Ind. Eng. Chem. Res. 2002, 41, 871-881
871
Mixing Molar Volume of Liquid n- and i-Alkanes Based on the Law of Corresponding States with Potential between Functional Groups Yutaka Tada,* Setsuro Hiraoka, Naoki Nishida, Naoya Tanuma, and Young-Sei Lee† Department of Applied Chemistry, Nagoya Institute of Technology, Nagoya 466, Japan
The excess Helmholtz free energy of a liquid alkane mixture is expanded around that of a reference liquid, the pair potential of which is the sum of a hard-sphere potential and an attractive potential, with the difference between the potentials. The characteristic length of the mixture is chosen such that the first-order perturbation term vanishes. The molar volume of the mixture reduced by the characteristic length is expressed by the sum of those of the reference and the second-order perturbation term in the corresponding states. The equation of the mixing molar volume is expressed by the corresponding states equations of the mixture and the pure components with the characteristic length and energy without any binary interaction parameters and reproduces the observed mixing molar volume well. The characteristic parameters are expressed by the parameters of the potential between the functional groups in the molecules, which are not fluid specific. 1. Introduction Liquid alkane mixtures are often used in many chemical industries, the thermodynamic properties of which are very important for chemical process design. Although it is desirable to use the observed properties, the properties of all of the alkane mixtures for the various compositions and operating conditions are not available. If we have the equations of the mixing properties, we can estimate the thermodynamic properties of the mixtures with those of the pure components. The mixing molar volumes for the liquid alkanes, which is one of the most fundamental properties, were measured and correlated well with polynomial-type equations or theoretical equations based on the VanPatterson theory1 or the Flory theory.2,3 However, the coefficients in the polynomial equations, the interaction parameter in the Van-Patterson theory X12, and the interchange energy parameter in the Flory theory X12 are system specific. Thus, for the mixture for which the experimental mixing molar volume is not available, these coefficients and the parameters cannot be evaluated and the mixing molar volume cannot be estimated with just the use of the molar volume of the pure components. The aim of this paper is to derive the estimation equation for the mixing molar volume of the liquid alkanes including the isomers without the use of system-specific interaction parameters based on the law of corresponding states. The excess Helmholtz free energy of a liquid alkane mixture of interest is expanded around a reference liquid, the pair potential of which is the sum of a hardsphere potential and an attractive potential, with the difference between the potentials. The characteristic length of the mixture is chosen such that the first-order perturbation term vanishes. The characteristic length and energy are expressed as simple mixing rules of those of the pure components, which are given in terms of the parameters of the potentials between the func* Corresponding author. Phone: 81-52-735-5231. Fax: 8152-735-5255. E-mail:
[email protected]. † Current address: Department of Chemical Engineering, Sangju National University, Kyung-Buk 742-711, Korea.
tional groups in the molecules (the intergroup potential parameters). The molar volume of the mixture reduced by the characteristic length is expressed by the sum of those of the reference and the second-order perturbation term in the corresponding states. The equation of the mixing molar volume is derived from the corresponding state equations of the mixture and the pure components with the characteristic parameters without any binary interaction parameters. The calculated mixing molar volume is compared to the observed ones and to the calculated ones from the equations with the use of the intermolecular potential parameters determined independently without the use of the intergroup potential parameters. 2. Perturbation Expansion of Thermodynamic Properties of Mixtures We consider a mixture of alkanes. We presume that the mixture has Lennard-Jones (12-6) intermolecular pair potential v for molecules i and j, which is represented by the sum of intergroup potentials between functional groups l and m.
vij(r) ) 4ij
[( ) ( ) ] σij r
12
-
σij
6
)
r
4lm ∑l ∑ m
[( ) ( )] σlm
12
-
rlm
σlm rlm
6
(1)
The sum is taken over all of the functional groups l and m in the molecules i and j, respectively. ij and σij are the intermolecular potential parameters, and r is the separation distance between the molecules i and j. lm and σlm are the intergroup potential parameters, and rlm is the separation distance between the functional groups l and m. When the value of the intermolecular potential at a characteristic length dij between the molecules i and j is equated to that of the sum of the intergroup potentials, ij and σij are expressed by eqs 2 and 3 with lm, σlm, dij, and distance correction parameters ∆lm (see Appendix A). ∆lm depends on the location N and the number n of the groups l and m (see Appendix
10.1021/ie0103916 CCC: $22.00 © 2002 American Chemical Society Published on Web 01/22/2002
872
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002
Table 1. Intergroup Potential Parameters Elm and σlm for n- and i-Alkanes lm × 1022 [J]
pair of groups l-m
σlm × 1010 [m]
n-Alkanes 17.737 7.1867 1.0580 42.810 0.063597
3.9844 4.5697 5.2574 3.7542 8.2883
i-Alkanes CH3-CH3 1.2377 44.000 CH3-CH2, CH3-CH CH2-CH2, CH2-CH, CH-CH 0.073800
5.7364 3.6648 8.6892
CH4-CH4(methane) CH3-CH3(ethane) CH3-CH3 CH3-CH2 CH2-CH2
B). lm, σlm, and ∆lm were evaluated such that the molar volume and the vapor pressure of the pure liquid alkanes were correlated in the corresponding states (see Appendix C), the values of which are shown in Tables 1 and 2. The molar volume and the vapor pressure were correlated well with the root square mean deviations (RMSDs) 1.86% and 2.53% (in log p), respectively.
