serves as a “touch-stone” for any predictive scheme of activity coefficients as UNIFAC or COSMO-RS. Further information for the systems studied in this work is obtained from high-level ab initio calculations of the reaction participants (methanol, 2-methyl-butene-1, 2-methyl-butene-2, and TAME) by using the Gaussian-03 program package. For the species included in this study, calculations have been performed at the G2(MP2), G3(MP2), and G3 levels concerning total electronic energy, chemical structure, normal frequencies, and moments of inertia from which values of KP,id.gas as well as ∆rH°id.gas have been obtained using basic statistical thermodynamic relationships.

tions was observed indicating that the chemical equilibrium was established. The compositions of the reaction mixtures were determined chromatographically using a Hewlett-Packard gas chromatograph 5890 Series II equipped with a flame ionization detector. A capillary column HP-5 (stationary phase cross linked 5% PH ME silicone) was used with a column length of 50 m, an inside diameter of 0.32 mm, and a film thickness of 0.25 µm. Response factors of all reagents were determined using calibration mixtures of the corresponding components prepared gravimetrically. The equilibrium ratios Kx of reaction eqs 1 and 2 in the liquid phase were determined as a ratio of mole fractions xi of the reaction participants:

Kx(eq 1) )

2. Experimental 2.1. Materials. Methanol, TAME, and 2-methyl-butene-2 (2MB2) were of commercial origin (Merck). GC analyses of the as-purchased samples of methanol and TAME gave purities >99.9% in agreement with their specifications. According to our GC-analysis, the commercial product 2MB2 was a mixture of isomeric pentenes (isoamylenes) containing mass percentages 86.4% of 2MB2 and 13.6% of 2-methyl-butene-1 (2MB1). As a catalyst, the cation exchange resin in the H+ form (Amberlist15, Aldrich) has been used. Prior to the experiments, the cation exchange resin Amberlist-15 in the H+ form was dried for 8 h at 383 K in a vacuum oven at reduced pressure. 2.2. Chemical Equilibrium Study in the Liquid Phase. Glass vials with screwed caps were filled two-thirds with the initial liquid mixture of methanol and isoamylenes. The cationexchange resin Amberlist-15 was added as a solid catalyst. The quantity of catalyst was approximately 10% weight of the mixture. The vials were thermostatted at the given temperature with an uncertainty of (0.1 K and periodically shaken. After definite time intervals, the vial was cooled rapidly in ice and opened. A sample for the GC analysis was taken from the liquid phase using a syringe. After thermostatting the vial at the original temperature, the procedure was pursued and the samples were taken successively until no further change of the composi-

xTAME xMeOH‚x2MB2

(16)

x2MB2 x2MB1

(17)

K′x(eq 2) )

Mole fractions xi at equilibrium in the liquid phase and values of Kx are listed in Table 2. The true thermodynamic constant Ka of the reaction eqs 1 and 2 is expressed by

Ka(eq 1) )

γΤΑΜΕ xTAME xTAME ) Kγ γΜeΟΗ‚γ2ΜΒ2 xMeOH‚x2MB2 xMeOH‚x2MB2 (18)

and

K′a(eq 2) )

γ2MB2 x2MB2 x2MB2 ) Kγ γ2MB1 x2MB1 x2MB1

(19)

and the values of γi were calculated using UNIFAC15 and COSMO-RS.19 The results presented in Table 2 show that the Ka values of eq 1 depend much less on the composition than the Kx values. However, the calculated Ka values are not really constant at a given temperature. This reflects a certain deficiency

10978 J. Phys. Chem. B, Vol. 111, No. 37, 2007

Heintz et al. TABLE 3: Results from Equilibrium Study of the TAME Synthesis Reaction and Isomerization Reaction (Equation 2) in the Gaseous Phasea T (K) w yMeOH y2MB1 y2MB2 yTAME Kφ(eq 1)

Figure 1. Apparatus for the study of the equilibrium reactions (eqs 1 and 2) in the gaseous phase: (1) entrance for the initial feed of the equilibrium mixture; (2) temperature sensor; (3) catalysator zone; (4) vaporization zone and reactor exit; (5) entrance and exit to the thermostate unit; (6) cold trap; (7) ice bath.

of COSMOS-RS and UNIFAC to predict activity coefficients correctly in the reaction mixture. UNIFAC seems to work slightly better than COSMO-RS. The K x′ values of eq 2 are almost constant at a given temperature justifying the assumption that K x′ is approximately equal to K a′ (K γ′ ≈ 1). 2.3. Chemical Equilibrium Study in the Gaseous Phase. The equilibrium studies in the gaseous phase were performed in a tubular flow-type isothermal reactor at atmospheric pressure in the temperature range 355-378 K using Amberlist-15 as catalyst. The apparatus used is sketched in Figure 1. At a given temperature, the initial feed mixture was dropped into the reactor (1) at a constant flow rate using a syringe pump. The liquid feed mixture evaporates in the upper part of zone (4), which is filled with broken porcelain pieces. The vapor mixture then passes the catalytic zone (3) where the chemical equilibrium in the gaseous state is established before the equilibrium mixture was collected in a cold trap (6). The flow rate passing the catalyst should be not too slow in order to avoid possible side reactions. On the other hand, it should be not too fast in order to reach the chemical equilibrium. We optimized the flow rate by studying a flow rate interval of 0.4-7.6 mL‚kg-1‚s-1. It was experimentally established that flow rates of 0.4-2.7 mL‚kg-1‚s-1 ensured that the reacting vapor mixture reached equilibrium at each specified temperature. At a given temperature and constant flow rate, three samples were withdrawn and analyzed chromatographically. The averaged data have been used. The equilibrium constants KP of reaction eqs 1 and 2 in the gaseous phase were calculated from the experimental mole fractions yi of the reaction participants in the gaseous phase according to eqs 8 and 12. The values of φi were calculated from the volumeexplicit virial equation of state truncated after the second term according to eq 9. Data for the second virial coefficients Bii and Bij have been attained using an approximative procedure. The summations were performed over all mixture components i and j independently. The values Bij were estimated from the generalized method given by VanNess and Abbot20 modified for polar molecules by Tsonopoulos.21 The algorithm of calculations and input data were taken from the compilation given by Reid et al.22 Mole fractions of the gaseous equilibrium

