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On the Veytsman statistics as applied to non-random lattice fluid equations of state
Fluid Phase Equilibria 187–188 (2001) 433–441
On the Veytsman statistics as applied to non-random lattice fluid equations of state C.S. Lee a,∗ , K.-P. Yoo b , B.H. Park a , J.W. Kang a a
Department of Chemical Engineering, Korea University, 1 Anamdong-5ka, Sungbuk-ku, Seoul 136-701, South Korea b Department of Chemical Engineering, Sogang University, Seoul 121-742, South Korea Received 17 July 2000; accepted 31 October 2000
Abstract Thermodynamic properties of hydrogen-bonding (HB) lattice fluids have been represented by superposing Veytsmann statistics to physically interacting lattice models. In the present work the validity of the superposition was questioned. The normalization of Veytsman statistics was proposed to properly represent HB effect for use with physical interaction models. Also to circumvent the difficulties associated with the solution of a set of non-linear equations representing the number of HB pairs, an expansion method was proposed for explicit computation. This expansion together with You et al.’s explicit non-random lattices fluid model for physical interactions constitutes a complete explicit model. The applicability of the explicit model was illustrated for vapor–liquid equilibria of pentane–pentanol mixture. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Alkanol; Associating system; Equation of state; Mixture; Vapor–liquid equilibria
1. Introduction Veytsman [1,2] proposed a free energy expression for hydrogen-bonding (HB) lattice fluids. Panayiotu and Sanchez [3] put the free energy in a readily usable form. Since the partition function is compatible with lattice statistics, it has been used by various investigators [3–5] to represent HB effects combined with physical contributions in a lattice statistics. A product of physical and chemical contributions generally approximates the configurational lattice partition function. c Ω c = ΩPc ΩHB
(1)
In typical application [3–5], the physical contribution is from the quasi-chemical (QC) theory [6] and the HB contribution is due to Veytsmann. c cV ΩHB = ΩHB ∗
Corresponding author. Fax: +82-2-922-5820.
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(2)
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C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
Veytsman statistics is based on the concept that donors and acceptors are distributed in lattice cells. A donor–acceptor pair is defined as a donor with an adjacent acceptor that is associated. A donor can be found adjacent to an acceptor even if HB free energy is zero. In the original formulation of Veytsman, this non-associated pair is still counted as a hydrogen-bond. This does not cause any difficulties as long as no other contributions are present. However, when Veytsmann statistics is used with QC theory to represent associating real fluids, we have both HB and QC pairs. HB pairs constitute a subset of QC pairs, since donors or acceptors are segments of molecular chains that also interact physically. In the limit of zero free energy, all HB pairs are, in fact, QC pairs. Still a pair is count both as HB and QC pair. Thus, Eq. (2) is embedded with this redundancy in counting that needs to be corrected. We propose a method in this study to remove this redundancy. The other problem we like to address is associated with the difficulty of solution of the Veytsman statistics. It yields a set of simultaneous non-linear equations when there are more than one type of hydrogen-bonds. In this study, an expansion method is also proposed for explicit computation and discussed.
2. Partition function Veytsmann statistics as proposed is based on donors and acceptors in a lattice and contributions due to molecular chains are not properly represented. Thus we begin with the total configurational partition function represented by the original Veytsmann statistics with a normalizing factor C. cV Ω c = CΩHB
(3)
In the limit of zero HB free energy, the configurational partition function is due entirely to pure physical interactions. Thus cV (AHB ΩPc = CΩHB ij = 0)
(4)
The normalizing factor is eliminated from Eqs. (3) and (4) to give Ωc =
cV ΩPc ΩHB cV ΩHB (AHB ij = 0)
(5)
This equation may be compared with Eq. (1) to give c ΩHB =
cV ΩHB cV ΩHB (AHB ij = 0)
(6)
Thus, Eq. (6) needs to be used instead of Eq. (2) to properly represent the HB contribution when Veytsmann statistics is combined with physical interaction models. To derive the expression for Eq. (6) by Veytsmann statistics, we consider a c component mixture in the lattice of fixed cell volume VH . Component i in a c component mixture has ri segments and may have hydrogen donor groups and/or acceptor groups. The number of hydrogen donor groups of type k in j species i is dki and the number of acceptor groups of type l in species j is al . The total number of donor
C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
435
types is m and that of acceptor types is n. Then the total number of donor groups of type l(Ndl ) and that of acceptor groups of types k(Nak ) are given by Ndk
c = dki Ni , i
Nal
c j = al Nj
(7)
j
HB ) and unpaired We also define the total number of donor–acceptor pair (NHB ), unpaired donors (Ni0 HB HB acceptors (N0j ) as functions of the number of i–j hydrogen-bond pair (Nij ).