[∑∑ ( ) ] ∑∑ ( ) ∑∑ ( ) ∑∑ ( ) σlm
lm
l
ij )
lm
σlm
12
[ ] l
m
lm
l
σij )
6 2
dij + ∆lm
m
m
lm
l
m
12 1/6
dij + ∆lm σlm
6
(3)
dij + ∆lm
The intermolecular potential vij is separated into a repulsive part vij0 and an attractive part wij by the WCA method.4 We select a reference liquid, the intermolecular potential of which vref is the sum of a hard-sphere potential vhs and a reference attractive potential w. 0
vij(r) ) vij (r) + wij(r)
(4)
vij0(r) ) vij(r) + ij, r e rm,ij ) 21/6σij ) 0, r > rm,ij
n-Alkane N
∆CH3(N)-CH3(N) × 1010 [m]
N
∆CH3(N)-CH2(n) × 1010 [m]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
0 0 -0.24517 -0.41849 -1.2947 -1.6261 -1.9540 -2.2014 -2.4341 -2.6813 -2.8192 -2.9858 -3.1574 -3.2863 -3.3969 -3.5235 -3.6330 -3.7354 -3.8261 -3.9268 -4.0197
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
0 0 0.051 675 0.005 502 9 0.092 679 -0.031 489 -0.158 52 -0.294 75 -0.430 27 -0.612 61 -0.687 32 -0.804 74 -0.949 33 -1.043 9 -1.115 6 -1.225 8 -1.316 0 -1.400 7 -1.463 4 -1.569 8 -1.667 2
N-N′
∆CH3(N)-CH3(N′) × 1010 [m]
N-n
∆CH3(N)-CH3(n) × 1010 [m]
1-1 1.5-1.5 2-2 2.5-2.5 3-3 3.5-3.5 4-4 1-2 1.5-2.5 2-3 2.5-3.5 3-4 1-3 1.5-3.5 2-4 1-4
-0.22916 -0.65751 -1.0597 -1.3690 -1.6498 -1.9003 -2.1235 -1.1085 -1.3939 -1.6921 -1.9526 -2.1586 -1.7075 -1.9611 -2.1854 -2.1854
1-1 1.5-2 2-3 2.5-4 3-5 3.5-6 4-7 1-3 1.5-4 2-5 2.5-6 3-7 1-5 1.5-6 2-7 1-7
0.140 77 0.329 29 0.285 09 0.180 75 0.045 276 -0.102 81 -0.239 68 0.234 29 0.197 21 0.031 083 -0.110 39 -0.232 94 -0.010 576 -0.149 24 -0.269 51 -0.305 2
vref(r) ) vhs(r) + w(r)
(5)
2
d3 ) (6)
) 0, r > d
ref
) v (r), r > rm
(8)
Dd ≡
∆CH2(n)-CH2(n) × 1010 [m]
1 2 3 4 5 6 7
0.48803 0.10603 2.1421 1.8671 2.4598 2.9119 2.9237
2
(10) (11)
{[( ) ][( ) ] [( ) ][( ) ] [( ) ][( ) ]} ∑i,j,k)1 ∑∑
xixjxk
dij
We make a perturbation expansion of the excess Helmholtz free energy of the mixture around that of the reference liquid in terms of the difference between the potentials. When we choose the hard-sphere diameter of the reference liquid d given by eq 10 as the charac-
n
xixjdij3 ∑ ∑ i)1 j)1
dij
2
(9)
0 0.099 815 0.013 119 0.394 47 0.316 65 0.379 81 0.279 28 0.250 35 0.321 73 0.124 92 0.108 82 0.264 98 0.156 08 -0.165 46 -0.074 273 -0.119 58 -0.154 90 -0.367 18 -0.227 00 -0.154 62
ref
where
w(r) ) -, r e rm ) 21/6σ
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
ˆ ex ) A ˆ ex + C ˆ A(T ˆ ) Dd βAex ≡ A
(7)
vhs(r) ) ∞, r e d
∆CH2(n)-CH2(n) × 1010 [m]
teristic length of the mixture and when the perturbation terms higher than the second order are neglected, the first-order perturbation term vanishes and the excess Helmholtz free energy is expressed by eq 11 (see Appendix D).
wij(r) ) -ij, r e rm,ij ) vij(r), r > rm,ij
n
i-Alkane
(2)
dij + ∆lm σlm
Table 2. Distance Correction Parameter ∆lm for n- and i-Alkanes
d
3
-1
djk d
d
dik
3
-1
3
-1 +
d
dik d
3
-1 +
3
-1
djk d
3
-1
(12)
T ˆ ≡ kT/
(13)
A ˆ exref is the reduced excess Helmholtz free energy for the reference liquid. dij is the characteristic length between molecules i and j. Dd is the parameter of the difference
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002 873
between the characteristic lengths dij. The characteristic length between like molecules dii is given by eq 14 (see Appendix E), and that between unlike molecules dij (i * j) is given by a combining rule in the later section.
Table 3. RMSDs (×108 [m3/mol]) of Mixing Molar Volume of Liquid Alkanes RMSD of ∆V
dii/σii ) 0.95732 + 5.0204 × 10-2 ln(ii/kT) + 5.1572 × 10-3(ii/kT) (14) Equation 11 means that the Helmholtz free energy with the second-order perturbation term follows the law of corresponding states. Thus, the thermodynamic properties derived from the Helmholtz free energy, molar volume, vapor pressure, and surface tension follow the law of corresponding states when d and are used as the characteristic parameters. These thermodynamic properties along the saturated curve can be related to the following universal functions if the density dependency of the d value is neglected and the perturbation terms higher than the second order are neglected. 3
ref
V ˆ ≡ V/(NAd ) ) V ˆ
+C ˆ V(T ˆ ) Dd
(15)
ˆ p(T ˆ ) Dd pˆ ≡ pd3/ ) pˆ ref + C
(16)
ˆ γ(T ˆ ) Dd γˆ ≡ γd2/ ) γˆ ref + C
(17)
where V ˆ ref, pˆ ref, and γˆ ref are the reduced molar volume, vapor pressure, and surface tension for the reference fluid, respectively. 3. Mixing and Combining Rules for Characteristic Parameters The characteristic parameters d and are needed to evaluate the thermodynamic properties of the mixture. From a viewpoint of prediction, it is useful to express the characteristic parameters in terms of the parameters of the pure components, that is, mixing and combining rules. The characteristic length d is given by eq 10 in the last section. Because the reference is the simple liquid that has the hard-sphere and attractive potentials characterized with d and , we use simple mixing and combining rules for them. The mixing rule for is from a one-fluid model, and the combining rules are the Lorentz rule for d12 and the Berthelot rule for 12. 2
n-octane + n-dodecaned n-octane + n-tridecane n-octane + n-pentadecane n-decane + n-tridecane n-decane + n-pentadecane n-dodecane + n-tridecane n-dodecane + n-pentadecane n-tridecane + n-pentadecane n-heptane + 2-methylpentanee n-heptane + 3-methylpentane n-hexadecane + 2-methylnonane n-hexadecane + 3-methylnonane n-hexadecane + 4-methylnonane average
with lm, σlm, and ∆lm valuea
with lm, σlm, and ∆lm functionb
with and σc
0.758 1.200 2.616
1.219 1.729 3.044
3.380 1.741 8.925
3.938 0.926
5.223 1.159
7.193 1.018
2.654
2.931
6.227
0.813
0.646
4.747
1.882
2.556
1.546
1.746
1.768
1.924
1.829
2.462
5.172
2.653
0.679
0.531
1.554
1.812
2.488
1.600
2.124
7.439
1.859
2.104
a
4.025 lm
Calculated with intergroup potential parameters and σlm and distance correction parameter ∆lm values. b Calculated with intergroup potential parameters lm and σlm and distance correction parameter ∆lm functions. c Calculated with intermolecular potential parameters and σ. d Data of n-alkane + n-alkane mixtures are at 293.15 and 298.15 K. e Data of n-alkane + i-alkane mixtures are at 298.15 K.