355.2 0.4 1.3 0.4 1.3 0.4 1.3 2.7 4.2 7.6 358.6 0.4 1.3 2.7 362.6 0.4 0.4 0.4 0.4 365.6 1.3 1.3 1.3 1.3 370.2 0.4 0.4 0.4 0.4 373.0 1.3 1.3 1.3 1.3 1.3 376.8 1.3 1.3 1.3 1.3 378.7 1.3 1.3

0.575 0.556 0.741 0.737 0.832 0.838 0.840 0.846 0.841 0.668 0.843 0.777 0.876 0.665 0.791 0.745 0.754 0.734 0.769 0.801 0.957 0.675 0.861 0.801 0.795 0.805 0.713 0.803 0.798 0.857 0.820 0.741 0.844 0.887 0.823

0.045 0.048 0.027 0.027 0.017 0.016 0.016 0.017 0.017 0.036 0.017 0.024 0.014 0.039 0.023 0.029 0.029 0.031 0.027 0.023 0.005 0.041 0.017 0.024 0.026 0.024 0.036 0.024 0.025 0.019 0.023 0.034 0.020 0.015 0.023

0.331 0.348 0.194 0.198 0.125 0.120 0.119 0.103 0.090 0.259 0.119 0.172 0.096 0.267 0.164 0.201 0.198 0.212 0.185 0.157 0.034 0.265 0.111 0.161 0.167 0.158 0.233 0.160 0.164 0.116 0.147 0.213 0.126 0.092 0.145

0.049 0.048 0.038 0.038 0.026 0.026 0.025 0.034 0.052 0.037 0.021 0.027 0.014 0.029 0.022 0.025 0.019 0.023 0.019 0.019 0.004 0.019 0.011 0.014 0.012 0.013 0.018 0.013 0.013 0.008 0.010 0.012 0.010 0.006 0.009

1.0002 1.0003 0.9992 0.9993 0.9988 0.9987 0.9987 0.9986 0.9985 0.9996 0.9988 0.9991 0.9987 0.9996 0.9990 0.9992 0.9992 0.9993 0.9992 0.9990 0.9986 0.9994 0.9989 0.9991 0.9991 0.9991 0.9993 0.9991 0.9991 0.9990 0.9991 0.9993 0.9990 0.9990 0.9991

KPP0/Ptotal (eq 1) KP(eq 2) 0.257 0.248 0.264 0.260 0.250 0.259 0.250 0.390b 0.687b 0.214 0.209 0.202 0.166 0.163 0.170 0.167 0.127 0.148 0.134 0.151 0.123 0.106 0.115 0.109 0.090 0.102 0.108 0.101 0.099 0.080 0.083 0.076 0.094 0.074 0.075

7.36 7.25 7.19 7.33 7.35 7.50 7.44 6.06b 5.29b 7.19 7.00 7.17 6.86 6.85 7.13 6.93 6.83 6.84 6.85 6.83 6.80 6.46 6.53 6.71 6.42 6.58 6.47 6.67 6.56 6.11 6.39 6.26 6.30 6.13 6.30

a

Experimentally determined mole fractions yi of the equilibrium mixtures are at a given temperature T, w is the feed flow in millileters per kilogram second of the reagents through the catalyst layer in the tubular reactor, and KP values in the gaseous phase are calculated according to eq 8 with Kφ(eq 1) ) φTAME/(φMeOH‚φ2MB2) and Kφ(eq 2) ≈ 1. b Values were not taken into account for calculations of the thermodynamic functions of reaction eqs 1 or 2 as nonequilibrated due to the high feed flow rates.

mixture yi and equilibrium constants in the gaseous phase KP are listed in Table 3. The contribution of the fugacity coefficients φi to Kp is also presented in Table 3 and defined as follows:

Kφ(eq 1) )

φTAME φMeOH‚φ2MB2

(20)

φ2MB2 ≈1 φ2MB1

(21)

and

K′φ(eq 2) )

3. Equilibrium Constants and Enthalpies of Reaction 3.1. Equilibrium Constants Ka in the Liquid and KP in the Gaseous Phase of the TAME Synthesis Reaction. Figures 2 and 3 show Kx and Ka of reaction eq 1 as a function of the methanol fraction xMeOH at 313 and 353 K. The experimental values of Kx measured in this work are in close agreement with those obtained by Rihko et al.1 and Piccoli and Lovisi.2 It is obvious from these results that the TAME synthesis mixture behaves strongly nonideal, especially when the mole fractions

Equilibria of tert-Amyl Methyl Ether Synthesis

Figure 2. Equilibrium ratios Kx and equilibrium constants Ka ) Kx‚ Kγ for reaction eq 1 with Kγ calculated with COSMO-RS and UNIFAC as a function of methanol mole fraction x in the equilibrium mixture at T ) 313.2 K. The solid line is the Ka calculated from KP using eq 22 (see text): (∆) Kx (exp); (2) Ka from Kp; (b) Ka from UNIFAC; (O) Ka from COSMO-RS.

J. Phys. Chem. B, Vol. 111, No. 37, 2007 10979

Figure 4. Enlarged presentation of equilibrium constants Ka ) Kx‚Kγ for reaction eq 1 with Kγ calculated using UNIFAC and COSMO-RS as a function of methanol mole fraction x in the equilibrium mixture at T ) 353.2 K (the experimental data are only those from Table 2). The solid line is the Ka calculated from KP using eq 22 (see text): (2) Ka from Kp; (b) Ka from UNIFAC; (O) Ka from COSMO-RS.