NHB =
n m i
NijHB ,
HB Ni0
=
Ndi
−
j
n
NijHB ,
HB N0j
=
Naj
m − NijHB
j
(8)
i
Now Veytsman statistics [1,3,5] allows us to write the partition function for HB contribution in a lattice with fixed cell volume. m n m n j 1 Ndi ! Na ! 1 cV = NHB ΩHB exp(−βNijHB AHB (9) ij ) HB HB HB N ! N ! N ! Nr ij i0 0j i j i j where HB HB AHB ij = Uij − TSij
Nr = N0 +
c
(10)
ri Ni
(11)
i=1
where N0 denotes the number of holes and the superscript HB the property change on hydrogen-bond formation. For HB contribution, the maximization conditions lead to n m HB HB HB HB i HB j HB Nij Nr = Ni0 N0j exp(−βAij ) = Nd − Na − Nik Nkj exp(−βAHB ij ) (i = 1, 2, . . . , m,
j = 1, 2, . . . , n)
k=1
k=1
(12)
The condition reduces to the following expression when association free energy is zero. n m HB0 HB0 Naj − N0j = Ndi − NikHB0 NkjHB0 NijHB0 Nr = Ni0 (i = 1, 2, . . . , m,
k=1
j = 1, 2, . . . , n)
k=1
(13)
Eqs. (12) and (13) define sets of simultaneous non-linear equations. They yield quadratic equations for HB0 and N11 when applied to a general mixture with a single HB fluid. For more than one group of donors or acceptors, Eq. (12) has to be solved numerically. With these quantities, we finally have the expression for HB contribution HB N11
c ΩHB =
m n HB0 m n HB0 HB0 cV ! N0j ! Nij ! NrNHB0 Ni0 ΩHB = exp(−βNijHB AHB ij ) cV HB HB HB NHB N ! N ! N ! ΩHB (AHB = 0) Nr ij ij i0 0j i j i j
(14)
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3. Explicit approximation An expansion method is practicable when the reference value of HB pairs can be obtained explicitly. For the reference HB free energy that does not depend on the type of hydrogen-bonds, an explicit solution is obtained regardless of type of bonds. By rearranging Eq. (12), we have n m HB HB i j HB HB Nr Nij exp(βAij ) = Nd − Na − Nik Nkj (15) k=1
k=1
Double summing both sides with respect to i and j for a reference AHB (R) that does not depend on ij pair. Nr NHB(R) exp(βAHB (R) ) = (Nd − NHB(R) )(Na − NHB(R) )
(16)
where (R) denotes a value at the reference free energy and Nd =
m Ndi ,
Na =
i=1
n Naj
(17)
j =1
This quadratic equation is readily solved for NHB(R) to give HB 2 1/2 NHB(R) = 21 [Nr exp(βAHB ] (R) ) + Na + Nd − {(Nr exp(βA(R) ) + Na + Nd ) − 4Na Nd }
(18)
NHB0 is obtained from the same expression by setting AHB (R) = 0. HB HB For Aij different from A(R) , we consider the following expansion. NijHB =
NijHB(R)
+
HB n m ∂Nij k=1 l=1
∂AHB kl
HB (AHB kl − A(R) ) + · · ·
(19)
(R)
After lengthy algebra, we have the expression for coefficients of expansion from Eq. (11). The procedure is outlined in the Appendix A.