corresponding states equation V ˆ (T ˆ ), and that of the pure component Vi is given by the equation with Dd ) 0 because dij ) d for the pure components. Therefore,
∆Vcalc ) [V ˆ ref(T ˆ) + C ˆ V(T ˆ ) Dd]NAd3 2
xi V ˆ ref(T ˆ i) NAdii3 ∑ i)1
(22)
The reduced temperature for the mixture T ˆ and that for the pure component T ˆ i are given by eqs 13 and 23, respectively.
2
∑ ∑xixjij i)1 j)1
(18)
T ˆ i ≡ kT/ii
d11 + d22 d12 ) 2
(19)
ˆ ) is expressed as a function in terms of T ˆ eq 24, V ˆ ref(T which was determined for the reduced molar volume of liquid propane in Appendix C.
12 ) x1122
(20)
V ˆ ref ) 0.6019 + 0.5093T ˆ + 3.948 × 10-4 exp(5.137T ˆ) (24)
Because the observed molar volume of some alkane mixtures is available and observed vapor pressure and surface tension are not, the mixing molar volume, which is defined by eq 21, is investigated in this work.
The mixing molar volume ∆Vcalc calculated from eq 22 is compared with the observed volume ∆Vobs for eight n-alkane + n-alkane mixtures at 293.15 and 298.15 K5 and five n-alkane + i-alkane mixtures at 293.15 K.1,2 The mixtures investigated are shown in Table 3. C ˆ V in eq 22 is given by a linear function of 1/T ˆ for simplicity.
)
4. Mixing Molar Volume
2
∆V ≡ Vmix -
xiVi ∑ i)1
(21)
The molar volume of the mixture Vmix is given by the
ˆ) ) C ˆ V(T
a +b T ˆ
(23)
(25)
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Table 4. Coefficients a and b in the Second-Order Perturbation Term C ˆ V(T ˆ ) for Mixing Molar Volume ∆Vcalc a × 102 lm,
σlm,
∆lm
valuea
with and n-alkane + n-alkane n-alkane + i-alkane with lm, σlm, and ∆lm functionb n-alkane + n-alkane n-alkane + i-alkane with and σc n-alkane + n-alkane n-alkane + i-alkane
b × 102
4.265 2.805
-0.7847 3.565
3.471 2.787
0.9638 3.640
4.809 5.235
0.009545 0.1728
a Calculation of ∆V lm calc with intergroup potential parameters and σlm and distance correction parameter ∆lm values. b Calculation of ∆Vcalc with intergroup potential parameters lm and σlm and distance correction parameter ∆lm functions. c Calculation of ∆Vcalc with intermolecular potential parameters and σ.
Figure 2. Mixing molar volume ∆V for n-alkane + i-alkane mixtures with intergroup potential parameters lm and σlm and distance correction parameter ∆lm values.
298.15 K, too, which is not shown in Figure 1. The RMSD is shown in Table 3. The deviation is not divided by ∆Vobs because the deviation of ∆Vcalc relative to ∆Vobs is very large when ∆Vobs happens to take a value near zero, which may lead to misunderstanding. 6. Mixing Molar Volume Calculated with Intergroup Potential Parameters and Separation Distance Correction Parameter Functions The distance correction parameter ∆lm can be expressed by functions of the location N and the number n of groups l and m in a molecule (eqs 26-31), Figure 1. Mixing molar volume ∆V for n-alkane + n-alkane mixtures with intergroup potential parameters lm and σlm and distance correction parameter ∆lm values.