PMeOH,0‚P2MB2,0 PTAME,0 BMeOH‚PMeOH,0 + B2MB2‚P2MB2,0 - BTAME‚PTAME,0 exp RT (22)

K a ) KP

(

Figure 3. Equilibrium ratios Kx and equilibrium constants Ka for reaction eq 1 as a function of methanol mole fraction x in the equilibrium mixture at T ) 353.2 K. The solid line is the Ka calculated from KP using eq 22 (see text): (4) Kx from this work; (2) Ka from Kp; (b) Ka from UNIFAC; (O) Ka from COSMO-RS; (]) Kx from Rihko et al.;1 (+) Kx from Serda et al.;3 (/) Kx from Piccoli and Lovisi.2

of methanol are low. Inspection of Figures 2 and 3 and Table 2 shows that the Kx values are almost independent of the mole fraction of methanol for xMeOH > 0.5. Calculated results of Ka using γi values predicted by COSMO-RS and UNIFAC include our experimental data of Kx and those of other authors.1-3 The Ka values are not really constant at 313 K. The constancy is somewhat better at 353 K (see Figure 3). However, as it can be seen in Figure 4, upscaling of the results at 353 K reveals that deviations arise apparently from the prediction of Kγ. The COSMO-RS model seems to work not quite as well as the UNIFAC model. For comparison, a further independent method of determining Ka values has been applied. Using values from the chemical equilibrium study of the TAME synthesis reaction in the gaseous phase, the procedure works as follows. The true thermodynamic constants Ka of reaction eq 1 in the liquid phase can be obtained from experimental values of KP in the gaseous phase using eqs 3 and 4 together with eq 7:

)

where Pi0 are the saturated vapor pressures of the pure components. The exponential term is the ratio of fugacity coefficients of the pure components according to eq 9. Saturated vapor pressures for methanol, 2MB1, 2MB2, and TAME are available in the literature23-25 (see the Supporting Information). Measured values of KP for the gaseous reaction eqs 1 and 2 presented in Table 3 reveal that KP is independent of the composition of the equilibrium mixtures and therefore can be assumed to be equal to the thermodynamic equilibrium constants in the gaseous phase. These values were used for calculating true thermodynamic constants Ka in the liquid phase according to eq 22 using values of Pi0, Bii, and Bij taken from Tables 1 and 2 (Supporting Information). The results for calculated Ka values are presented as solid lines in Figures 2-4. If we refer to these data as the most reliable reference data for Ka in the liquid phase, Figures 2-4 show that the general conclusion drawn from the comparison with the method of determining Ka from experimental Kx values in the liquid phase and accounting for the reality of the mixture using COSMO-RS and UNIFAC is the following. Both methods give only roughly the same Ka values. UNIFAC works somewhat better at higher temperatures then COSMO-RS (see Figures 3 and 4); however, at lower temperatures (see Figure 2), COSMO-RS provides more reliable activity coefficients. 3.2. Equilibrium Constants Ka in the Liquid Phase and KP in the Gaseous Phase of the Reaction 2MB1 / 2MB2. The side reaction 2MB1 / 2MB2 (eq 2) takes place in the liquid phase as well as in the gaseous phase simultaneously with the TAME synthesis reaction. Experimental results for equilibrium constants of reaction eq 2 are presented in Table 2. The measured equilibrium ratio Kx was found to be independent of the mixture composition. This means that the ratio of activity coefficients of the isomers (see Table 2) is close to unity, i.e., Kx = Ka. Comparison with the data in the liquid phase available from the literature is given in Figure 6. The scattering of the literature data of Kx in the liquid phase is quite large; however, on the average, the data are in agreement with the equilibrium

10980 J. Phys. Chem. B, Vol. 111, No. 37, 2007

Figure 5. Equilibrium constant Ka and equilibrium ratio Kx of the TAME synthesis reaction (eq 1) in the liquid phase as a function of temperature: (∆) Kx for x > 0.5; (2) Ka from Kp; (b) Ka from UNIFAC (average); (O) Ka from COSMO-RS.

Heintz et al.

Figure 7. Experimental thermodynamic equilibrium constants KP of the isomerization reaction 2MB1 / 2MB2 (eq 2) in the gaseous phase as a function of temperature: (4) this work; (2) Radyuk and Kabo.26

ln K ) -

Figure 6. Experimental equilibrium constants KP of the reaction 2MB1 / 2MB2 (eq 2) in the gaseous state as a function of temperature: (4) this work, (]) Rihko et al.;1 (/) Piccoli and Lovisi;2 (+) Serda et al.;3 (O) Seyed et al.5

constants measured in this work. We made the attempt to study reaction eq 2 independently from reaction eq 1 under similar conditions as the TAME synthesis reaction. Unfortunately, the polymerization rate of methyl-butenes on Amberlist-15 in the absence of methanol was too high and made it impossible to obtain reasonable results. However, the neat isomerization reaction was studied successfully in the gaseous phase by Radyuk and Kabo26 in the temperature range 542-582 K where polymerization is suppressed. Our results of reaction eq 2 obtained simultaneously with eq 1 in the gaseous phase are in good agreement with those from Radyuk and Kabo26 in spite of the very different temperature range (see Figure 7). 3.3. Enthalpies and Entropies of Reactions of the TAME Synthesis Reaction and the Isomerization Reaction in the Liquid and in the Gaseous Phase. Generally, the enthalpy of reaction, ∆rH°, is defined as the stoichiometric difference of the enthalpies of the products and educts in the pure states. The values of ∆rH °liquid for the reaction in the liquid state and ∆rH °id.gas or the reaction in the ideal gaseous state can be obtained from the temperature dependence of the corresponding equilibrium constants Ka and KP according to eqs 13 and 14, i.e., a linear plot of experimental values of ln Ka or ln KP versus 1/T respectively provides ∆rH° from the slope of the plot and the reaction entropy ∆rS° from the section of the ln K axis according to the following:

∆rH° ∆rS b + ) +a RT R T

(23)

Figures 5-7 show the results for the TAME synthesis reaction eq 1 and the isomerization reaction eq 2. Results of the equilibrium ratio Kx of the reaction eq 1 are also shown in Figures 5 at mole fractions of xMeOH > 0.5 where Kx is nearly independent of x. The parameters a and b as well as values of ∆rH° and ∆rS° obtained by the linear fitting procedure are given in Table 4. ∆rH° obtained from Kx with xMeOH > 0.5 and from Ka obtained from KP values (see eq 22) and heats of vaporization according to eq 15 agrees surprisingly well for the TAME synthesis reaction in spite of the fact that Kx is not the true equilibrium constant in the liquid state. This has already been observed in cases of other ether synthesis reactions12-14 leading to the conclusion that the study of ether synthesis reactions at compositions with a considerable excess of the alcoholic component allows determination at ∆rH° without additional measurements of activity coefficients γi.12-14 The least-square treatment of the temperature dependence of the equilibrium constants previously published by Rozhnov et al.18 gave ∆rH °id.gas ) -60.7 kJ‚mol-1 for the reaction eq 1 and ∆rH °id.gas ) -8.0 kJ‚mol-1 for reaction eq 2. These results are in acceptable agreement with the results from our extended study presented in Table 4. The result for the gaseous isomerization reaction is ∆rH° ) -(8.1 ( 0.5) kJ‚mol-1 obtained by Radyuk and Kabo26 in the temperature range 542-582 K is in excellent agreement with our result -(7.8 ( 0.6) kJ‚mol-1. 3.4. Comparison of the ∆rH° Obtained from Equilibrium Studies and from Calorimetric Combustion Experiments. Enthalpies of chemical reactions, ∆rH°, can also be determined from calorimetric measurements.14 In order to test the thermodynamic consistency, the data of ∆rH° of reactions determined in this work for the liquid as well as for the gaseous phase from the slope of ln K vs 1/T plots have been compared with results obtained from differences of the enthalpies of formation of the reaction participants available from calorimetry. For the latter purpose, the standard molar enthalpies of formation, ∆fH °m, of TAME, methanol, 2MB2, and 2MB1 are required. The value ∆fH °m(l) ) -(336.8 ( 1.7) kJ‚mol-1 of TAME was measured in our previous work18,27 and has been confirmed by Steele et al.28 recently: ∆fH °m(l) ) -(335.03 ( 0.66) kJ‚mol-1. Using the enthalpy of vaporization ∆gl H °m ) (35.27 ( 0.39) kJ‚mol-1 of TAME,27 the value of the gaseous enthalpy of formation of TAME, ∆fH °m(g) ) -(301.5 ( 1.7) kJ‚mol-1, has been obtained recently27 and used in further calculations. Other

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J. Phys. Chem. B, Vol. 111, No. 37, 2007 10981

TABLE 4: Experimental Thermodynamic Functions ∆rH° and ∆rS° of the TAME Synthesis Reaction (Equation 1) and the Isomerization Reaction (Equation 2) in the Liquid and in the Gaseous Phase at T ) 298.15 Ka ∆rH°(equilibrium)b (kJ·mol-1)

∆rH°(calorimetry)c (kJ·mol-1)

from Kφ(eq 1) from Ka(eq 1) (UNIFAC) from Ka(eq 1) (COSMO-RS) from Kx(eq 1)d from Ka(eq 2)

-28.4 ( 1.5 -27.7 ( 3.0

-29.2 ( 2.1

-28.1 ( 0.6 -7.1 ( 0.3

from KP(eq 1) from KP(eq 2)

-57.5 ( 2.2 -7.8 ( 0.6

b

temp range (K)

∆rS° b (J·K-1‚mol-1)

Liquid State -9.0 -8.8

3415.1 3327.1

298.2-393.2 298.2-393.2

-75.0 ( 4.2 -73.4 ( 9.0

-8.3

3243.7

298.2-393.2

-69.0 ( 4.1

-7.1 ( 1.5

-8.1 -0.27

3383.0 858.0

298.2-393.2 298.2-393.2

-67.6 ( 1.7 -2.2 ( 0.8

-58.5 ( 2.2 -6.0 ( 1.5

Gaseous Phase -32.4 -0.64

6914.9 934.8

355.2-378.7 355.2-378.7

-173.4 ( 4.3 -5.3 ( 1.7

a

-27.0 ( 1.4

a Coefficients a and b were obtained from fitting ln K ) a + b‚(T/K)-1 to experimental values of Ka, Kx, or KP as indicated in the first column. Derived from the temperature dependence of Kx, Ka, or KP. c Calculated from the enthalpies of formation of the reaction participants measured by combustion calorimetry. d Obtained in this work from Kx for xMeOH > 0.5.

b

TABLE 5: Calculated Relative Energies, Enthalpies and Gibbs Energies, and Populations for Stable Conformers of TAME at T ) 298.15 K G2MP2

G3MP2

CBS-Q

conformer i ∆Ea (kJ‚mol-1) ∆E0b ∆Hc ∆Gd pie (%) ∆Ea (kJ‚mol-1) ∆E0b ∆Hc ∆Gd pie (%) ∆Ea (kJ‚mol-1) ∆E0b ∆Hc ∆Gd pie(%) GG TG TT GT

