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An analytic equation-of-state for mixture of square-well chain fluids of variable well width
Fluid Phase Equilibria 179 (2001) 231–243
An analytic equation-of-state for mixture of square-well chain fluids of variable well width Márcio L.L. Paredes a,1 , Ronaldo Nobrega a , Frederico W. Tavares b,∗ a
b
Programa de Engenharia Qu´ımica/COPPE, Universidade Federal do Rio de Janeiro, Caixa Postal: 68502, 21945-970 Rio de Janeiro, RJ, Brazil Escola de Qu´ımica, Universidade Federal do Rio de Janeiro, Caixa Postal: 68542, 21949-900 Rio de Janeiro, RJ, Brazil Received 20 October 2000; accepted 3 November 2000
Abstract An analytic perturbation theory equation of state for a mixture of freely-jointed square-well fluids of variable well width (1 ≤ λ ≤ 2) is developed. The equation of state is based on second-order Barker and Henderson perturbation theory to calculate the thermodynamic properties of the reference sphere fluid, and on first-order Wertheim thermodynamic perturbation theory to account for the connectivity of spheres to form chains. A real function expression for the radial distribution function of hard spheres and a one-fluid type mixing rule are used to obtain an analytic, closed-form expression, for the Helmohltz free energy of mixtures of square-well spheres. Good results were obtained when this equation of state was used with temperature-independent parameters to correlate vapor–liquid equilibrium data of pure substances and mixtures. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Equation of state; Mixture; Polymer; Square-well fluid
1. Introduction Many papers have been published describing molecules as a chain of spheres [1–20]. This molecular modeling can be useful for representing macromolecules as polymers and proteins, as well as for low molecular weight substances. Hall and co-workers [1,2] and Chapman and co-workers [3,4] have presented equations of state for chains of freely-jointed tangent spheres. The equation of state presented by Chapman and co-workers [3,4] is based on the Wertheim [5] theory to account for the connectivity of spheres, and is called the TPT1 equation of state. The Wertheim theory has been used to account for the Abbreviations: RMSADy, root-mean-square absolute deviations in vapor composition; RMSRDP, root-mean-square relative deviations in vapor pressure ∗ Corresponding author. Tel.: +55-21-562-7650; fax: +55-21-562-7567. E-mail addresses: [email protected] (R. Nobrega), [email protected] (F.W. Tavares). 1 Tel.: +55-21-590-7135; fax: +55-21-590-7134. 0378-3812/01/$20.00 © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 0 ) 0 0 5 0 4 - 5
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connectivity of spheres in forming chains in different ways [6–22]. Many studies have been published describing molecules as a chain of hard spheres [1,2,5–8] or a chain of attractive spheres interacting by the Lennard–Jones potential [12–15] and by the square-well potential [15–22]. The van der Waals approximation [3,4,9–11] and perturbation theories [12–22] are used in the literature to calculate the thermodynamic properties of the reference fluid (non-bonded segments) that include the attractive part. Efforts have been made in literature to account for the influence of the attractive part of the square-well potential on the bond formation term [16,19–22]. Banaszak et al. [16] have applied first-order perturbation Barker and Henderson theory to the radial distribution function of square-well spheres [23,24] to account for attractivity in the Wertheim theory, for a square-well width λ = 1.5. Galindo et al. [15] have proposed an equation of state for square-well chain mixtures, taking into account the effect of attractivity in bond formation, using the mean value theorem to evaluate Barker and Henderson’s integrals. Adidharma and Radosz [22] have presented an equation of state for mixtures of copolymers based on the Wertheim theory and on the work of Galindo et al. [15]. Tavares et al. [20] have extended the Banaszak et al. [16] equation of state to variable square-well width using Barker and Henderson’s logarithmic expansion approximation (LEA) [23,24]. Though using only a first-order expansion function, the introduction of this term generates a more consistent model for attractive chains. The square-well chain properties calculated with this equation of state are in good agreement with Monte Carlo simulation data of Tavares et al. [19], and in better agreement with simulation than the analogous equation of state that does not take into account the effect of attractivity in bond formation. In this equation of state, Barker and Henderson perturbation theory was evaluated using Chang and Sandler’s hard-sphere radial distribution function expression [25,26]. Gulati and Hall [21] and Hino and Prausnitz [18] have applied the one-fluid type mixing rule to mixtures of square-well chains, using the Barker and Henderson [23,24] perturbation theory, along with the Chang and Sandler hard sphere radial distribution function [25,26]. In the present work, an analytic equation of state for freely-jointed square-well mixtures of variable well width was proposed. The importance of taking into account the effect of attractivity in bond formation was also confirmed for mixtures by comparing the equation of state results with Monte Carlo simulation data [27]. The equation of state parameters were obtained for a large set of pure substances. Also, two binary adjustable parameters were determined for some mixtures. The equation of state was found to be very sensitive to the parameters. This sensitivity came to be useful for correlating and predicting the thermodynamic properties of real mixtures.
2. The equation-of-state for mixtures In this section, the analytic equation of state for mixtures of square-well chains is presented. The interaction potential Φ ij between two non-adjacent spheres i and j at distance rij is defined as follows: rij < σij ∞, Φij = −εij , σij ≤ rij < λij σij (1) 0, rij ≥ λij σij where σ ij is defined in Eq. (2) and εij and λij are defined in Eq. (3), following standard combining rules. σij =
σi + σj 2
(2)
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p √ εij = εi εj (1 − φij ) and λij = λi λj (1 − ϕij )
233
(3)
where, for component i, σ i is the segment diameter, εi the square-well depth and λi is the square-well width; φ ij and ϕ ij are binary adjustable parameters, set equal to zero when i = j . The one-fluid type mixing rule is used in this work in a very similar way to that of Hino and Prausnitz [18] and Gulati and Hall [21]. It can be expressed by the average quantities of the equation of state parameters kept within the parentheses in the following equation:
m σ f (ξ3 ; ε, λ) = 2
3
nc X nc X
xi xj mi mj σij3 f (ξ3 ; εij , λij )
(4)
i=1 j =1
where nc is the number of components, i and j denote the component indexes, x is the mole fraction and f(ξ 3 ; εij , λij ) denotes an arbitrary function of mixture packing fraction (ξ 3 ), εij and λij . The mixture packing fraction is defined as follows: ξk =
nc π X ρ xj mj σjk where k = 0, 1, 2, 3 6 j =1
(5)
The proposed equation of state for square-well chain mixtures is Zchain = 1 + (mZ
R,HS
)mix + (mZ )mix − att
nc X
xi (mi − 1)ξ3
i=1
∂ ln giiSWS ∂ξ3
where according to Boubl´ık [28] and Mansoori et al. [29] ξ23 (3 − ξ3 ) 6 3ξ1 ξ2 ξ0 ξ3 + + (mZR,HS )mix = πρ 1 − ξ3 (1 − ξ3 )2 (1 − ξ3 )3
(6)
(7)
By applying the mixing rule (Eq. (2)) to calculate the attractive contribution to the compressibility factor, the following equation is obtained: nc X nc ε 2 ξ (1 − ξ )4 ∂lij εij π X ij 3 3 3 xi xj mi mj σij 12 lij + ξ3 (mZ )mix = − ρ +6 6 i=1 j =1 kT ∂ξ3 kT (1 + 2ξ3 )2 (1 − ξ3 )3 (1 − 5ξ3 − 20ξ32 − 12ξ33 ) ∂lij ∂ 2 lij ∂lij × 2 + ξ3 2 + lij + ξ3 ∂ξ3 (1 + 2ξ3 )4 ∂ξ3 ∂ξ3 att
(8)
where lij = l(ξ3 ; λij )
(9)
and l is a integral function [20,26]. The radial distribution function at the contact point of square-well spheres giiSWS is calculated by ln giiSWS = ln giiHS +
εii g1,ii kT giiHS
(10)
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where the perturbation term g1,ii is expressed in Eq. (11). ∂lij + λ3ii g HS (λσ + ) g1,ii = −3 lij + ξ3 ∂ξ3
(11)
In Eq. (10), the Boubl´ık [28] and Mansoori et al. [29] radial distribution function of hard spheres giiHS is used, as follows: giiHS =
1 3 ξ2 σii 1 (ξ2 σii )2 + + 1 − ξ3 2 (1 − ξ3 )2 2 (1 − ξ3 )3
(12)
By substituting Eqs. (7), (8) and (10) into Eq. (6), the TPT1M model for square-well chain mixtures is obtained. The TPT1 model for square-well chain mixtures used in this work is obtained by setting g1,ii equal to zero in the TPT1M model, leading to ln giiSWS ← ln giiHS
(13)
Using the hard sphere radial distribution function at the contact point to calculate attractive sphere chain properties, information about attractiveness is missed in the TPT1 bond formation term. For this reason, the main advantage of the TPT1M equation of state for mixtures of square-well chain fluids is that it takes into account the influence of attractivity on the bond formation term.