The coefficients a and b in C ˆ V are determined by the least-squares fitting of the data for the mixing volume and shown in Table 4. 5. Mixing Molar Volume Calculated with Intergroup Potential Parameters and Separation Distance Correction Parameter Values The mixing molar volume ∆Vcalc is calculated from eq 22 with the help of eqs 2, 3, 10, 12-14, 18-20, and 2325 with the use of the intergroup potential parameters lm and σlm and the values of the distance correction parameter ∆lm, where the values are shown in Tables 1 and 2. An example of the calculation procedure is shown in Appendix F. Figures 1 and 2 show the comparison of the calculated volume ∆Vcalc to the observed volume ∆Vobs for n- + n-alkane mixtures at 293.15 K and n- + i-alkane mixtures at 298.15 K, respectively. ∆Vcalc agrees well with ∆Vobs, as shown in Figure 1. ∆Vcalc can represent the difference of the isomers, that is, the order of the ∆Vobs values n-heptane + 2-methylpentane < + 3-methylpentane and n-hexadecane + 2-methylnonane < + 4-methylnonane < + 3-methylnonane in Figure 2. ∆Vcalc agrees well with ∆Vobs for n- + n-alkanes at
For n-alkanes ∆CH3(N)-CH3(N) ) 2.885 × 10-11 - 1.489 × 10-10 ln(2N - 1.993) [m], N g 1 (26) ∆CH3(N)-CH2 ) 3.315 × 10-10 - 1.031 × 10-10 exp(-0.3042N2) - 1.965 × 10-10 ln(N + 2.655) [m], N g 1 (27) ∆CH2-CH2 ) 0
(28)
For i-alkanes ∆CH3(N)-CH3(N′) ) 3.202 × 10-10 - 2.261 × 10-10 ln[2.551 + N + N′ + 1.1(N - N′)] 10
-10
) 3.202 × 10-10 - 2.261 × ln(2.551 + 2.1N - 0.1N′) [m], N g N′ (29)
∆CH3(N)-CH2(n) ) 3.340 × 10-10 - 1.870 × 10-10 ln(N + 1.578 + 0.427m + 0.085m2) 1.230 × 10-10 exp[-1.81(N - 2.555 + 1.11m 0.0925m2)] [m] (30) ∆CH2(n)-CH2(n) ) -2.687 × 10-11 + 1.345 × 10-10 ln(n + 0.1424) [m] (31) where
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002 875
m)
n+1 -N+2 2
(32)
N and N′ are the locations of the group CH3. For example, in the case of 3-methylhexane, N and N′ ) 2.5 and 1.5 for eq 29, N ) 2.5 and 1.5 for eq 30, and n ) 4 for eqs 30 and 31. The number n of the group CH2 is uniquely determined by the location N of the group CH3 in n-alkanes; thus, n is not included in eq 27. Figures 3 and 4 show the comparison of ∆Vcalc, which is calculated from eq 22 with the help of eqs 2, 3, 10, 12-14, 18-20, and 23-25 with the lm, σlm, and ∆lm functions in eqs 26-31, to the observed ∆Vobs for the alkane mixtures. The RMSD of ∆Vcalc with the ∆lm functions and the coefficients a and b of C ˆ V in eq 22 are shown in Tables 3 and 4, respectively. Although the RMSD of ∆Vcalc with the ∆lm functions is a little larger than that with the ∆lm values, the agreement of ∆Vcalc
Figure 5. Mixing molar volume ∆V for n-alkane + n-alkane mixtures with intermolecular potential parameters and σ.
and ∆Vobs is satisfactory, and ∆Vcalc with the ∆lm functions can represent the difference of the isomers as well as that with the ∆lm values mentioned in the last section. 7. Comparison of Mixing Molar Volume with the Use of Intergroup Potential Parameters to That with the Use of Intermolecular Potential Parameters
Figure 3. Mixing molar volume ∆V for n-alkane + n-alkane mixtures with intergroup potential parameters lm and σlm and distance correction parameter ∆lm functions.
Figure 4. Mixing molar volume ∆V for n-alkane + i-alkane mixtures with intergroup potential parameters lm and σlm and distance correction parameter ∆lm functions.
The molar volume and the vapor pressure of the pure alkanes were correlated in the corresponding states with the use of intermolecular potential parameters and σ without the use of intergroup potential parameters (see Appendix G). The values of and σ are shown in Table 5 in Appendix G. The RMSDs of the correlation for n-alkanes C1-C21 and i-alkanes i-C4-C10 with and σ are 3.8% and 12.7% for the molar volume and the vapor pressure (in log p), respectively, which are worse than those with lm and σlm. Figures 5 and 6 show ∆Vcalc calculated from eq 22 with the help of eqs 10, 12-14, 18-20, and 23-25 with the use of the intermolecular potential parameters and σ and ∆Vobs for the alkane mixtures. ∆Vcalc of n-tridecane + n-pentadecane shows the sign opposite to that of ∆Vobs. ∆Vcalc cannot represent the difference of the isomers, that is, the order of the mixing molar volume for n-hexadecane + 2-methylnonane, + 3-methylnonane, and + 4-methylnonane. The ˆ) RMSD and the values of the coefficients a and b in C ˆ V(T are shown in Tables 3 and 4, respectively. The RMSDs with the intergroup potential parameters lm and σlm and the distance correction parameter ∆lm values and functions are better than that with the intermolecular potential parameters and σ. This is because good corresponding states correlations are obtained when lm, σlm, and ∆lm are used, and good correlations are not obtained when and σ are used as shown above. It is important for the good results of ∆Vcalc to obtain the good corresponding states correlations of the pure liquid alkanes. The mixing rules for the characteristic length and energy (eqs 10 and 18) and the combining rules (eqs 19
876
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Table 5. Intermolecular Potential Parameters E and σ for n- and i-Alkanes methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-undecane n-dodecane n-tridecane n-tetradecane n-pentadecane n-hexadecane n-heptadecane n-octadecane n-nonadecane n-eicosane n-heneicosane
× 1021 [J]
σ × 1010 [m]
1.7737 2.8747 3.5045 4.0522 4.5021 4.9066 5.2098 5.5562 5.8061 6.0928 6.4093 6.6565 6.8559 7.0737 7.4941 7.6044 7.8127 8.0321 8.3960 8.4993 8.7074
3.9844 4.5697 5.0627 5.4745 5.8417 6.1689 6.4649 6.7594 7.0149 7.3369 7.5375 7.7773 8.0166 8.2224 8.4525 8.6473 8.8421 9.0361 9.2218 9.4112 9.6024
× 1021 [J]
σ × 1010 [m]
3.8999 4.4057 4.7511 4.8295 5.1157 5.2386 5.4233 5.4984 5.4644 5.7180 5.6477 5.7063 5.9429 5.9604 5.8913 5.9387
5.4986 5.8354 6.1647 6.1413 6.4660 6.4480 6.7403 6.7240 6.7192 7.0129 6.9802 6.9783 7.2566 7.2225 7.2271 7.2306
2-methylpropane 2-methylbutane 2-methylpentane 3-methylpentane 2-methylhexane 3-methylhexane 2-methylheptane 3-methylheptane 4-methylheptane 2-methyloctane 3-methyloctane 4-methyloctane 2-methylnonane 3-methylnonane 4-methylnonane 5-methylnonane
expressed by the functions of the location and the number of the functional groups in the molecules. The use of the intermolecular potential parameters and σ without the intergroup potential parameters reproduces the mixing volume worse than the use of the intergroup potential parameters and cannot predict the order of the mixing molar volume for n-hexadecane + 2-methylnonane, + 3-methylnonane, and + 4-methylnonane. Appendix A It is assumed that the separation distance dlm between group l in molecule i and group m in molecule j is expressed by eq A1 when the molecules i and j are at separation distance dij.