0.00 1.33 3.68 4.05

0.00 1.17 3.41 3.98

0.00 1.27 3.64 4.13

0.00 0.64 2.25 2.79

40.0 30.9 16.1 13.0

0.00 1.43 3.71 4.06

0.00 1.27 3.43 3.98

0.00 1.37 3.67 4.14

0.00 0.76 2.25 2.78

40.6 29.8 16.4 13.2

0.00 2.70 5.00 5.36

0.00 2.54 4.72 5.27

0.00 2.64 4.96 5.45

0.00 1.99 3.52 3.85

52.6 23.6 12.7 11.1

a Electronic energy. b Sum of electronic and zero-point energy. c Sum of electronic and thermal enthalpy enthalpy at 298.15 K. d Sum of electronic and thermal free energy. e Relative populations (percents) at 298.15 K are calculated according to eq 29 (relative energy values are based on the absolute energies of the most stable conformer (GG)).

experimental data necessary for comparison are available in the literature. The enthalpy of formation for 2MB2 in the liquid phase,29 ∆fH °m(l) is -(68.1 ( 1.3) kJ‚mol-1, and the enthalpy of vaporization30 ∆gl H °m of 2MB2 is 27.34 kJ‚mol-1 from which ∆fH °m(g) ) -(40.8 ( 1.3) kJ‚mol-1 is obtained. ∆fH °m(l) of 2MB129 is -(60.96 ( 0.84) kJ‚mol-1, and the enthalpy of vaporization30 ∆gl H °m for 2MB1 is 26.19 kJ‚mol-1 from which ∆fH °m(g) ) -(34.77 ( 0.84) kJ‚mol-1 is obtained. The enthalpy of formation for methanol in the liquid phase,31 ∆fH °m(l), is -(239.5 ( 0.2) kJ‚mol-1, and the enthalpy of vaporization32 ∆gl H °m is (37.30 ( 0.08) kJ‚mol-1 from which ∆fH °m(g) ) -(202.20 ( 0.22) kJ‚mol-1 is obtained. These data were used to calculate independently ∆rH °m(calorimetry) of the TAME synthesis reactions in the liquid phase as well as in the gaseous phase by the following relationship:

∆rH°(1)(calorimetry) ) ∆fH°m(liq,TAME) - ∆fH°m(liq,MeOH) ∆fH°m(liq,2MB2) ) -(29.2 ( 2.1) kJ‚mol-1 (24) ∆rH°(g)(calorimetry) ) ∆fH°m(gas,TAME) - ∆fH°m(gas,MeOH) ∆fH°m(gas,2MB2) ) - (7.1 ( 1.5) kJ‚mol-1 (25) The comparison with experimental values obtained from chemical equilibrium study is given in Table 4. The calculated values of the ∆rH °m(calorimetry) values for the reaction eqs 1 and 2 are in close agreement, i.e., within the boundaries of experimental uncertainties, with those derived from the chemical equilibrium studies confirming the thermodynamic consistency of the whole procedure. 4. Theoretical Calculations 4.1. Methods of Calculations. High level ab initio calculations have been performed on the basis of the G2(MP2), G3-

(MP2), G3, and CBS-Q methods, as well as density functional theory (DFT) to obtain ideal gas thermodynamic functions for the TAME synthesis reaction (eq 1) and for the isomerization reaction according to eq 2. It has been demonstrated recently, that both G* methods39 and DFT methods40 are able to predict enthalpies of formation of oxygenated hydrocarbons accurately. In addition to the TAME synthesis reaction (eq 1), calculations for the thermodynamic functions of similar ether synthesis reactions for methyl tert-butyl ether (MTBE) (eq 27) and methyl cumyl ether (MCE) (eq 28) have also been performed:

iso-butene + methanol / MTBE

(27)

R-methyl styrene + methanol / MCE

(28)

(The CAS number of MCE is [935-67-1].) The geometry optimization was done on the HF/6-31G(d), and the MP2/6-31G(d,p) levels, since they are part of the Gaussian-N and CBS-approaches. Harmonic vibrational frequencies and zero-point vibrational energies (ZPE) were computed on the basis of HF/6-31G(d) theory. DFT calculations are known to provide accurate geometries and reasonable energies for a variety of organic compounds. We have used B3LYP/6-31G(d,p) and B3LYP/6-311+G(2d,2p) methods for the calculation of the geometry parameter and the ZPE values (see the Supporting Information). The methods B3LYP/6-311+G(2d,2p) and B3LYP/6-311+G(3df,2p) have been used for single point energy calculations to check if these larger basis sets show an improvement toward B3LYP/6-31G(d,p) energies. The details on G*-methods and the CBS-Q method can be found in the literature.33-38 The calculations were performed using the Gaussian-03 program suite.42 4.2. Low Energy Conformers and Relative Energies of TAME. Potential barriers for the internal rotations of TAME were calculated by using the MM2-force field. The potential

10982 J. Phys. Chem. B, Vol. 111, No. 37, 2007

Heintz et al.

Figure 9. Experimental and ab initio equilibrium constants KP of the TAME synthesis reaction (eq 1) in the gaseous phase as a function of temperature: (4) experimental values from KP; (2) G2MP2; (b) G3MP2; (×) G3MP2 without the conformational approach (see text); (O) CBS-Q. Figure 8. Four most stable conformations of TAME (a) gauchegauche, (b) trans-gauche, (c) trans-trans, and (d) gauche-trans and the values of dihedral angles τ1 and τ2 (only τ2 is indicated).