3. Results 3.1. Pure component parameters In this section the parameters of the TPT1 and TPT1M equations of state are obtained for some pure normal alkanes, ethers and esters. The models present four parameters: ε and λ are related to intermolecular attractions and σ and m are related to molecular size and shape. These parameters are determined so that the equations of state give the experimental critical point [30] and the minimum relative mean square deviations of the smoothed vapor pressures [30] over temperature range of 0.6 ≤ T /T c ≤ 1.0. The fitting procedure was carried out in two loops: in the inner loop, parameters ε and σ were used to give the experimental critical point, and in the outer loop, parameters λ and m were determined by fitting the vapor pressure. The parameters obtained for alkanes are plotted against the number of carbons while the parameters for esters are plotted against the molecular weight (mol) of the substances. Since the TPT1 and TPT1M models attempt to represent chain molecules, the parameters related to the segment are expected to be independent of molecular weight. Parameter m (number of segments) is expected to vary linearly with molecular weight. Since the properties of the segments should not depend on molecular weight, the mol/m ratio is expected to be independent of molecular weight. In Fig. 1, the parameters for linear alkanes, from methane to dodecane, are plotted against the number of carbons, for the TPT1 and TPT1M equations of state. In Fig. 1(a) and (b), the TPT1M parameters (ε and σ ) are almost constant from pentane to dodecane, as expected, while the TPT1 parameters are dependent on chain size. In Fig. 1(c), the TPT1 parameter (λ) decreases with chain size, while the TPT1M parameters decrease from methane to butane and increase from pentane to dodecane. This result, leading
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Fig. 1. TPT1 and TPT1M parameters for linear alkanes as a function of number of carbons. (a) ε/k; (b) σ ; (c) λ; and (d) m.
to the TPT1M λ parameters increasing with chain size for alkanes greater than pentane, is not expected. In Fig. 1(d), the TPT1M parameter (m) behaves as a linear function of the chain size from pentane to dodecane, as expected, and the TPT1 parameter behaves as an almost linear function of the chain size. The slopes of the curves in Fig. 1(d) indicate that, when hydrocarbon molecules are represented as chains of tangent attractive hard spheres, a lower number of segments is obtained than the number of carbon atoms in the hydrocarbon molecule. The consequence of underestimating the number of segments in the chain is an overestimation of the effect of the number of segments on the chain properties. This fact may indicate that the description of a molecule as a chain of fused spheres would be more convenient to describe the behavior of real substances [22]. The diameter of the segments in a chain of n (number of carbons) segments that gives the same molecular volume generated by the obtained σ and m parameters, is defined here as an equivalent diameter σ ∗ , and can be expressed by: r π π m ∗ 3 3 ∗ n(σ ) = mσ ⇒ σ = σ 3 (14) 6 6 n As shown in Fig. 2(a), the obtained values of σ ∗ for the TPT1M equation of state are almost constant from methane to dodecane, showing that the model produces coherent values for molecular volume. For the TPT1 equation of state, the values of σ ∗ are not constant, suggesting a lower molecular modeling consistency in comparison with the TPT1M model. In Fig. 2(b), the mol/m ratio is plotted against chain size. As occurs for ε and σ parameters, the values obtained for the TPT1M model are constant from pentane to dodecane, while the values obtained for the TPT1 model are dependent on chain size.
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Fig. 2. TPT1 and TPT1M parameters for linear alkanes as a function of number of carbons. (a) Diameter (σ ∗ ) of each segment in a chain of n (number of carbons) segments that gives the same molecular volume generated by the obtained parameters. (b) mol/m ratio.