dlm ) dij + ∆lm
Figure 6. Mixing molar volume ∆V for n-alkane + i-alkane mixtures with intermolecular potential parameters and σ.
and 20) are very simple because the mixture is projected on the simple reference system and the equation of the mixing molar volume is based on the law of corresponding states. The combining rules do not need any binary interaction parameters. Thus, the mixing molar volume can be obtained just from the parameters for the pure liquid alkanes; that is, eq 22 can predict the mixing molar volume of the other alkane mixtures that are not investigated in this work with the use of lm, σlm, and ∆lm values or functions.
(A1)
It is reasonable to assume that the repulsive and attractive parts of the intermolecular potential are equal to the sum of the repulsive and attractive parts of the intergroup potential, respectively.
4ij
() ()
4ij
σij
12
)
dij
σij
dij
6
)
( ) ( )
4lm ∑l ∑ m
∑l ∑ m
4lm
σlm
12
dlm σlm
dlm
(A2)
6
(A3)
Equations A1-A3 give eqs 2 and 3 in the text for the intermolecular potential parameters ij and σij with the intergroup potential parameters lm and σlm and the distance correction parameter ∆lm.
8. Conclusion The mixing molar volume for 13 mixtures of n- + n-alkanes and n- + i-alkanes can be reproduced well by the equation based on the law of corresponding states with the use of the intergroup potential parameters lm and σlm and the separation distance correction parameters ∆lm without the use of any binary interaction parameters. The equation can distinguish the mixing molar volume for n-heptane + 2-methylpentane and + 3-methylpentane and n-hexadecane + 2-methylnonane, + 3-methylnonane, and + 4-methylnonane. ∆lm’s are
Appendix B The distance correction parameters are ∆CH3(N)-CH3(N), ∆CH3(N)-CH2(n), and ∆CH2(n)-CH2(n) for n-alkanes and ∆CH3(N)-CH3(N′), ∆CH3(N)-CH2(n) ) ∆CH3(N)-CH(n), and ∆CH2(n)-CH2(n) ) ∆CH2(n)-CH(n) ) ∆CH(n)-CH(n) for i-alkanes, where N and N′ mean the locations of the group CH3 and n means the number of the groups CH2 and CH in a molecule. It is assumed that the distance correction for the group CH3 just depends on the location N and the groups that are not at the end of the molecule, CH2
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002 877
and CH, are not distinguished from each other for simplicity. From the latter assumption, the parameters between the functional groups CH3-CH2 and CH3-CH and CH2-CH2, CH2-CH, and CH-CH are equal to each other as shown above and the correction just depends on the number of the groups n. N is defined as the location of the carbon atom counted from the center in the main chain of the molecule. N is an integer when the center is at a carbon atom, and N has a fraction 0.5 when the center is at a bond between two carbon atoms. For example, N ) 2 and n ) 3 for n-pentane and N ) 2.5 and n ) 4 for n-hexane. For 3-methylheptane, N ) 3 for two CH3 groups at the end of the main chain, N′ ) 2 for the other CH3 group, and n ) 5.
characteristic length d. Finally the observed values of the molar volume and the vapor pressure are reduced with d and and are correlated with the reduced temperature. Appendix D The reduced excess Helmholtz free energy of a mixture is expressed in terms of diagram8
Appendix C The intergroup potential parameters are CH3-CH3, CH2-CH2, σCH3-CH3, σCH3-CH2, and σCH2-CH2 for n-alkanes and CH3-CH3, CH3-CH2 ) CH3-CH, CH2-CH2 ) CH2-CH ) CH-CH, σCH3-CH3, σCH3-CH2 ) σCH3-CH, and σCH2-CH2 ) σCH2-CH ) σCH-CH for i-alkanes. Propane is used for the determination of the universal functions because it is the smallest molecule that has both groups CH3 and CH2. The observed molar volume and vapor pressure of propane6 are reduced by eqs 15 and 16 with Dd ) 0 because of the pure liquid using the characteristic parameters d and , where d is evaluated by eq 14 with the intermolecular potential parameters and σ in the literature.7 The reduced thermodynamic properties are fitted by eqs C1 and C2, which are the universal functions. CH3-CH2,
The Mayer f bond is defined by
f ) exp(-βv) - 1
(D2)
and it is divided into two parts: the f ref bond for the reference system and the f b bond which was called the blip function by Andersen et al.:9
f ) f ref + f b
(D3)
f ref ) exp(-βvref) - 1
(D4)
The intermolecular potential v and vref are given by eqs 4 and 7 in the text, respectively. Introduction of eq D3 into eq D1 yields
V ˆ ref ) 0.6019 + 0.5093T ˆ + 3.948 × 10-4 exp(5.137T ˆ) (C1) log pˆ ref ) 1.458 - 3.187/T ˆ - 0.3761/T ˆ2
(C2)
The values of the intergroup potential parameters for n-alkane lm and σlm (l and m ) CH3 and CH2) are determined such that the molar volume and the vapor pressure reduced with the parameters d and are correlated in the corresponding states with eqs C1 and C2 for propane, n-butane, and n-pentane6 with the leastsquares method. The values for i-alkane are determined such that the thermodynamic properties are correlated for i-butanes, i-pentanes, and i-hexanes6 in a similar manner. Methane and ethane have just CH4 and CH3 groups, respectively, and do not have any CH2 groups. Thus, the values of the intergroup parameters CH4-CH4, σCH4-CH4, CH3-CH3, and σCH3-CH3 for methane and ethane are determined such that their molar volume and the vapor pressure are correlated with eqs C1 and C2. The values of the distance correction parameter ∆lm are determined such that the molar volume and the vapor pressure of the pure liquid alkanes are correlated with eqs C1 and C2. The molecular size of methane and ethane is small, and their shape is almost a sphere; thus, the distance correction parameters for methane and ethane ∆CH4-CH4 and ∆CH3-CH3 are set to zero. The procedure for the correlations is as follows. The approximate intermolecular potential parameters ′ and σ′ are evaluated from eqs 2 and 3 with ∆lm ) 0 using the intergroup potential parameters lm and σlm. The approximate characteristic length d′ is evaluated from eq 14 with ′ and σ′. Equations 2 and 3 with lm, σlm, d′, and ∆lm give the desired intermolecular potential parameters and σ, with which eq 14 gives the desired
where the dotted bond represents the f b bond. The first square-bracketed term in eq D5 is the sum of simple irreducible diagrams involving only one f b bond. The second one is that involving two f b bonds that are connected by a black circle, and the third one is that involving two f b bonds that are connected at least by two black circles and an f b bond. The characteristic length of the mixture d is chosen such that the first square-bracketed term in eq D5 vanishes.