energy was calculated as a function of the six dihedral angles along the C-C and C-O bonds by varying the torsion angles in 15° intervals and allowing other parameters to be optimized. The located geometries of each stable conformer were then reoptimized by ab initio and DFT calculations. Also, energy calculations have been performed which were based on these geometries using the methods mentioned above. The described procedure predicts the existence of four stable low energy conformers of TAME, which can be characterized by the two independent dihedral angles O(1)-C(2)-C(3)-C(4) and C(5)-O(1)-C(2)-C(3), i.e. gauche-gauche (GG), transgauche (TG), trans-trans (TT), and gauche-trans (GT). Relative energies obtained by the G-models are almost identical and on the average 1.2 kJ‚mol-1 lower than those calculated by the CBS-Q model. The conformers are shown in Figure 8 where 3-dimensional structures as well as Newman projections illustrate the different geometrical situations. The energy differences of the four conformers related to the most stable conformer (GG) are given in Table 5. The thermal population of the conformers at T ) 298.15 K is given by the following:

pi )

e-∆Gi/RT n

1 + ∑ e-∆Gi/RT

; ∆Gi ) Gi - GGG

(29)

i)1

Results of pi are presented in Table 5. Single point energy calculations using DFT at the B3LYP/ 6-31G(d,p)//B3LYP/6-31G(d,p) and B3LYP/6-311+G(3df,2p)// B3LYP/6-311+G(2d,2p) levels of the four conformers have also been performed providing relative Gibbs energies which were very close to those obtained by the G and CBS methods. 4.3. Thermodynamic Functions. Thermodynamic functions ∆rH°, ∆rG°, ∆rS°, and KP were calculated for the gas-phase reaction eqs 1, 2, 27, and 28. The calculated values for reaction eq 1 are given in Table 6 accounting for the equilibrium mixture of the four conformers of TAME with populations of the four conformers according to the results given in Table 5. ∆rH° and ∆rS° turn out to be nearly independent of temperature; therefore, only results at 298.15 K are presented. The least deviations of the theoretical results from the experimental enthalpies of reactions ∆rH° are observed using the most accurate G3MP2 and G3 methods, where differences are not larger than 2-6

Figure 10. Experimental and ab initio equilibrium constant KP of the isomerization reaction 2MB1 / 2MB2 (eq 2) in the gaseous phase as a function of temperature: (4) experimental values from KP; (2) G2MP2; (b) G3MP2; (O) G3.

kJ‚mol-1 (see Table 6). Calculated values of KP for eq 1 are in acceptable agreement with the experimental values at the G3MP2 level while they differ by a factor of 10 at the G2MP2 and CBS-Q levels. Thermodynamic functions calculated on the basis of the most stable TAME conformer GG are less accurate than those based on the different populations of conformers in comparison with the experimental values (see Table 6 and Figure 9). For the isomerization reaction according eq 2, the G3MP2 and G3 methods provide values for KP and ∆rH° close to the experimental values. The G2MP2 model shows larger deviations but is still in acceptable agreement with the experiments (see Figure 10 and Table 6). DFT calculations based on the B3LYP/6-31G(d,p)//B3LYP/ 6-31G(d,p) level provide reasonable values for ∆rH° only, while calculated values of KP differ by a factor of 10 for both reaction eqs 1 and 2 from the experimental values (see Table 6). 4.4. Enthalpy of Formation of TAME. In order to complete the thermodynamic study of the TAME synthesis reaction we have also performed theoretical calculations of the molar enthalpy of formation ∆fH °m (gas, 298.15 K) of TAME in the gaseous phase. There are several schemes to convert theoretically total energies to the corresponding standard enthalpies of formation. The conventional method uses the atomization energies in conjunction with experimental values of ∆fH °m (298.15 K) of the consisting atoms for deriving enthalpies of formation for the corresponding molecules. An alternative procedure to obtain values of ∆fH °m (gas, 298.15 K), which is usually more accurate, especially for larger systems, uses

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J. Phys. Chem. B, Vol. 111, No. 37, 2007 10983

isodesmic or homodesmic reactions rather than atomization energies. In an isodesmic reaction, the number of chemical bonds of each type is conserved, so the cancellation of possible errors arising from insufficient treatment of electron correlation and incompleteness of the basis set might be expected. An isodesmic reaction will lead to even more accurate results if chemical groups are also conserved in the reaction, because next nearest neighbor interactions are also conserved. The quality of both methods for the determination of ∆fH °m (gas, 298.15 K) using the Gaussian-N and CBS approaches has been illustrated in several previous studies,33-40 and an accuracy of 1-8 kJ‚mol-1 can be expected. In this work, both methods were applied. Enthalpies of formation ∆fH °m (gas, 298.15 K) of TAME were calculated on the basis of the atomization reaction

CH3-CH2-C(CH3)2-O-CH3(g) f 6C(g) + 14H(g) + O(g) (30) as well as on the basis of the selected isodesmic reaction:

CH3-CH2-C(CH3)2-O-CH3(g) + CH3OH(g) f CH3-CH2-C(CH3)2-OH(g) + CH3-O-CH3(g) (31) Results are given in Table 7. In both cases, populations of the conformers of TAME (given in Table 5) have been taken into consideration for calculating of ∆fH °m (gas, 298.15 K). Using enthalpies of reactions according to eqs 30 and 31, calculated by ab initio methods, together with enthalpies of formation, ∆fH °m (gas, 298.15 K), for CH3-OH, CH3-CH2-C(CH3)2OH, and CH3-O-CH3 (listed in the Supporting Information),

TABLE 6: Experimental and Predicted Values for Thermodynamic Functions of the TAME Synthesis Reaction (Equation 1) and Isomerization Reaction (2MB1 / 2MB2, Equation 2) as Well as for Two Other Similar Ether Synthesis Reactions (Equations 27 and 28) reaction eq 1

reaction eq 1

eq 2

eq 27

eq 28

method Ka from Kp Ka ) Kx‚Kγ using UNIFAC Ka ) Kx‚Kγ using COSMO-RS

∆rH °298 (kJ‚mol-1) Liquid Phase -28.4 ( 1.5a -27.8 ( 3.0a -27.0 ( 1.4a ∆rH °298 (kJ‚mol-1)

method experimental G2MP2 G3MP2 CBS-Q B3LYP/6-31G(d,p)//B3LYP/6-31G(d,p) experimental G2MP2 G3MP2 G3 B3LYP/6-31G(d,p)//B3LYP/6-31G(d,p) experimental G2MP2 G3MP2 G3 CBS-Q experimental G3MP2