∂NijHB ∂AHB kl
= (R)
HB(R) βN0j
Nr exp(βAHB (R) ) + Na − NHB(R) HB(R) HB(R) Ni0 Nkl HB(R) × Nil δik − Nr exp(βAHB (R) ) + Nd + Na − 2NHB(R) HB(R) βNi0 + Nr exp(βAHB (R) ) + Nd − NHB(R) HB(R) HB(R) N N 0j kl × NkjHB(R) δjl − − NijHB(R) δik δjl Nr exp(βAHB ) + N + N − 2N d a HB(R) (R)
(20)
C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
where δ ij is the Kronecker delta and j Ndi Na Na − NHB(R) HR(R) Nij = 1− Nr exp(βAHB Nr exp(βAHB (R) ) (R) ) + Na − NHB(R) Nd − NHB(R) × 1− Nr exp(βAHB (R) ) + Nd − NHB(R)
437
(21)
The expression gives NijHB0 by setting AHB (R) = 0 and N HB = N HB0 . 4. HB contributions to thermodynamic properties With NijHB and NijHB0 known, Helmholtz free energy due to the HB contribution is readily written in the Stirling approximation. c βAcHB = −ln ΩHB = (NHB − NHB0 )(ln Nr + 1) +
m
HB HB HB0 HB0 (Ni0 ln Ni0 − Ni0 ln Ni0 )
i n n m HB HB HB0 HB0 HB HB HB HB0 + (N0i ln N0i − N0i ln N0i )+ (βAHB ln NijHB0 ) ij Nij +Nij ln Nij −Nij j
i
j
(22) The free energy allows us to derive thermodynamic properties. Since the volume is given by V = VH Nr
(23)
the HB correction to the lattice equation of state (EOS) follows by the standard method. c
∂AHB 1 ∂AcHB νHB ρ PHB = − =− = ∂V T VH ∂N0 T βVH
(24)
where ρ=
c
ρi ,
ρi =
i=1
Ni ri Nr
(25)
NHB − NHB0 νHB = c i=1 Ni ri
(26)
The chemical potential is derived using the well-known relation c
∂A µi = Na + ri Na VH P ∂Ni T,Nj =i
(27)
to give after some algebra m
n
HB HB Nk0 N0k µHB i = dki ln HB0 + aki ln HB0 RT Nk0 N0k k=1 k=1
(28)
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C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
The internal energy due to HB is
c βUHB
∂βAcHB =− ∂ ln T
NHB − NHB0 = Nr
∂Nr ∂ ln T
+
∂βAHB ij ∂ ln T
NijHB
(29)
5. Applications and discussions Pressure, internal energy and chemical potential may be represented by sums of physical interaction term and association correction term. The association term is explicitly presented in the present study. The non-random lattice fluid theory of You et al. [7,8] gives explicit physical interaction terms. Thus, with explicitly calculated number of HB pairs, we have an explicit non-random lattice-fluid HB EOS applicable to general multi-component mixtures with any number of HB types. Since the HB free energy is much greater than that due to physical interactions, a first order expansion from zero free energy gives unrealistic approximations. We propose the reference free energy as the mean value for 1-alkanol and water. For 1-alkanol the literature value is −25.1 kJ/mol for energy and HB HB −26.5 J/mol K for entropy. In this way, we set U(R) = −20.3 kJ/mol and S(R) = −21.55 J/mol K. In the method of You et al. [5,7,8], the model is applied with two temperature-dependent parameters representing physical interaction energy (ε) and segment number (r). They are fitted to saturated liquid volume and vapor pressure. For a binary mixture, the binary parameter kij is defined by εij = (εii εjj )1/2 (1 − kij ). Unit cell volume (V H = 9.75 cm3 /mol) and the coordination number (z = 10) are fixed. The HB term as originally proposed gives essentially identical results with the normalized HB term as presented here NLF–HB normalized). It is interesting to compare results of NLF–HB expanded with those of NLF–HB normalized as shown in Figs. 1 and 2 for 1-alkanols. The figures show that the dependence of reduced density and segment number on temperature is closely approximated by NLF–HB normalized
Fig. 1. Temperature-dependence of HB fraction on reduced density for 1-alkanol at r1 = 4.0.
C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
439
Fig. 2. Temperature-dependence of HB fraction on segment number for 1-alkanol at ρ = 0.8.
and expanded when used with the proposed reference free energy. However, the expansion with zero reference free energy shows larger deviation. With the HB contribution obtained by expansion, physical parameters for pure components are refitted. In this way, the non-random lattice fluid theory with the normalized Veytsman statistics is made to represent real fluids. Then, they are used for mixture equilibrium calculations with a binary interaction parameter. Fig. 3 shows the comparison of data [9] with calculated results for pentane–pentanol mixture. Expansion method is seen to give results comparable with the rigorous method.
Fig. 3. Comparison of calculated vapor–liquid equilibria with data for pentanol–pentane system at 303.15 K.