Equation D6 can be described with the radial distribution function for molecules i and j as eq D7 (see Appendix E).
∑i ∑j ∫gij(r) exp(βvij)fijb drb ) 0 exp(βvij)fijb can be rewritten in the following way.
(D7)
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exp(βvij)fijb ) exp(βvij)[exp(-βvij) - exp(-βvijref)] ) 1 - exp[β(vij0 - vhs + wij - w)] (D8) Assumption 1. Because wij and w are the attractive part of the intermolecular potential for the mixture and reference, respectively, the difference between them is small and should be neglected. Assumption 1 reduces eq D8 to eq D9.
exp(βvij)fijb ) 1 - exp[β(vij0 - vhs)]
[( ) ( ) ] 12
σij r
vij0 )
-
σij r
6
(D9)
+ ij, r e rm,ij ) 21/6σij
) 0, r > rm,ij
(D10)
vhs ) ∞, r e d ) 0, r > d
(D11)
fijb (i, j ) 1, 2) is nonzero only near r ) dij, which is the characteristic length between the components i and j, and gij is not so sensitive to r in the vicinity of r ) dij. Thus, the following assumption is reasonable. Assumption 2. The radial distribution function gij is constant near r ) dij in eq D7, and the g values near r ) dij are equal to each other irrespective of the combination of i and j.
g11 ) g12 ) g21 ) g22 ) gav
(D12)
Using eqs D9 and D12, eq D7 becomes 2
2
∑ ∑ i)1 j)1
xixjgav
{
d3
+
3
∫d∞r2[1 - exp(βvij0)] dr
}
)0 (D13)
The integral over d to ∞ in eq D13 is divided into two parts: the integral over d to dij and that over dij to ∞. The latter is evaluated as -dij3/3 (see Appendix E). Thus, eq D13 is rewritten as 2
2
∑ ∑xixjg i)1 j)1
av
{
d3 3
-
dij3
+
3
∫
dij 2 r [1 d
0
}
- exp(βvij )] dr ) 0 (D14)
dij is given by eqs 14 and 19 in the text. Because the value of d is between the minimum and the maximum of the four dij values (i, j ) 1, 2), at least one of the four integrals over d to dij in eq D14 has a contrast sign to the other integrals. Thus, the following is assumed. Assumption 3. The sum of the integrals in eq D14 contributes little to eq D14. Assumption 3 reduces eq D14 to 2
d3 )
where d is the e bond, exp(-βvij). The second square-bracketed term in eq D5 is expressed by eq D17, using the function g3. Here the sum is taken over all molecules in the mixture, and a relation ∫dr bi ) V is used.
the second square-bracketed term in eq D5 ) 1 [g3(ij,ik) exp(βvij) exp(βvik)fijb fikb + V i j k g3(ij,jk) exp(βvij) exp(βvjk)fijb fjkb +
∑∑ ∑ ∫
bi dr bj dr bk ) g3(ik,jk) exp(βvik) exp(βvjk)fikb fjkb] dr
∑i ∑j ∑k [∫g3(ij,ik) exp(βvij) exp(βvik)fijb fjkb
∫g3(ij,jk) exp(βvij) exp(βvjk)fijb fjkb dr bij dr bik + ∫g3(ik,jk) exp(βvik)
dr bij dr bik +
exp(βvjk)fikb fjkb dr bik dr bjk] (D17) The factor exp(βvij)fijb in eq D17 is rewritten as in eq D9. Assumption 4. The function g3(ij,ik) takes a constant value g3av near rij ) dij and rik ) dik, which does not depend on the species in the mixture. Using assumption 4 with the help of eqs D9-D11, eq D17 can be written by
the second square-bracketed term in eq D5 ) 2
∑i,j,k)1 ∑ ∑NiNjNkg3av[∫exp(βvij)fijb
∫
dr bij exp(βvik)fikb dr bik +
∫exp(βvjk)fjkb drbjk + ∫exp(βvik)fikb drbik ∫exp(βvjk)fjkb drbjk]
(D15)
which is eq 10 in the text. The function g3(12,13) defined by eq D16 is proportional to the probability density that molecules 1-3 exist at positions r1, r2, and r3, respectively, and interact via e bonds, exp(-βv12) and exp(-βv13).