∆rS °298 (J‚mol-1‚K-1)

Ka (298 K)

-75.0 ( 4.2 -73.4 ( 9.0 -69.0 ( 4.1

11.4 10.4 13.4

∆r S °298 (J‚mol-1‚K-1)

Gaseous Phase -57.5 ( 2.2a -68.7 -62.8 (-64.4)e -68.5 -55.0 -7.8 ( 0.6a -4.8 -5.6 -5.7 -11.2 -64.2 ( 1.7b -72.9 -67.5 -71.3 -75.7 -58.4 ( 2.7c -57.9

Kp (298 K)

-173.4 ( 4.3 -183.6 -183.7 (-185.7)e -184.1 -182.5 -5.3 ( 1.7 4.2 4.2 4.2 2.9

11.0 270 26 (37)e 240 1.3‚(6.2)f 12.3a (14.1)d 11.4 15.5 16.4 129.6

-197.20 ( 0.23c -178.1

a Results from this work. b Calculated from the data in ref 24. c Calculated from the data in ref 11. d Results from the data in ref 26 extrapolated to 298.15 K. e Values in parenthesis are derived without conformational analysis (see text). f Value in parenthesis derived by a hindered rotor model (see text).

TABLE 7: Calculated Individually Enthalpies of Formation ∆f H °298 for the Most Stable TAME Conformers and for the Mixture G2MP2 conformer

∆fH°298 (kJ‚mol-1)

G3MP2 ∆fH°298‚Pi (kJ‚mol-1)

∆fH°298 (kJ‚mol-1)

CBS-Q ∆fH °298Pi (kJ‚mol-1)

∆fH°298 (kJ‚mol-1)

GG TG TT GT Σ(mixture)

From Atomization Energies (eq 30): CH3-CH2-C(CH3)2-O-CH3(g) f 6C(g) + 14H(g) + O(g) -310.56 -124.10 -305.75 -124.07 -309.57 -309.29 -95.65 -304.39 -90.76 -306.93 -306.91 -49.48 -302.08 -49.46 -304.61 -306.43 -39.80 -301.61 -39.91 -304.12 -309.04 -304.20

GG TG TT GT Σ(mixture)

From Isodesmic Group Balanced Energies (eq 31): CH3-CH2-C(CH3)2-O-CH3(g) + CH3OH(g) f CH3-CH2-C(CH3)2-OH(g) + CH3-O-CH3(g) -302.49 -122.75 -301.12 -89.78 -298.82 -48.92 -298.35 -39.48 -300.93

∆fH°298‚Pi (kJ‚mol-1) -162.75 -72.38 -38.74 -33.84 -307.71

10984 J. Phys. Chem. B, Vol. 111, No. 37, 2007 enthalpies of formation of TAME have been calculated (see Table 7). These values are in acceptable agreement with ∆fH °m(g) ) -(301.5 ( 1.7) kJ‚mol-1 obtained experimentally.27 DFT calculations at the B3LYP/6-31G(d,p)//B3LYP/631G(d,p) level gave ∆fH °298(gas) ) -292.25 kJ‚mol-1 on the basis of the isodesmic reaction (eq 31), and this value is in substantially less agreement with experimental data.27 5. Conclusion Experimentally determined chemical equilibrium data of the TAME synthesis reaction (eq 1) and the simultaneously occurring isomerization reaction (eq 2) in the liquid and in the gaseous phase offer to test theoretical models such as UNIFAC and COSMO-RS in the liquid phase and theoretical results obtained from ab initio calculations in the gaseous phase. In particular, the results obtained for the gaseous phase demonstrate that ab initio calculation methods available today allow predicting chemical equilibrium data in the gaseous phase with good quality considering the fact that Gibbs reaction enthalpies and reaction enthalpies are obtained as the difference of very large energy values of the single molecules involved into the reactions. Deviations still observed are most probably due to incompleteness of the basis set used as well as to the harmonic vibrational approximation in calculating of normal frequencies. The present work demonstrates the usefulness of ab initio calculations in predicting chemical equilibrium properties of reactions being of considerable technical importance. Acknowledgment. We are grateful to Dr. Andreas Klamt (COSMOlogic GmbH & Co. KG) for calculating of the activity coefficients by the COSMO-RS. S.P.V. acknowledges gratefully financial support from the Research Training Group “New Methods for Sustainability in Catalysis and Technique” of the German Science Foundation (DFG). Supporting Information Available: Values for the second virial coefficients, vapor pressures of the participating substances of the TAME-synthesis reactions, ideal gas thermodynamic functions for the TAME synthesis reactions, enthalpies of formation ∆fH °298 for species involved in isodesmic reactions, and atomization energies calculations; zero-point vibration energies (ZPE), ZPE-corrected electronic energies, enthalpies H298, and Gibbs energies for the species studied; UNIFAC parameters. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Rihko, L. K.; Linnekoski, J. A.; Krause, A. O. J. Chem. Eng. Data. 1994, 39, 700-704. (2) Piccoli, R. L.; Lovisi, H. R. Ind. Eng. Chem. Res. 1995, 34, 510515. (3) Serda, J. A.; Izquierdo, J. F.; Tejero, J.; Cunill, F.; Iborra, M. Thermochim. Acta 1995, 259, 111-120. (4) Sola, L.; Pericas, M. A.; Cunill, F.; Izquierdo, J. P. Ind. Eng. Chem. Res. 1997, 36, 2012-2018. (5) Syed, F. H.; Egelston, C.; Datta, R. J. Chem. Eng. Data 2000, 45, 319-323. (6) Thiel, C.; Hoffmann, U. Chem. Ing. Tech. 1996, 68, 1317-1320. (7) Oost, C.; Hoffmann, U. Chem. Eng. Technol. 1996, 51, 329-40. (8) Oost, C.; Hoffmann, U. Chem. Eng. Technol. 1995, 18, 203-209. (9) Oost, C.; Sundmacher, K.; Hoffmann, U. Chem. Eng. Technol. 1995, 18, 110-117. (10) Oost, C.; Hoffmann, U. Chem. Eng. Technol. 1996, 51, 329-340. (11) Heintz, A.; Verevkin, S. P. Fluid Phase Equilib. 2001, 179, 85100. (12) Verevkin, S. P.; Heintz, A. J. Chem. Eng. Data. 2001, 46, 41-46. (13) Verevkin, S. P.; Heintz, A. J. Chem. Eng. Data. 2001, 46, 984990.