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C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
6. Conclusions To correctly represent HB effect for physically interacting lattice fluids, a normalization of Veytsman statistics was proposed. Also based on the normalized statistics, an expansion method was proposed for explicit solution of the number of HB pairs. When expanded around an appropriate reference HB free energy, the expansion was found to closely approximate the rigorous results. The applicability of the expansion method was illustrated for phase equilibria of pentane–pentanol mixture. The method was found to describe the behavior of the real fluid mixture accurately.
Appendix A We sum both sides of Eq. (11) once in i and once in j to obtain m m Nr exp(βAHB NijHB(R) = (Nd − NHB(R) ) Naj − NkjHB(R) (R) ) i=1 n
Nr exp(βAHB (R) )
NijHB(R)
=
(A.1)
k=1
Ndi
j =1
n HB(R) − (Na − NHB(R) ) Nik
(A.2)
k=1
HB(R) and nj=1 NijHB(R) in terms of known NHB(R) . Note that These equations readily give m i=1 Nij NijHB(R) is obtained explicitly using Eqs. (3) and (11) as represented by Eq. (17). This procedure also applies to obtain NijHB0 . To obtain the first order derivative with respect to AHB ij , we rewrite Eq. (7) Ndi − NijHB Nr exp(βAHB ij ) =
n
m j HB Nij Na − NijHB
j
(i = 1, 2, . . . , m,
j = 1, 2, . . . , n)
i
(A.3) Differentiating Eq. (A.3) with respect to AHB kl , we obtain
HB(R) NijHB Nr exp(βAHB exp(βAHB (R) ) + βNr Nij (R) )δik δjl n m m n
= − NijHB Naj − NijHB(R) − NijHB Ndi − NijHB(R) j
i
i
(A.4)
j
where F = (∂F /∂AHB kl )(R) . Summing Eq. (A.4) with respect to i and j and rearranging, we have n m
NijHB = − i
j
HB(R) βNr exp(βAHB (R) )Nkl m HB(R) n HB(R) Nr exp(βAHB + j N0j i Ni0 (R) ) +
(A.5)
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441
where HB(R) Ni0 = Ndi −
n NijHB(R) j
and
HB(R) N0j = Naj −
m NijHB(R)
(A.6)
i
Summing Eq. (A.4) once in i, once in j and then replacing the double summed derivative with Eq. (A.5), we have m
NijHB =
i
n
NijHB
j
βNr exp(βAHB (R) ) m HB(R) Nr exp(βAHB i Ni0 (R) ) + HB(R) HB(R) N0j Nkl HB(R) × δjl m HB(R) n HB(R) − Nkj Nr exp(βAHB ) + N + N i j 0j i0 (R)
βNr exp(βAHB (R) ) = n HB(R) Nr exp(βAHB j N0j (R) ) + HB(R) HB(R) Ni0 Nkl HB(R) × δik m HB(R) n HB(R) − Nil Nr exp(βAHB + j N0j i Ni0 (R) ) +
(A.7)
(A.8)
Putting Eqs. (A.5), (A.7) and (A.8) into Eq. (A.4) and rearranging, we obtain Eq. (16) for the expansion coefficients. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
B.A. Veytsman, J. Phys. Chem. 94 (1990) 8499. B.A. Veytsman, J. Phys. Chem. 97 (1993) 7144. C.G. Panayiotou, I.C. Sanchez, Macromolecules 24 (1991) 6231. I.C. Sanchez, C.G. Panayiotou, in: S.I. Sanlder (Ed.), Models for Thermodynamic and Phase Equilibria Calculations, Marcel Dekker, New York, 1993, p. 187. M.S. Yeom, K.-P. Yoo, B.H. Park, C.S. Lee, Fluid Phase Equilb. 158 (1999) 143. E.A. Guggenheim, Mixture, Clarendon, Oxford, 1952. S.S. You, K.-P. Yoo, C.S. Lee, Fluid Phase Equilb. 93 (1994) 193. S.S. You, K.-P. Yoo, C.S. Lee, Fluid Phase Equilb. 93 (1994) 215. M. Ronc, G.R. Ratcliff, Can. J. Chem. Eng. 54 (1976) 326.