(D18)
Ni means the number of the molecules of component i, and subscript ij means a pair of components i and j. One of the factors in eq D18 is rewritten by
∫exp(βvij)fijb drbij ) 4π{∫0drij2 drij + ∫dd rij2[1 ∞ exp(βvij0)] drij + ∫d rij2[1 - exp(βvij0)] drij} (D19) ij
ij
The second term in the right-hand side of eq D19 can be neglected by assumption 3, and the third term is evaluated as -dij3/3 (see Appendix E). Thus, eq D19 becomes
2
xixjdij3 ∑ ∑ i)1 j)1
∫exp(βvij)fijb drbij
∫
(
exp(βvij)fijb dr bij ) 4π
)
3
d3 dij 3 3
(D20)
Substitution of eq D20 into D18 gives
the second square-bracketed term in eq D5 ) π 3 av 6 N g3 d Dd (D21) 6
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where
ref
[( )( ) ( )( ) ( )( )] 2
Dd )
∑i,j,k)1 ∑ ∑xixjxk 3
dij
d3
dij3 d
3
-1
3
-1
djk
d3
dik3 d
3
-1 +
3
-1 +
dik
d3
A ˆ ex ) A ˆ ex + C ˆ A(T ˆ ) Dd
djk3
-1
d3
-1
(D22)
ˆ ) kT/. Assumption 5. g3avd6 depends only on T Using assumption 5, eq D21 is rewritten by
the second square-bracketed term in eq D5 ) ˆ ) Dd (D23) C ˆ A(T
(D29)
In the approximations stated above, assumptions 1-3 are used so that the characteristic length d is obtained and the first-order perturbation term vanishes. Assumptions 4-6 are used so that the second-order perturbation term is expressed by the molecular size difference. Assumption 7 is for order perturbation terms higher than 2. Appendix E The radial distribution function for molecules 1 and 2, the positions of which are r1 and r2, is expanded by a series of diagrams.8
where
π C ˆ A(T ˆ ) ) N3g3avd6 6
(D24)
Equation D25 defines a function g4(12,34), which is proportional to the probability density that molecules 1-4 exist at positions r1, r2, r3, and r4, respectively, and interact via e bonds, exp(-βv12) and exp(-βv34).
The third square-bracketed term in eq D5 is expressed by eq D26, using the function g4.
the third square-bracketed term in eq D5 ) 1 g4(ij,kl) exp(βvij) exp(βvkl)fijb fklb i j k l V
∑∑ ∑ ∑ ∫
dr bi dr bj dr bk dr bl )
∑i ∑j ∑k ∑l V∫g4(ij,kl) exp(βvij) b
b
exp(βvkl)fij fkl dr bij dr bkl (D26) Assumption 6. The function g4(ij,kl) takes a constant value g4av near rij ) dij and rkl ) dkl, which does not depend on the species in the mixture. Equation D26 can be expressed by the following equation, using assumption 6. 2
2
∑ ∑ i)1 j)1
NiNj
∫
2
exp(βvij)fijb dr bij
2
∑ ∑NkNl
k)1l)1
∫exp(βvij)fklb drbkl
∫gii(r) exp(βvii)fiib drb ) 0
(E8)
fiib ) exp(-βvii) - exp(-βviiref)
(E9)
where
exp(βvii)fiib ) exp(βvii)[exp(-βvii) - exp(-βviiref)] ) 1 - exp[β(vii0 - viihs)] (E10) Equation E8 has the blip function which is effectively nonzero only in the vicinity of r ) dii. On the contrary, g(r) is not sensitive to r in this range, and it may be assumed that g(r) is constant irrespective of r. This assumption yields
∫d∞[exp(βvii0) - 1]r2 dr
dii3 ) 3
ii
(E11)
The dii value is numerically calculated from eq E11, and the resultant values are fitted by the function 14 in the text.
the third square-bracketed term in eq D5 ) g4av
Equation D6 for the characteristic length of the mixture d can be described with the radial distribution function g(1,2) in eq E7 as eq D7. The characteristic length of like molecules dii is determined from eq D7 for the pure liquid of component i.
Appendix F
(D27)
Equation D27 vanishes when the characteristic length is chosen such that eq D15 is satisfied.
the third square-bracketed term in eq A5 ) 0 (D28) Because the f b bond is nonzero only in a small range of the intermolecular potential, we assume following. Assumption 7. The higher order perturbed terms that contain at least three f b bonds are neglected. The reduced excess Helmholtz free energy A ˆ ex is given by eq D29, using eqs D5, D6, D23, and D28 and assumption 7.
The mixing molar volume ∆Vcalc for n-octane (1) + n-undecane (2) at x1 ) 0.5 and 293.15 K is calculated as follows. 1. The approximate intermolecular potential parameters between the like molecules ii′ and σii′ (i ) 1, 2) are calculated from eqs 2 and 3 with ∆lm ) 0 using the intergroup potential parameters CH3-CH3 ) 1.0580 × 10-22 J, CH3-CH2 ) 4.2810 × 10-21 J, CH2-CH2 ) 6.3597 × 10-24 J, σCH3-CH3 ) 5.2574 × 10-10 m, σCH3-CH2 ) 3.5742 × 10-10 m, and σCH2-CH2 ) 8.2883 × 10-10 m, where the values are from Table 1.
11′)5.4877 × 10-21 J, 22′ ) 6.6434 × 10-21 J
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Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002
σ11′ ) 6.3820 × 10-10 m, σ22′ ) 6.7119 × 10-10 m 2. The approximate characteristic lengths between the like molecules dii′ (i ) 1, 2) are calculated from eq 14 in the text with ii′ and σii′ obtained above.
d11′ ) 6.1626 × 10-10 m, d22′ ) 6.5356 × 10-10 m 3. The desired intermolecular potential parameters between the like molecules ii and σii (i ) 1, 2) are calculated from eqs 2 and 3 in the text with the approximate characteristic length dii′ (i ) 1, 2) obtained above, the intergroup potential parameters CH3-CH3, CH3-CH2, CH2-CH2, σCH3-CH3, σCH3-CH2, and σCH2-CH2 shown above, and the separation distance correction parameters ∆CH3(3.5)-CH3(3.5) ) -2.2014 × 10-10 m, ∆CH3(3.5)-CH2(6) ) -2.9475 × 10-11 m, and ∆CH2(6)-CH2(6) ) 2.7928 × 10-11 m for 11 and σ11 and ∆CH3(5)-CH3(5) ) -2.8192 × 10-10 m, ∆CH3(5)-CH2(9) ) -6.8732 × 10-11 m, and ∆CH2(9)-CH2(9) ) 1.2492 × 10-11 m for 22 and σ22, where the values are from Table 2.