Heintz et al. (14) Verevkin, S. P.; Heintz, A. J. Chem. Soc., Perkin Trans. 2002, 2, 728-733. (15) UNIFAC program from Sandler, S. I. Chemical and Engineering Thermodynamics, second ed.; J. Wiley & Sons, Inc.: New York, 1989. (16) Klamt, A. J. Phys. Chem. 1995, 99, 2224-2235. (17) Klamt, A.; Eckert, F. Fluid Phase Equilib. 2000, 172, 43-72. (18) Rozhnov, A. M.; Safronov, V. V.; Verevkin, S. P.; Sharonov, K. G.; Alenin, V. I. J. Chem. Thermodyn. 1991, 23, 629-635. (19) Calculations performed using program described in Eckert, F.; Klamt, A. COSMOtherm, version C1.2, release 01.03; COSMOlogic GmbH&Co. KG: Leverkusen, Germany, 2003. (20) Van Ness, H. C.; Abbott, M. M. McGraw-Hill: New York, 1982. (21) Tsonopoulos, C. AIChE J. 1975, 21, 827-829. (22) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1988. (23) de Reuck, K. M.; Craven, R. J. B. Methanol-International Thermodynamic Tables of the Fluid State-12; IUPAC Chemical Data Series; Blackwell Scientific Publications: Oxford, 1993; no. 38, pp 62-63. (24) Afeefy, H. Y.; Liebman, J. F.; Stein, S. E. NIST Standard Reference Database Number 69; Linstrom, P. J., Mallard, W. G., Eds.; National Institute of Standards and Technology: Gaithersburg, MD, June 2005; http:// webbook.nist.gov. (25) Kra¨henbu¨hl, M. A.; Gmehling, J. J. Chem. Eng. Data 1994, 39, 29-32. (26) Radyuk, Z. A.; Kabo, G. Y.; Andreevskii, D. N. Neftekhimiya 1973, 13, 356-360. (27) Verevkin, S. P. J. Chem. Eng. Data 2002, 47, 1071-1097. (28) Steele, W. V.; Chirico, R. D.; Cowell, A. B.; Knipmeyer, S. E.; Nguyen, A. J. Chem. Eng. Data 2002, 47, 667-688. (29) Good, W. D.; Smith, N. K. J. Chem. Thermodyn. 1979, 11, 111118. (30) Majer, V.; Svoboda, V. Enthalpies of Vaporization of Organic Compounds: A Critical ReView and Data Compilation; Blackwell Scientific Publications: Oxford, 1985; pp 1-300. (31) Chao, J.; Rossini, F. D. J. Chem. Eng. Data 1965, 10, 374-379. (32) Wadso, I.; Acta Chem. Scand. 1966, 20, 544. (33) Curtiss, L. A.; Raghavachari, K. In Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy; Langhoff, S. R., Ed.; Kluwer Academic Press: Netherlands, 1995; p 139. (34) Raghavachariand, K.; Curtiss, L. A. In Modern Electronic Structure Theory; Yarkony, D. R., Ed.; World Scientific: Singapore, 1995; 991. (35) Curtiss, L. A.; Raghavachari, K.; Redfern, P. C.; Rassolov, V.; Pople, J. A. J. Chem. Phys. 1998, 109, 7764-7776. (36) Curtiss, L. A.; Raghavachari, K.; Redfern, P. C.; Baboul, A. G.; Pople, J. A. Chem. Phys. Lett. 1999, 314 (1-2), 101-107. (37) Curtiss, L. A.; Redfern, P. C.; Raghavachari, K.; Pople, J. A. Chem. Phys. Lett. 1999, 313 (3-4), 600-607. (38) Curtiss, L. A.; Raghavachari, K. Theor. Chem Acc. 2002, 108, 6170. (39) Dorofeeva, O. V.; Yungman, V. S.; Varushchenko, R. M.; Druzhinina, A. I. Int. J. Thermophys. 2004, 25, 1097-1113. (40) Sebbar, N.; Bozzelli, J. W.; Bockhorn, H. Int. J. Chem. Kinet. 2005, 37, 633-648. (41) Stull, D. R.; Westrum, E. F.; Sinke, G. C. Chemical Thermodynamics of Organic Compounds; J. Wiley&Sons Inc.: New York, 1969. (42) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A., Jr.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.: Peng, C. Y.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Andres, J. L.; Gonzalez, C.; Head-Gordon, M.; Replogle, E. S.; Pople, J. A.; Gaussian 03, revision B.04; Gaussian, Inc.: Pittsburgh, PA, 2003. (43) Pedley, J. B. Thermochemical Data and Structures of Organic Compounds; Thermodynamics Research Center: College Station, TX, 1994; Vol. 1.