On the Veytsman statistics as applied to non-random lattice fluid equations of state C.S. Lee a,∗ , K.-P. Yoo b , B.H. Park a , J.W. Kang a a
Department of Chemical Engineering, Korea University, 1 Anamdong-5ka, Sungbuk-ku, Seoul 136-701, South Korea b Department of Chemical Engineering, Sogang University, Seoul 121-742, South Korea Received 17 July 2000; accepted 31 October 2000
Abstract Thermodynamic properties of hydrogen-bonding (HB) lattice fluids have been represented by superposing Veytsmann statistics to physically interacting lattice models. In the present work the validity of the superposition was questioned. The normalization of Veytsman statistics was proposed to properly represent HB effect for use with physical interaction models. Also to circumvent the difficulties associated with the solution of a set of non-linear equations representing the number of HB pairs, an expansion method was proposed for explicit computation. This expansion together with You et al.’s explicit non-random lattices fluid model for physical interactions constitutes a complete explicit model. The applicability of the explicit model was illustrated for vapor–liquid equilibria of pentane–pentanol mixture. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Alkanol; Associating system; Equation of state; Mixture; Vapor–liquid equilibria
1. Introduction Veytsman [1,2] proposed a free energy expression for hydrogen-bonding (HB) lattice fluids. Panayiotu and Sanchez [3] put the free energy in a readily usable form. Since the partition function is compatible with lattice statistics, it has been used by various investigators [3–5] to represent HB effects combined with physical contributions in a lattice statistics. A product of physical and chemical contributions generally approximates the configurational lattice partition function. c Ω c = ΩPc ΩHB
(1)
In typical application [3–5], the physical contribution is from the quasi-chemical (QC) theory [6] and the HB contribution is due to Veytsmann. c cV ΩHB = ΩHB ∗
Corresponding author. Fax: +82-2-922-5820.
0378-3812/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 1 ) 0 0 5 5 5 - 6
(2)
434
C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
Veytsman statistics is based on the concept that donors and acceptors are distributed in lattice cells. A donor–acceptor pair is defined as a donor with an adjacent acceptor that is associated. A donor can be found adjacent to an acceptor even if HB free energy is zero. In the original formulation of Veytsman, this non-associated pair is still counted as a hydrogen-bond. This does not cause any difficulties as long as no other contributions are present. However, when Veytsmann statistics is used with QC theory to represent associating real fluids, we have both HB and QC pairs. HB pairs constitute a subset of QC pairs, since donors or acceptors are segments of molecular chains that also interact physically. In the limit of zero free energy, all HB pairs are, in fact, QC pairs. Still a pair is count both as HB and QC pair. Thus, Eq. (2) is embedded with this redundancy in counting that needs to be corrected. We propose a method in this study to remove this redundancy. The other problem we like to address is associated with the difficulty of solution of the Veytsman statistics. It yields a set of simultaneous non-linear equations when there are more than one type of hydrogen-bonds. In this study, an expansion method is also proposed for explicit computation and discussed.
2. Partition function Veytsmann statistics as proposed is based on donors and acceptors in a lattice and contributions due to molecular chains are not properly represented. Thus we begin with the total configurational partition function represented by the original Veytsmann statistics with a normalizing factor C. cV Ω c = CΩHB
(3)
In the limit of zero HB free energy, the configurational partition function is due entirely to pure physical interactions. Thus cV (AHB ΩPc = CΩHB ij = 0)
(4)
The normalizing factor is eliminated from Eqs. (3) and (4) to give Ωc =
cV ΩPc ΩHB cV ΩHB (AHB ij = 0)
(5)
This equation may be compared with Eq. (1) to give c ΩHB =
cV ΩHB cV ΩHB (AHB ij = 0)
(6)
Thus, Eq. (6) needs to be used instead of Eq. (2) to properly represent the HB contribution when Veytsmann statistics is combined with physical interaction models. To derive the expression for Eq. (6) by Veytsmann statistics, we consider a c component mixture in the lattice of fixed cell volume VH . Component i in a c component mixture has ri segments and may have hydrogen donor groups and/or acceptor groups. The number of hydrogen donor groups of type k in j species i is dki and the number of acceptor groups of type l in species j is al . The total number of donor
C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
435
types is m and that of acceptor types is n. Then the total number of donor groups of type l(Ndl ) and that of acceptor groups of types k(Nak ) are given by Ndk
c = dki Ni , i
Nal
c j = al Nj
(7)
j
HB ) and unpaired We also define the total number of donor–acceptor pair (NHB ), unpaired donors (Ni0 HB HB acceptors (N0j ) as functions of the number of i–j hydrogen-bond pair (Nij ).