11 ) 6.0614 × 10-21 J, 22 ) 7.3870 × 10-21 J σ11 ) 6.7426 × 10-10 m, σ22 ) 7.4409 × 10-10 m 4. The desired characteristic lengths between the like molecules dii (i ) 1, 2) are calculated from eq 14 in the text with ii and σii obtained above.
d11 ) 6.5395 × 10-10 m, d22 ) 7.2781 × 10-10 m 5. The characteristic parameters between the unlike molecules d12 and 12 are calculated from the combining rules (eqs 19 and 20) with dii and ii (i ) 1, 2) obtained above.
d12 ) 6.9088 × 10-10 m, 12 ) 6.6914 × 10-21 J 6. The characteristic parameters of the mixture d and are calculated from the mixing rules (eqs 10 and 18) with dij and ij (i and j ) 1 and 2) obtained above.
d ) 6.9186 × 10-10 m, ) 6.7078 × 10-21 J 7. The reduced temperatures T ˆ for the mixture and T ˆ i (i ) 1, 2) for the pure components are calculated from eqs 13 and 23 with and ii (i ) 1, 2) obtained above.
T ˆ ) 0.603 39, T ˆ 1 ) 0.667 73, T ˆ 2 ) 0.547 91 8. The mixing molar volume ∆Vcalc is calculated from eq 22 with the help of eqs 24 and 25 with the coefficients a ) 4.265 × 10-2 and b ) -7.847 × 10-3 in Table 4 and with T ˆ , d, T ˆ i, and dii (i ) 1, 2) obtained above.
∆Vcalc ) 5.790 × 10-8 m3 Appendix G The intermolecular potential parameters of the pure alkanes and σ can be directly determined such that the molar volume and the vapor pressure reduced by eqs 15 and 16 with Dd ) 0 due to the pure liquids are
expressed by the reference functions (eqs C1 and C2). The values of and σ determined are shown in Table 5. Nomenclature A ) Helmholtz free energy a, b ) coefficients in CV C ) coefficient of the second-order perturbation term Dd ) parameter of difference between characteristic length dij’s defined by eq 12 d ) characteristic length of mixture dij ) characteristic length between molecules i and j f ) Mayer f bond f b ) f b bond (blip function) g ) radial distribution function g3 ) function defined by eq D16 g4 ) function defined by eq D25 i, j, k ) molecules or molecular species k ) Boltzmann’s constant l, m ) functional groups N ) number of molecules NA ) Avogadro’s number p ) vapor pressure rlm ) separation distance between functional groups l and m rm ) separation distance at which intermolecular potential takes the minimum value r ) separation distance between molecules V ) molar volume v ) intermolecular potential w ) attractive potential X12 ) interaction parameter in Van-Patterson theory and in Flory theory x ) mole fraction Greek Symbols β ) reciprocal temperature ()1/kT) γ ) surface tension ∆lm ) distance correction parameter between functional groups l and m ∆V ) mixing molar volume ) intermolecular potential parameter of reference liquid or pure liquid lm ) intergroup potential parameter (potential parameter between functional groups l and m) ij ) intermolecular potential parameter between molecules i and j σ ) intermolecular potential parameter of reference liquid or pure liquid σlm ) intergroup potential parameter (potential parameter between functional groups l and m) σij ) intermolecular potential parameter between molecules i and j Superscripts av ) average ex ) excess hs ) hard-sphere potential ref ) reference system ∧ ) reduced form f ) vector 0 ) repulsive potential Subscripts A ) Helmholtz free energy calc ) calculated value mix ) mixture
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002 881 obs ) observed value p ) vapor pressure V ) molar volume
Literature Cited (1) Awwad, A. M.; Hassan, F. A.; Salman, M. A. Volume of Mixing of Decane Isomers with Normal Hexadecane at 298.15K. An Interpretation in Terms of the Van-Patterson Theory. Fluid Phase Equilib. 1987, 38, 291-298. (2) Kimura, F.; Benson, G. C. Excess Volumes of Binary Mixtures of n-Heptane with Hexane Isomers. J. Chem. Eng. Data 1983, 28, 387-390. (3) Jain, D. V. S.; Dhar, N. S. Excess Molar Volumes of Binary Mixtures of Toluene and Ethylbenzene with Propylbenzene, Butylbenzene, and Hexylbenzene at 298.15, 308.15 and 318.15 K. Fluid Phase Equilib. 1994, 102, 293-303. (4) Weeks, J. D.; Chandler, D.; Andersen, H. C. Role of Repulsive Forces in Determining the Equilibrium Structure of Simple Liquids. J. Chem. Phys. 1971, 54, 5237-5247.
(5) Wu, J.; Asfour, A. F. A. Densities and Excess Molar Volumes of Eight N-Alkane Binary Systems at 293.15 and 298.15K. Fluid Phase Equilib. 1994, 102, 305-315. (6) Daubert, T. E.; Danner, R. P. Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation; Taylor and Francis: Philadelphia, PA, 1989-1999. (7) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Sons: New York, 1954; revised edition 1964. (8) Morita, M.; Hiroike, K. A New Approach of the Theory of Classical Fluids. III. General Treatment of Classical Systems. Prog. Theor. Phys. 1961, 25, 537-578. (9) Andersen, H. C.; Weeks, J. D.; Chandler, D. Relationship between the Hard-sphere Fluid and Fluids with Realistic Repulsive Forces. Phys. Rev. 1971, A4, 1597-1607.
Received for review April 30, 2001 Revised manuscript received November 9, 2001 Accepted November 15, 2001 IE0103916