The parameters for linear alkanes, from methane to dodecane, are shown in Table 1. In this table, the root-mean-square relative deviations in vapor pressure (RMSRDP) are also presented. In general, the deviations generated by the TPT1 model are lower than those generated by the TPT1M model. This fact would be due to the low accuracy of the Barker and Henderson first order logarithmic expansion approximation, used to calculate the radial distribution function of square-well spheres (Eq. (10)) in a wide temperature range. Even though this term is not accurate in low temperatures and close to the critical point, this correction was able to give good results with temperature- and molecular weight-independent parameters from pentane to dodecane. The TPT1 and TPT1M models were derived for a chain of segments of the same type, a homonuclear chain. It would have been more correct to consider the ether and ester molecules as heteronuclear chains; nevertheless, the TPT1M model was applied to the ethers and esters without any additional consideration. Thus, the parameters obtained here were a result of the contributions of the different types of segments to the real molecule. The results of the root-mean-square relative deviations in vapor pressure (RMSRDP) and the obtained TPT1M parameters are reported in Table 2. In Fig. 3, the TPT1M model parameters for some esters are plotted against molecular weight. Similar results were obtained for ethers. The parameter σ (Fig. 3(b)) and the mol/m ratio (Fig. 3(d)) show a tendency to rise with an increase in molecular weight. Good results for the RMSRDP were obtained for these substances, except for dipropyl ether, dibutyl ether and ethyl-butyl ether. In this table, parameter λ for esters stays between 1.30 and 1.34 and ε/k stays above 400 K, except for methyl formate, ethyl formate and propyl formate. Although the molecular description adopted is not the best description for these substances, good results were obtained in correlating vapor pressures of pure substances. However, the TPT1M parameters obtained here are not molecular weight-independent. 3.2. Results for binary mixtures The results of prediction and correlation of vapor composition and pressure for some binary mixtures using the TPT1M equation of state are presented in Tables 3 and 4. For all cases studied, the phase equilibria coexistence condition was obtained by specifying temperature and liquid composition and
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Table 1 Pure alkane parameters for TPT1 and TPT1M equation of state obtained from fitting vapor pressure and critical point experimental data Substance
Model
ε/k (K)
σ (Å)
λ
m
RMSRDPa
Methane
TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1
172.5092 169.5855 313.4284 250.5443 384.0507 288.0665 427.4073 352.4030 454.3382 394.8549 456.2159 424.7874 455.0411 478.1280 456.6275 516.0807 460.0856 552.0281 456.7366 572.5572 454.3867 601.1574 454.6768 621.9875
4.2868 4.2562 5.2781 4.4856 5.9103 4.7298 6.2431 5.2758 6.4805 5.6236 6.5108 5.8648 6.4979 6.3563 6.5273 6.6917 6.5694 7.0012 6.5627 7.1635 6.5510 7.3970 6.5972 7.5938
1.3887 1.3909 1.3164 1.3574 1.2984 1.3468 1.2974 1.3184 1.2982 1.3017 1.3121 1.2914 1.3240 1.2716 1.3316 1.2585 1.3366 1.2468 1.3459 1.2416 1.3527 1.2330 1.3583 1.2284
1.0000 1.0128 1.0000 1.3150 1.0831 1.5244 1.2263 1.5242 1.4078 1.6098 1.6570 1.7236 1.9264 1.7035 2.1848 1.7480 2.4358 1.7941 2.6968 1.8767 2.9895 1.9365 3.2159 1.9946
0.47 0.47 0.92 0.50 1.03 0.52 0.97 0.57 1.23 0.72 1.25 0.66 1.28 0.66 1.73 0.78 2.36 1.01 2.57 1.03 4.14 2.20 3.90 1.67
Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Undecane n-Dodecane
RMSRDP: root-mean-square relative deviations in vapor pressure = 100 experimental points. a
q
Pnexp i=1
exp
exp 2
Pi − Pi /Pi
/nexp ; nexp : number of
calculating vapor composition and pressure using the TPT1M equation of state. The deviations presented in the tables do not include the pure component calculations. The TPT1M model for mixtures, Eq. (6), was used for some binary mixtures with no adjustable binary parameters (Table 3). In general, the TPT1M model predictions were in good agreement with the experimental data [31], except for the mixtures of heptane–ethylbutyrate, methyl acetate–ethyl acetate, methyl formate–hexane, dipropyl ether–octane and heptane–dibutyl ether. In Fig. 4, the TPT1M vapor composition and pressure predictions for the mixtures of pentane–hexane (Fig. 4(a)) and hexane–octane (Fig. 4(b)) are presented, showing the very good agreement of the TPT1M model calculations with the experimental data for these alkane mixtures. The binary interaction parameters φ 12 and ϕ 12 , Eq. (3), were obtained for some binary mixtures and are shown in Table 4. The strategy used to obtain these parameters was to minimize the root-mean-square absolute deviations in vapor composition (RMSADy) and the root-mean-square relative deviations in vapor pressure (RMSRDP) as the objective functions. In general, parameters φ 12 are one order greater than parameters ϕ 12 , except for the ethyl formate–methyl acetate mixture. An exploratory investigation showed that the calculation of the vapor–liquid equilibria is more sensitive to parameter φ 12 than to
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Table 2 TPT1M parameters obtained from fitting vapor pressure and critical point experimental data of some ethers, esters and benzene Substance
ε/k (K)
σ (Å)
λ
m
RMSRDPb
Ethers Dimethyl ether Diethyl ether Dipropyl ether Dibutyl ethera Methyl ethyl ether Methyl propyl ether Methyl butyl ether Ethyl butyl ether Diisopropyl ether
418.2824 415.2228 456.7472 443.7798 392.7807 456.5583 462.3876 447.4148 419.2838
5.6553 5.9236 6.2920 6.1719 5.5197 6.1949 6.2748 6.1585 6.2209
1.2867 1.3180 1.3134 1.3344 1.3229 1.2972 1.3079 1.3164 1.3269
1.1544 1.6347 1.9592 2.6371 1.5176 1.4723 1.7030 2.0835 1.9113
1.22 1.15 3.95 6.13 2.58 1.81 2.03 4.75 1.25
Esters Methyl formate Methyl acetate Methyl propionate Methyl butyrate Ethyl formate Ethyl acetate Ethyl propionate Ethyl butyrate Propyl formate Propyl acetate Propyl propionate Propyl butyrate Isobutyl formate Isobutyl acetate Isobutyl propionate Isobutyl butyrate
374.1712 451.8207 456.4575 437.8970 370.6363 419.2081 442.9431 445.8006 389.9736 432.6884 447.0451 451.7059 473.9284 452.9002 455.2652 458.4358
4.6569 5.5451 5.7612 5.7587 4.9207 5.5452 5.8647 6.0359 5.2228 5.7962 5.9961 6.1276 5.9548 6.0821 6.1498 6.3333
1.3664 1.3098 1.3166 1.3389 1.3781 1.3372 1.3283 1.3362 1.3751 1.3385 1.3362 1.3378 1.3176 1.3294 1.3328 1.3389
1.8799 1.7517 1.8988 2.1683 2.0679 2.0776 2.1634 2.3119 2.1664 2.2059 2.3205 2.5227 1.9253 2.2342 2.4425 2.5740
1.17 2.57 3.01 1.29 1.56 1.32 3.19 1.64 1.94 1.50 1.70 2.35 2.30 2.26 3.04 2.79
Benzene Benzene
536.3333
6.0643
1.3104
1.3396
1.29
a
0.624 < T r < 1.
RMSRDP: root-mean-square relative deviations in vapor pressure = 100 experimental points. b
qP
nexp i=1 (Pi
exp
exp
− Pi /Pi )2 /nexp ; nexp : number of
parameter ϕ 12 . For the ethyl formate–methyl acetate mixture, parameter φ 12 would be set equal to zero with a small variation in the correlated properties. The TPT1M vapor composition and pressure correlation and predictions for the heptane–ethyl butyrate and methyl acetate–ethyl acetate mixtures are presented in Fig. 4(c) and (d), respectively, showing the good agreement of the TPT1M correlation with the experimental data for these mixtures. For the hexane–benzene mixture, the binary parameters were determined at 60◦ C and were used to predict the properties of the mixture at 25, 55 and 70◦ C. This procedure gave good results for vapor pressure and reasonable results for vapor composition. At 55, 60 and 70◦ C, the predictions generated with φ ij and ϕ ij obtained at 60◦ C are in better agreement with the experimental data than the predictions with no binary parameters (φ 12 and ϕ 12 set equal to zero). At 25◦ C, the vapor composition predictions with
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239
Fig. 3. TPT1M parameters for esters as a function of molecular weight. (a) ε/k; (b) σ ; (c) λ; and (d) mol/m ratio. Table 3 Prediction of composition and pressure for some binary mixtures by using the TPT1M equation of state Mixture Pentane–hexane at 298.15 K Hexane–heptane at 303.15 K Hexane–octane at 328.15 K Hexane–benzene at 298.15 K Hexane–benzene at 328.15 K Hexane–benzene at 333.15 K Hexane–benzene at 343.15 K Heptane–ethylbutyrate at 373.15 K Methyl acetate–ethyl acetate at 313.15 K Methyl formate–hexane at 101.3 K Pa Ethyl formate–methyl acetate at 101.3 K Pa Diisopropyl ether–heptane at 101.3 K Pa Dipropyl ether–octane at 363.15 K Dipropyl ether–nonane at 363.15 K Heptane–dibutyl ether at 363.15 K
RMSADya
RMSRDPb
0.0031 1.39 0.0127 3.53 0.0062 1.15 0.0419 4.68 0.0425 2.17 0.0372 4.81 0.0307 4.63 0.0508 12.53 0.0483 9.38 0.2098 41.84 0.0078 1.74 0.0082 4.18 0.0190 9.99 0.0092 5.16 0.0150 12.30 qP nexp exp a RMSADy: root-mean-square absolute deviations in vapor composition = (yi − yi )2 /nexp . qP i=1 nexp exp exp 2 b RMSRDP: root-mean-square relative deviations in vapor pressure = 100 i=1 (Pi − Pi /Pi ) /nexp ; nexp : number of experimental points.