NHB =
n m i
NijHB ,
HB Ni0
=
Ndi
−
j
n
NijHB ,
HB N0j
=
Naj
m − NijHB
j
(8)
i
Now Veytsman statistics [1,3,5] allows us to write the partition function for HB contribution in a lattice with fixed cell volume. m n m n j 1 Ndi ! Na ! 1 cV = NHB ΩHB exp(−βNijHB AHB (9) ij ) HB HB HB N ! N ! N ! Nr ij i0 0j i j i j where HB HB AHB ij = Uij − TSij
Nr = N0 +
c
(10)
ri Ni
(11)
i=1
where N0 denotes the number of holes and the superscript HB the property change on hydrogen-bond formation. For HB contribution, the maximization conditions lead to n m HB HB HB HB i HB j HB Nij Nr = Ni0 N0j exp(−βAij ) = Nd − Na − Nik Nkj exp(−βAHB ij ) (i = 1, 2, . . . , m,
j = 1, 2, . . . , n)
k=1
k=1
(12)
The condition reduces to the following expression when association free energy is zero. n m HB0 HB0 Naj − N0j = Ndi − NikHB0 NkjHB0 NijHB0 Nr = Ni0 (i = 1, 2, . . . , m,
k=1
j = 1, 2, . . . , n)
k=1
(13)
Eqs. (12) and (13) define sets of simultaneous non-linear equations. They yield quadratic equations for HB0 and N11 when applied to a general mixture with a single HB fluid. For more than one group of donors or acceptors, Eq. (12) has to be solved numerically. With these quantities, we finally have the expression for HB contribution HB N11
c ΩHB =
m n HB0 m n HB0 HB0 cV ! N0j ! Nij ! NrNHB0 Ni0 ΩHB = exp(−βNijHB AHB ij ) cV HB HB HB NHB N ! N ! N ! ΩHB (AHB = 0) Nr ij ij i0 0j i j i j
(14)
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C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
3. Explicit approximation An expansion method is practicable when the reference value of HB pairs can be obtained explicitly. For the reference HB free energy that does not depend on the type of hydrogen-bonds, an explicit solution is obtained regardless of type of bonds. By rearranging Eq. (12), we have n m HB HB i j HB HB Nr Nij exp(βAij ) = Nd − Na − Nik Nkj (15) k=1
k=1
Double summing both sides with respect to i and j for a reference AHB (R) that does not depend on ij pair. Nr NHB(R) exp(βAHB (R) ) = (Nd − NHB(R) )(Na − NHB(R) )
(16)
where (R) denotes a value at the reference free energy and Nd =
m Ndi ,
Na =
i=1
n Naj
(17)
j =1
This quadratic equation is readily solved for NHB(R) to give HB 2 1/2 NHB(R) = 21 [Nr exp(βAHB ] (R) ) + Na + Nd − {(Nr exp(βA(R) ) + Na + Nd ) − 4Na Nd }
(18)
NHB0 is obtained from the same expression by setting AHB (R) = 0. HB HB For Aij different from A(R) , we consider the following expansion. NijHB =
NijHB(R)
+
HB n m ∂Nij k=1 l=1
∂AHB kl
HB (AHB kl − A(R) ) + · · ·
(19)
(R)
After lengthy algebra, we have the expression for coefficients of expansion from Eq. (11). The procedure is outlined in the Appendix A.