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Table 4 Correlation of composition and pressure for some binary mixtures by using the TPT1M equation of state Mixture
φ 12 × 102
ϕ 12 × 102
RMSADya
RMSRDPb
−0.1368 0.0219 5.72 −0.1368 0.0131 1.93 −0.1368 0.0297 1.63 −0.1368 0.0221 2.11 −0.0101 0.0114 1.77 −0.1755 0.0100 0.37 0.7175 0.0494 7.98 −1.3833 0.0502 0.96 qP nexp exp a RMSADy: root-mean-square absolute deviations in vapor composition = (yi − yi )2 /nexp . qP i=1 nexp exp exp 2 b RMSRDP: root-mean-square relative deviations in vapor pressure = 100 i=1 (Pi − Pi /Pi ) /nexp ; nexp : number of experimental points.
Hexane–benzene at 298.15 K Hexane–benzene at 328.15 K Hexane–benzene at 333.15 K Hexane–benzene at 343.15 K Heptane–ethylbutyrate at 373.15 K Methyl acetate–ethyl acetate at 313.15 K Methyl formate–hexane at 101.3 K Pa Ethyl formate–methyl acetate at 101.3 K Pa
1.4301 1.4301 1.4301 1.4301 2.6627 2.0901 10.1626 3.1581
Fig. 4. Experimental vapor–liquid equilibrium data and TPT1M prediction and correlation curves as a function of composition. (a) Pentane and hexane at 25◦ C. (b) Hexane and octane at 55◦ C. (c) Heptane and ethyl butyrate at 100◦ C. (d) Methyl acetate and ethyl acetate at 40◦ C.
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241
Fig. 5. Experimental hexane and benzene vapor–liquid equilibrium data and TPT1M prediction and correlation curves as a function of composition. The correlation curves are generated with the binary parameters obtained at 60◦ C.
φ 12 and ϕ 12 obtained at 60◦ C were in better agreement with the experimental data than the predictions with φ 12 and ϕ 12 set equal to zero, but for the pressure predictions the contrary was observed. This result indicates that these binary parameters are not temperature-independent, although they can be used in a relatively wide temperature range. The vapor composition and pressure correlation (φ 12 and ϕ 12 obtained at 60◦ C) and predictions (φ 12 and ϕ 12 set equal to zero) using the TPT1M equation of state for the hexane–benzene mixture are presented in Fig. 5. For several binary mixtures studied here, e.g. heptane–ethylbutyrate, methyl acetate–ethyl acetate and methyl formate–hexane, determining the binary parameters led to improved accuracy of the vapor pressure and composition calculations.
4. Conclusions An analytic perturbation theory equation of state for mixtures of square-well homonuclear chain fluids of variable well width was developed. This equation of state is an extension for chain mixtures of the previous work (Tavares et al.) and is based on Wertheim’s first-order thermodynamic perturbation theory to account for bond formation. The influence of the attractive part of the intersegment potential is evaluated using Barker and Henderson perturbation theory. The second-order perturbation theory is used to account for the influence of attractivity of square-well spheres and the first-order logarithmic expansion approximation
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is used to account for the influence of attractivity on the bond formation term. In this framework, the influence of attractivity on bond formation is accounted for in the TPT1M equation of state, but is not accounted for in the TPT1 equation of state. Both TPT1M and TPT1 equations of state were extended for mixtures by using the one-fluid type mixing rule. The TPT1M equation-of-state was used to correlate vapor–liquid equilibrium data and the critical points of some substances. For alkanes, the parameters obtained with the TPT1M equation showed to be constant after five carbons. The increase in the number of segments (m) per rise in number of carbons was less than one, leading to an overestimation of the chain length effect. This fact may indicate that the description of a molecule as a chain of fused spheres could be more convenient to describe the behavior of real substances. The TPT1M equation of state for mixtures was able to represent the vapor pressure and composition of some binary mixtures without binary adjustable parameters, especially for mixtures of alkanes. For some mixtures, two binary interaction parameters (φ ij and ϕ ij ) were obtained by fitting the calculated and experimental data. The results show that the TPT1M model, with two binary parameters, can be used in a relatively wide range of temperatures to predict vapor–liquid equilibrium data in a large range of pressure and composition. The model in this work represents a molecule as a homonuclear chain of tangent spheres and only takes into consideration dispersion forces (due to square-well interaction). This modeling does not attempt to account for the presence of polar segments. For this purpose the molecule would need to be represented as a heteronuclear chain. The results obtained indicate that the improvements in the TPT1 equation of state are leading towards an accurate description of real fluids on a thermodynamic statistic basis. List of symbols m number of segments n number of carbons nc number of components Greek symbols ε square-well depth (J) η pure component packing fraction λ square-well length σ hard-core diameter (Å) ξ3 mixture packing fraction Subscripts chain sphere chain mix mixture Superscripts R residual att attractive SWS square-well system HS hard sphere
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Acknowledgements The authors gratefully acknowledge the financial support of CAPES/Brazil, CNPq/Brazil, FAPERJ/ Brazil and PRONEX/Brazil. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
R. Dickman, C.K. Hall, J. Chem. Phys. 89 (1988) 3168. K.G. Honnel, C.K. Hall, J. Chem. Phys. 90 (1989) 1841. G. Jackson, W.G. Chapman, K.E. Gubbins, Mol. Phys. 65 (1988) 1. W.G. Chapman, G. Jackson, K.E. Gubbins, Mol. Phys. 65 (1988) 1057. M.S. Wertheim, J. Chem. Phys. 87 (1987) 7323. J. Chang, S.I. Sandler, Chem. Eng. Sci. 49 (1994) 2777. D. Ghonasgi, W.G. Chapman, J. Chem. Phys. 100 (1994) 6633. K.P. Shukla, W.G. Chapman, Mol. Phys. 93 (1998) 287. S.M. Lambert, Y. Song, J.M. Prausnitz, Macromolecules 28 (1995) 4866. Y. Song, T. Hino, S.M. Lambert, J.M. Prausnitz, Fluid Phase Equilibria 117 (1996) 69. G. Sadowski, Fluid Phase Equilibria 149 (1998) 75. R.L. Cotterman, B.J. Schwarz, J.M. Prausnitz, AIChE J. 32 (1986) 1787. M. Banaszak, Y.C. Chiew, R. O’Lenick, M. Radosz, J. Chem. Phys. 100 (1994) 3803. W.G. Chapman, K.E. Gubbins, G. Jackson, M. Radosz, Ind. Chem. Eng. Res. 29 (1990) 1709. A. Galindo, L.A. Davies, A. Gil-Villegas, G. Jackson, Mol. Phys. 93 (1998) 241. M. Banaszak, Y.C. Chiew, M. Radosz, Phys. Rev. E 48 (1993) 3760. S.-J. Chen, Y.C. Chiew, J.A. Gardecki, S. Nilsen, M. Radosz, J. Polym. Sci: Pt. B: Polym. Phys. 32 (1994) 1791. T. Hino, J.M. Prausnitz, Fluid Phase Equilibria 138 (1997) 105. F.W. Tavares, J. Chang, S.I. Sandler, Mol. Phys. 86 (1995) 1451. F.W. Tavares, J. Chang, S.I. Sandler, Fluid Phase Equilibria 140 (1997) 129. H.S. Gulati, C.K. Hall, J. Chem. Phys. 107 (1997) 3930. H. Adidharma, M. Radosz, Ind. Eng. Chem. Res. 37 (1998) 4453. J.A. Barker, D. Henderson, J. Chem. Phys. 47 (1967) 2856. J.A. Barker, D. Henderson, Rev. Mod. Phys. 48 (1976) 587. J. Chang, S. Sandler, Mol. Phys. 81 (1994) 735. J. Chang, S. Sandler, Mol. Phys. 81 (1994) 745. M.L.L. Paredes, R. Nobrega, F.W. Tavares, Fluid Phase Equilibria 179 (2000) 245–267. T. Boubl´ık, J. Chem. Phys. 53 (1970) 471. G.A. Mansoori, N.F. Carnahan, K.E. Starling, T.W. Leland, J. Chem. Phys. 54 (1971) 1523. R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, McGraw-Hill, New York, 1987. J. Gmelhing, U. Onken, W. Arlt, Vapor–liquid equilibrium data collection, DECHEMA Chemistry Data Series, Frankfurt, 1980.
An analytic equation-of-state for mixture of square-well chain fluids of variable well width Márcio L.L. Paredes a,1 , Ronaldo Nobrega a , Frederico W. Tavares b,∗ a
b
Programa de Engenharia Qu´ımica/COPPE, Universidade Federal do Rio de Janeiro, Caixa Postal: 68502, 21945-970 Rio de Janeiro, RJ, Brazil Escola de Qu´ımica, Universidade Federal do Rio de Janeiro, Caixa Postal: 68542, 21949-900 Rio de Janeiro, RJ, Brazil Received 20 October 2000; accepted 3 November 2000
Abstract An analytic perturbation theory equation of state for a mixture of freely-jointed square-well fluids of variable well width (1 ≤ λ ≤ 2) is developed. The equation of state is based on second-order Barker and Henderson perturbation theory to calculate the thermodynamic properties of the reference sphere fluid, and on first-order Wertheim thermodynamic perturbation theory to account for the connectivity of spheres to form chains. A real function expression for the radial distribution function of hard spheres and a one-fluid type mixing rule are used to obtain an analytic, closed-form expression, for the Helmohltz free energy of mixtures of square-well spheres. Good results were obtained when this equation of state was used with temperature-independent parameters to correlate vapor–liquid equilibrium data of pure substances and mixtures. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Equation of state; Mixture; Polymer; Square-well fluid
1. Introduction Many papers have been published describing molecules as a chain of spheres [1–20]. This molecular modeling can be useful for representing macromolecules as polymers and proteins, as well as for low molecular weight substances. Hall and co-workers [1,2] and Chapman and co-workers [3,4] have presented equations of state for chains of freely-jointed tangent spheres. The equation of state presented by Chapman and co-workers [3,4] is based on the Wertheim [5] theory to account for the connectivity of spheres, and is called the TPT1 equation of state. The Wertheim theory has been used to account for the Abbreviations: RMSADy, root-mean-square absolute deviations in vapor composition; RMSRDP, root-mean-square relative deviations in vapor pressure ∗ Corresponding author. Tel.: +55-21-562-7650; fax: +55-21-562-7567. E-mail addresses: [email protected] (R. Nobrega), [email protected] (F.W. Tavares). 1 Tel.: +55-21-590-7135; fax: +55-21-590-7134. 0378-3812/01/$20.00 © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 0 ) 0 0 5 0 4 - 5
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connectivity of spheres in forming chains in different ways [6–22]. Many studies have been published describing molecules as a chain of hard spheres [1,2,5–8] or a chain of attractive spheres interacting by the Lennard–Jones potential [12–15] and by the square-well potential [15–22]. The van der Waals approximation [3,4,9–11] and perturbation theories [12–22] are used in the literature to calculate the thermodynamic properties of the reference fluid (non-bonded segments) that include the attractive part. Efforts have been made in literature to account for the influence of the attractive part of the square-well potential on the bond formation term [16,19–22]. Banaszak et al. [16] have applied first-order perturbation Barker and Henderson theory to the radial distribution function of square-well spheres [23,24] to account for attractivity in the Wertheim theory, for a square-well width λ = 1.5. Galindo et al. [15] have proposed an equation of state for square-well chain mixtures, taking into account the effect of attractivity in bond formation, using the mean value theorem to evaluate Barker and Henderson’s integrals. Adidharma and Radosz [22] have presented an equation of state for mixtures of copolymers based on the Wertheim theory and on the work of Galindo et al. [15]. Tavares et al. [20] have extended the Banaszak et al. [16] equation of state to variable square-well width using Barker and Henderson’s logarithmic expansion approximation (LEA) [23,24]. Though using only a first-order expansion function, the introduction of this term generates a more consistent model for attractive chains. The square-well chain properties calculated with this equation of state are in good agreement with Monte Carlo simulation data of Tavares et al. [19], and in better agreement with simulation than the analogous equation of state that does not take into account the effect of attractivity in bond formation. In this equation of state, Barker and Henderson perturbation theory was evaluated using Chang and Sandler’s hard-sphere radial distribution function expression [25,26]. Gulati and Hall [21] and Hino and Prausnitz [18] have applied the one-fluid type mixing rule to mixtures of square-well chains, using the Barker and Henderson [23,24] perturbation theory, along with the Chang and Sandler hard sphere radial distribution function [25,26]. In the present work, an analytic equation of state for freely-jointed square-well mixtures of variable well width was proposed. The importance of taking into account the effect of attractivity in bond formation was also confirmed for mixtures by comparing the equation of state results with Monte Carlo simulation data [27]. The equation of state parameters were obtained for a large set of pure substances. Also, two binary adjustable parameters were determined for some mixtures. The equation of state was found to be very sensitive to the parameters. This sensitivity came to be useful for correlating and predicting the thermodynamic properties of real mixtures.
2. The equation-of-state for mixtures In this section, the analytic equation of state for mixtures of square-well chains is presented. The interaction potential Φ ij between two non-adjacent spheres i and j at distance rij is defined as follows: rij < σij ∞, Φij = −εij , σij ≤ rij < λij σij (1) 0, rij ≥ λij σij where σ ij is defined in Eq. (2) and εij and λij are defined in Eq. (3), following standard combining rules. σij =
σi + σj 2
(2)
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p √ εij = εi εj (1 − φij ) and λij = λi λj (1 − ϕij )
233
(3)
where, for component i, σ i is the segment diameter, εi the square-well depth and λi is the square-well width; φ ij and ϕ ij are binary adjustable parameters, set equal to zero when i = j . The one-fluid type mixing rule is used in this work in a very similar way to that of Hino and Prausnitz [18] and Gulati and Hall [21]. It can be expressed by the average quantities of the equation of state parameters kept within the parentheses in the following equation:
m σ f (ξ3 ; ε, λ) = 2
3
nc X nc X
xi xj mi mj σij3 f (ξ3 ; εij , λij )
(4)
i=1 j =1
where nc is the number of components, i and j denote the component indexes, x is the mole fraction and f(ξ 3 ; εij , λij ) denotes an arbitrary function of mixture packing fraction (ξ 3 ), εij and λij . The mixture packing fraction is defined as follows: ξk =
nc π X ρ xj mj σjk where k = 0, 1, 2, 3 6 j =1
(5)
The proposed equation of state for square-well chain mixtures is Zchain = 1 + (mZ
R,HS
)mix + (mZ )mix − att
nc X
xi (mi − 1)ξ3
i=1
∂ ln giiSWS ∂ξ3
where according to Boubl´ık [28] and Mansoori et al. [29] ξ23 (3 − ξ3 ) 6 3ξ1 ξ2 ξ0 ξ3 + + (mZR,HS )mix = πρ 1 − ξ3 (1 − ξ3 )2 (1 − ξ3 )3
(6)
(7)
By applying the mixing rule (Eq. (2)) to calculate the attractive contribution to the compressibility factor, the following equation is obtained: nc X nc ε 2 ξ (1 − ξ )4 ∂lij εij π X ij 3 3 3 xi xj mi mj σij 12 lij + ξ3 (mZ )mix = − ρ +6 6 i=1 j =1 kT ∂ξ3 kT (1 + 2ξ3 )2 (1 − ξ3 )3 (1 − 5ξ3 − 20ξ32 − 12ξ33 ) ∂lij ∂ 2 lij ∂lij × 2 + ξ3 2 + lij + ξ3 ∂ξ3 (1 + 2ξ3 )4 ∂ξ3 ∂ξ3 att
(8)
where lij = l(ξ3 ; λij )
(9)
and l is a integral function [20,26]. The radial distribution function at the contact point of square-well spheres giiSWS is calculated by ln giiSWS = ln giiHS +
εii g1,ii kT giiHS
(10)
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where the perturbation term g1,ii is expressed in Eq. (11). ∂lij + λ3ii g HS (λσ + ) g1,ii = −3 lij + ξ3 ∂ξ3
(11)
In Eq. (10), the Boubl´ık [28] and Mansoori et al. [29] radial distribution function of hard spheres giiHS is used, as follows: giiHS =
1 3 ξ2 σii 1 (ξ2 σii )2 + + 1 − ξ3 2 (1 − ξ3 )2 2 (1 − ξ3 )3
(12)
By substituting Eqs. (7), (8) and (10) into Eq. (6), the TPT1M model for square-well chain mixtures is obtained. The TPT1 model for square-well chain mixtures used in this work is obtained by setting g1,ii equal to zero in the TPT1M model, leading to ln giiSWS ← ln giiHS
(13)
Using the hard sphere radial distribution function at the contact point to calculate attractive sphere chain properties, information about attractiveness is missed in the TPT1 bond formation term. For this reason, the main advantage of the TPT1M equation of state for mixtures of square-well chain fluids is that it takes into account the influence of attractivity on the bond formation term.
3. Results 3.1. Pure component parameters In this section the parameters of the TPT1 and TPT1M equations of state are obtained for some pure normal alkanes, ethers and esters. The models present four parameters: ε and λ are related to intermolecular attractions and σ and m are related to molecular size and shape. These parameters are determined so that the equations of state give the experimental critical point [30] and the minimum relative mean square deviations of the smoothed vapor pressures [30] over temperature range of 0.6 ≤ T /T c ≤ 1.0. The fitting procedure was carried out in two loops: in the inner loop, parameters ε and σ were used to give the experimental critical point, and in the outer loop, parameters λ and m were determined by fitting the vapor pressure. The parameters obtained for alkanes are plotted against the number of carbons while the parameters for esters are plotted against the molecular weight (mol) of the substances. Since the TPT1 and TPT1M models attempt to represent chain molecules, the parameters related to the segment are expected to be independent of molecular weight. Parameter m (number of segments) is expected to vary linearly with molecular weight. Since the properties of the segments should not depend on molecular weight, the mol/m ratio is expected to be independent of molecular weight. In Fig. 1, the parameters for linear alkanes, from methane to dodecane, are plotted against the number of carbons, for the TPT1 and TPT1M equations of state. In Fig. 1(a) and (b), the TPT1M parameters (ε and σ ) are almost constant from pentane to dodecane, as expected, while the TPT1 parameters are dependent on chain size. In Fig. 1(c), the TPT1 parameter (λ) decreases with chain size, while the TPT1M parameters decrease from methane to butane and increase from pentane to dodecane. This result, leading
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Fig. 1. TPT1 and TPT1M parameters for linear alkanes as a function of number of carbons. (a) ε/k; (b) σ ; (c) λ; and (d) m.
to the TPT1M λ parameters increasing with chain size for alkanes greater than pentane, is not expected. In Fig. 1(d), the TPT1M parameter (m) behaves as a linear function of the chain size from pentane to dodecane, as expected, and the TPT1 parameter behaves as an almost linear function of the chain size. The slopes of the curves in Fig. 1(d) indicate that, when hydrocarbon molecules are represented as chains of tangent attractive hard spheres, a lower number of segments is obtained than the number of carbon atoms in the hydrocarbon molecule. The consequence of underestimating the number of segments in the chain is an overestimation of the effect of the number of segments on the chain properties. This fact may indicate that the description of a molecule as a chain of fused spheres would be more convenient to describe the behavior of real substances [22]. The diameter of the segments in a chain of n (number of carbons) segments that gives the same molecular volume generated by the obtained σ and m parameters, is defined here as an equivalent diameter σ ∗ , and can be expressed by: r π π m ∗ 3 3 ∗ n(σ ) = mσ ⇒ σ = σ 3 (14) 6 6 n As shown in Fig. 2(a), the obtained values of σ ∗ for the TPT1M equation of state are almost constant from methane to dodecane, showing that the model produces coherent values for molecular volume. For the TPT1 equation of state, the values of σ ∗ are not constant, suggesting a lower molecular modeling consistency in comparison with the TPT1M model. In Fig. 2(b), the mol/m ratio is plotted against chain size. As occurs for ε and σ parameters, the values obtained for the TPT1M model are constant from pentane to dodecane, while the values obtained for the TPT1 model are dependent on chain size.
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Fig. 2. TPT1 and TPT1M parameters for linear alkanes as a function of number of carbons. (a) Diameter (σ ∗ ) of each segment in a chain of n (number of carbons) segments that gives the same molecular volume generated by the obtained parameters. (b) mol/m ratio.
The parameters for linear alkanes, from methane to dodecane, are shown in Table 1. In this table, the root-mean-square relative deviations in vapor pressure (RMSRDP) are also presented. In general, the deviations generated by the TPT1 model are lower than those generated by the TPT1M model. This fact would be due to the low accuracy of the Barker and Henderson first order logarithmic expansion approximation, used to calculate the radial distribution function of square-well spheres (Eq. (10)) in a wide temperature range. Even though this term is not accurate in low temperatures and close to the critical point, this correction was able to give good results with temperature- and molecular weight-independent parameters from pentane to dodecane. The TPT1 and TPT1M models were derived for a chain of segments of the same type, a homonuclear chain. It would have been more correct to consider the ether and ester molecules as heteronuclear chains; nevertheless, the TPT1M model was applied to the ethers and esters without any additional consideration. Thus, the parameters obtained here were a result of the contributions of the different types of segments to the real molecule. The results of the root-mean-square relative deviations in vapor pressure (RMSRDP) and the obtained TPT1M parameters are reported in Table 2. In Fig. 3, the TPT1M model parameters for some esters are plotted against molecular weight. Similar results were obtained for ethers. The parameter σ (Fig. 3(b)) and the mol/m ratio (Fig. 3(d)) show a tendency to rise with an increase in molecular weight. Good results for the RMSRDP were obtained for these substances, except for dipropyl ether, dibutyl ether and ethyl-butyl ether. In this table, parameter λ for esters stays between 1.30 and 1.34 and ε/k stays above 400 K, except for methyl formate, ethyl formate and propyl formate. Although the molecular description adopted is not the best description for these substances, good results were obtained in correlating vapor pressures of pure substances. However, the TPT1M parameters obtained here are not molecular weight-independent. 3.2. Results for binary mixtures The results of prediction and correlation of vapor composition and pressure for some binary mixtures using the TPT1M equation of state are presented in Tables 3 and 4. For all cases studied, the phase equilibria coexistence condition was obtained by specifying temperature and liquid composition and
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Table 1 Pure alkane parameters for TPT1 and TPT1M equation of state obtained from fitting vapor pressure and critical point experimental data Substance
Model
ε/k (K)
σ (Å)
λ
m
RMSRDPa
Methane
TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1 TPT1M TPT1
172.5092 169.5855 313.4284 250.5443 384.0507 288.0665 427.4073 352.4030 454.3382 394.8549 456.2159 424.7874 455.0411 478.1280 456.6275 516.0807 460.0856 552.0281 456.7366 572.5572 454.3867 601.1574 454.6768 621.9875
4.2868 4.2562 5.2781 4.4856 5.9103 4.7298 6.2431 5.2758 6.4805 5.6236 6.5108 5.8648 6.4979 6.3563 6.5273 6.6917 6.5694 7.0012 6.5627 7.1635 6.5510 7.3970 6.5972 7.5938
1.3887 1.3909 1.3164 1.3574 1.2984 1.3468 1.2974 1.3184 1.2982 1.3017 1.3121 1.2914 1.3240 1.2716 1.3316 1.2585 1.3366 1.2468 1.3459 1.2416 1.3527 1.2330 1.3583 1.2284
1.0000 1.0128 1.0000 1.3150 1.0831 1.5244 1.2263 1.5242 1.4078 1.6098 1.6570 1.7236 1.9264 1.7035 2.1848 1.7480 2.4358 1.7941 2.6968 1.8767 2.9895 1.9365 3.2159 1.9946
0.47 0.47 0.92 0.50 1.03 0.52 0.97 0.57 1.23 0.72 1.25 0.66 1.28 0.66 1.73 0.78 2.36 1.01 2.57 1.03 4.14 2.20 3.90 1.67
Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Undecane n-Dodecane
RMSRDP: root-mean-square relative deviations in vapor pressure = 100 experimental points. a
q
Pnexp i=1
exp
exp 2
Pi − Pi /Pi
/nexp ; nexp : number of
calculating vapor composition and pressure using the TPT1M equation of state. The deviations presented in the tables do not include the pure component calculations. The TPT1M model for mixtures, Eq. (6), was used for some binary mixtures with no adjustable binary parameters (Table 3). In general, the TPT1M model predictions were in good agreement with the experimental data [31], except for the mixtures of heptane–ethylbutyrate, methyl acetate–ethyl acetate, methyl formate–hexane, dipropyl ether–octane and heptane–dibutyl ether. In Fig. 4, the TPT1M vapor composition and pressure predictions for the mixtures of pentane–hexane (Fig. 4(a)) and hexane–octane (Fig. 4(b)) are presented, showing the very good agreement of the TPT1M model calculations with the experimental data for these alkane mixtures. The binary interaction parameters φ 12 and ϕ 12 , Eq. (3), were obtained for some binary mixtures and are shown in Table 4. The strategy used to obtain these parameters was to minimize the root-mean-square absolute deviations in vapor composition (RMSADy) and the root-mean-square relative deviations in vapor pressure (RMSRDP) as the objective functions. In general, parameters φ 12 are one order greater than parameters ϕ 12 , except for the ethyl formate–methyl acetate mixture. An exploratory investigation showed that the calculation of the vapor–liquid equilibria is more sensitive to parameter φ 12 than to
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Table 2 TPT1M parameters obtained from fitting vapor pressure and critical point experimental data of some ethers, esters and benzene Substance
ε/k (K)
σ (Å)
λ
m
RMSRDPb
Ethers Dimethyl ether Diethyl ether Dipropyl ether Dibutyl ethera Methyl ethyl ether Methyl propyl ether Methyl butyl ether Ethyl butyl ether Diisopropyl ether
418.2824 415.2228 456.7472 443.7798 392.7807 456.5583 462.3876 447.4148 419.2838
5.6553 5.9236 6.2920 6.1719 5.5197 6.1949 6.2748 6.1585 6.2209
1.2867 1.3180 1.3134 1.3344 1.3229 1.2972 1.3079 1.3164 1.3269
1.1544 1.6347 1.9592 2.6371 1.5176 1.4723 1.7030 2.0835 1.9113
1.22 1.15 3.95 6.13 2.58 1.81 2.03 4.75 1.25
Esters Methyl formate Methyl acetate Methyl propionate Methyl butyrate Ethyl formate Ethyl acetate Ethyl propionate Ethyl butyrate Propyl formate Propyl acetate Propyl propionate Propyl butyrate Isobutyl formate Isobutyl acetate Isobutyl propionate Isobutyl butyrate
374.1712 451.8207 456.4575 437.8970 370.6363 419.2081 442.9431 445.8006 389.9736 432.6884 447.0451 451.7059 473.9284 452.9002 455.2652 458.4358
4.6569 5.5451 5.7612 5.7587 4.9207 5.5452 5.8647 6.0359 5.2228 5.7962 5.9961 6.1276 5.9548 6.0821 6.1498 6.3333
1.3664 1.3098 1.3166 1.3389 1.3781 1.3372 1.3283 1.3362 1.3751 1.3385 1.3362 1.3378 1.3176 1.3294 1.3328 1.3389
1.8799 1.7517 1.8988 2.1683 2.0679 2.0776 2.1634 2.3119 2.1664 2.2059 2.3205 2.5227 1.9253 2.2342 2.4425 2.5740
1.17 2.57 3.01 1.29 1.56 1.32 3.19 1.64 1.94 1.50 1.70 2.35 2.30 2.26 3.04 2.79
Benzene Benzene
536.3333
6.0643
1.3104
1.3396
1.29
a
0.624 < T r < 1.
RMSRDP: root-mean-square relative deviations in vapor pressure = 100 experimental points. b
qP
nexp i=1 (Pi
exp
exp
− Pi /Pi )2 /nexp ; nexp : number of
parameter ϕ 12 . For the ethyl formate–methyl acetate mixture, parameter φ 12 would be set equal to zero with a small variation in the correlated properties. The TPT1M vapor composition and pressure correlation and predictions for the heptane–ethyl butyrate and methyl acetate–ethyl acetate mixtures are presented in Fig. 4(c) and (d), respectively, showing the good agreement of the TPT1M correlation with the experimental data for these mixtures. For the hexane–benzene mixture, the binary parameters were determined at 60◦ C and were used to predict the properties of the mixture at 25, 55 and 70◦ C. This procedure gave good results for vapor pressure and reasonable results for vapor composition. At 55, 60 and 70◦ C, the predictions generated with φ ij and ϕ ij obtained at 60◦ C are in better agreement with the experimental data than the predictions with no binary parameters (φ 12 and ϕ 12 set equal to zero). At 25◦ C, the vapor composition predictions with
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239
Fig. 3. TPT1M parameters for esters as a function of molecular weight. (a) ε/k; (b) σ ; (c) λ; and (d) mol/m ratio. Table 3 Prediction of composition and pressure for some binary mixtures by using the TPT1M equation of state Mixture Pentane–hexane at 298.15 K Hexane–heptane at 303.15 K Hexane–octane at 328.15 K Hexane–benzene at 298.15 K Hexane–benzene at 328.15 K Hexane–benzene at 333.15 K Hexane–benzene at 343.15 K Heptane–ethylbutyrate at 373.15 K Methyl acetate–ethyl acetate at 313.15 K Methyl formate–hexane at 101.3 K Pa Ethyl formate–methyl acetate at 101.3 K Pa Diisopropyl ether–heptane at 101.3 K Pa Dipropyl ether–octane at 363.15 K Dipropyl ether–nonane at 363.15 K Heptane–dibutyl ether at 363.15 K
RMSADya
RMSRDPb
0.0031 1.39 0.0127 3.53 0.0062 1.15 0.0419 4.68 0.0425 2.17 0.0372 4.81 0.0307 4.63 0.0508 12.53 0.0483 9.38 0.2098 41.84 0.0078 1.74 0.0082 4.18 0.0190 9.99 0.0092 5.16 0.0150 12.30 qP nexp exp a RMSADy: root-mean-square absolute deviations in vapor composition = (yi − yi )2 /nexp . qP i=1 nexp exp exp 2 b RMSRDP: root-mean-square relative deviations in vapor pressure = 100 i=1 (Pi − Pi /Pi ) /nexp ; nexp : number of experimental points.
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Table 4 Correlation of composition and pressure for some binary mixtures by using the TPT1M equation of state Mixture
φ 12 × 102
ϕ 12 × 102
RMSADya
RMSRDPb
−0.1368 0.0219 5.72 −0.1368 0.0131 1.93 −0.1368 0.0297 1.63 −0.1368 0.0221 2.11 −0.0101 0.0114 1.77 −0.1755 0.0100 0.37 0.7175 0.0494 7.98 −1.3833 0.0502 0.96 qP nexp exp a RMSADy: root-mean-square absolute deviations in vapor composition = (yi − yi )2 /nexp . qP i=1 nexp exp exp 2 b RMSRDP: root-mean-square relative deviations in vapor pressure = 100 i=1 (Pi − Pi /Pi ) /nexp ; nexp : number of experimental points.
Hexane–benzene at 298.15 K Hexane–benzene at 328.15 K Hexane–benzene at 333.15 K Hexane–benzene at 343.15 K Heptane–ethylbutyrate at 373.15 K Methyl acetate–ethyl acetate at 313.15 K Methyl formate–hexane at 101.3 K Pa Ethyl formate–methyl acetate at 101.3 K Pa
1.4301 1.4301 1.4301 1.4301 2.6627 2.0901 10.1626 3.1581
Fig. 4. Experimental vapor–liquid equilibrium data and TPT1M prediction and correlation curves as a function of composition. (a) Pentane and hexane at 25◦ C. (b) Hexane and octane at 55◦ C. (c) Heptane and ethyl butyrate at 100◦ C. (d) Methyl acetate and ethyl acetate at 40◦ C.
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Fig. 5. Experimental hexane and benzene vapor–liquid equilibrium data and TPT1M prediction and correlation curves as a function of composition. The correlation curves are generated with the binary parameters obtained at 60◦ C.
φ 12 and ϕ 12 obtained at 60◦ C were in better agreement with the experimental data than the predictions with φ 12 and ϕ 12 set equal to zero, but for the pressure predictions the contrary was observed. This result indicates that these binary parameters are not temperature-independent, although they can be used in a relatively wide temperature range. The vapor composition and pressure correlation (φ 12 and ϕ 12 obtained at 60◦ C) and predictions (φ 12 and ϕ 12 set equal to zero) using the TPT1M equation of state for the hexane–benzene mixture are presented in Fig. 5. For several binary mixtures studied here, e.g. heptane–ethylbutyrate, methyl acetate–ethyl acetate and methyl formate–hexane, determining the binary parameters led to improved accuracy of the vapor pressure and composition calculations.
4. Conclusions An analytic perturbation theory equation of state for mixtures of square-well homonuclear chain fluids of variable well width was developed. This equation of state is an extension for chain mixtures of the previous work (Tavares et al.) and is based on Wertheim’s first-order thermodynamic perturbation theory to account for bond formation. The influence of the attractive part of the intersegment potential is evaluated using Barker and Henderson perturbation theory. The second-order perturbation theory is used to account for the influence of attractivity of square-well spheres and the first-order logarithmic expansion approximation
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is used to account for the influence of attractivity on the bond formation term. In this framework, the influence of attractivity on bond formation is accounted for in the TPT1M equation of state, but is not accounted for in the TPT1 equation of state. Both TPT1M and TPT1 equations of state were extended for mixtures by using the one-fluid type mixing rule. The TPT1M equation-of-state was used to correlate vapor–liquid equilibrium data and the critical points of some substances. For alkanes, the parameters obtained with the TPT1M equation showed to be constant after five carbons. The increase in the number of segments (m) per rise in number of carbons was less than one, leading to an overestimation of the chain length effect. This fact may indicate that the description of a molecule as a chain of fused spheres could be more convenient to describe the behavior of real substances. The TPT1M equation of state for mixtures was able to represent the vapor pressure and composition of some binary mixtures without binary adjustable parameters, especially for mixtures of alkanes. For some mixtures, two binary interaction parameters (φ ij and ϕ ij ) were obtained by fitting the calculated and experimental data. The results show that the TPT1M model, with two binary parameters, can be used in a relatively wide range of temperatures to predict vapor–liquid equilibrium data in a large range of pressure and composition. The model in this work represents a molecule as a homonuclear chain of tangent spheres and only takes into consideration dispersion forces (due to square-well interaction). This modeling does not attempt to account for the presence of polar segments. For this purpose the molecule would need to be represented as a heteronuclear chain. The results obtained indicate that the improvements in the TPT1 equation of state are leading towards an accurate description of real fluids on a thermodynamic statistic basis. List of symbols m number of segments n number of carbons nc number of components Greek symbols ε square-well depth (J) η pure component packing fraction λ square-well length σ hard-core diameter (Å) ξ3 mixture packing fraction Subscripts chain sphere chain mix mixture Superscripts R residual att attractive SWS square-well system HS hard sphere
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Acknowledgements The authors gratefully acknowledge the financial support of CAPES/Brazil, CNPq/Brazil, FAPERJ/ Brazil and PRONEX/Brazil. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
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