∂NijHB ∂AHB kl
= (R)
HB(R) βN0j
Nr exp(βAHB (R) ) + Na − NHB(R) HB(R) HB(R) Ni0 Nkl HB(R) × Nil δik − Nr exp(βAHB (R) ) + Nd + Na − 2NHB(R) HB(R) βNi0 + Nr exp(βAHB (R) ) + Nd − NHB(R) HB(R) HB(R) N N 0j kl × NkjHB(R) δjl − − NijHB(R) δik δjl Nr exp(βAHB ) + N + N − 2N d a HB(R) (R)
(20)
C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
where δ ij is the Kronecker delta and j Ndi Na Na − NHB(R) HR(R) Nij = 1− Nr exp(βAHB Nr exp(βAHB (R) ) (R) ) + Na − NHB(R) Nd − NHB(R) × 1− Nr exp(βAHB (R) ) + Nd − NHB(R)
437
(21)
The expression gives NijHB0 by setting AHB (R) = 0 and N HB = N HB0 . 4. HB contributions to thermodynamic properties With NijHB and NijHB0 known, Helmholtz free energy due to the HB contribution is readily written in the Stirling approximation. c βAcHB = −ln ΩHB = (NHB − NHB0 )(ln Nr + 1) +
m
HB HB HB0 HB0 (Ni0 ln Ni0 − Ni0 ln Ni0 )
i n n m HB HB HB0 HB0 HB HB HB HB0 + (N0i ln N0i − N0i ln N0i )+ (βAHB ln NijHB0 ) ij Nij +Nij ln Nij −Nij j
i
j
(22) The free energy allows us to derive thermodynamic properties. Since the volume is given by V = VH Nr
(23)
the HB correction to the lattice equation of state (EOS) follows by the standard method. c
∂AHB 1 ∂AcHB νHB ρ PHB = − =− = ∂V T VH ∂N0 T βVH
(24)
where ρ=
c
ρi ,
ρi =
i=1
Ni ri Nr
(25)
NHB − NHB0 νHB = c i=1 Ni ri
(26)
The chemical potential is derived using the well-known relation c
∂A µi = Na + ri Na VH P ∂Ni T,Nj =i
(27)
to give after some algebra m
n
HB HB Nk0 N0k µHB i = dki ln HB0 + aki ln HB0 RT Nk0 N0k k=1 k=1
(28)
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C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
The internal energy due to HB is
c βUHB
∂βAcHB =− ∂ ln T
NHB − NHB0 = Nr
∂Nr ∂ ln T
+
∂βAHB ij ∂ ln T
NijHB
(29)
5. Applications and discussions Pressure, internal energy and chemical potential may be represented by sums of physical interaction term and association correction term. The association term is explicitly presented in the present study. The non-random lattice fluid theory of You et al. [7,8] gives explicit physical interaction terms. Thus, with explicitly calculated number of HB pairs, we have an explicit non-random lattice-fluid HB EOS applicable to general multi-component mixtures with any number of HB types. Since the HB free energy is much greater than that due to physical interactions, a first order expansion from zero free energy gives unrealistic approximations. We propose the reference free energy as the mean value for 1-alkanol and water. For 1-alkanol the literature value is −25.1 kJ/mol for energy and HB HB −26.5 J/mol K for entropy. In this way, we set U(R) = −20.3 kJ/mol and S(R) = −21.55 J/mol K. In the method of You et al. [5,7,8], the model is applied with two temperature-dependent parameters representing physical interaction energy (ε) and segment number (r). They are fitted to saturated liquid volume and vapor pressure. For a binary mixture, the binary parameter kij is defined by εij = (εii εjj )1/2 (1 − kij ). Unit cell volume (V H = 9.75 cm3 /mol) and the coordination number (z = 10) are fixed. The HB term as originally proposed gives essentially identical results with the normalized HB term as presented here NLF–HB normalized). It is interesting to compare results of NLF–HB expanded with those of NLF–HB normalized as shown in Figs. 1 and 2 for 1-alkanols. The figures show that the dependence of reduced density and segment number on temperature is closely approximated by NLF–HB normalized
Fig. 1. Temperature-dependence of HB fraction on reduced density for 1-alkanol at r1 = 4.0.
C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
439
Fig. 2. Temperature-dependence of HB fraction on segment number for 1-alkanol at ρ = 0.8.
and expanded when used with the proposed reference free energy. However, the expansion with zero reference free energy shows larger deviation. With the HB contribution obtained by expansion, physical parameters for pure components are refitted. In this way, the non-random lattice fluid theory with the normalized Veytsman statistics is made to represent real fluids. Then, they are used for mixture equilibrium calculations with a binary interaction parameter. Fig. 3 shows the comparison of data [9] with calculated results for pentane–pentanol mixture. Expansion method is seen to give results comparable with the rigorous method.
Fig. 3. Comparison of calculated vapor–liquid equilibria with data for pentanol–pentane system at 303.15 K.
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C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
6. Conclusions To correctly represent HB effect for physically interacting lattice fluids, a normalization of Veytsman statistics was proposed. Also based on the normalized statistics, an expansion method was proposed for explicit solution of the number of HB pairs. When expanded around an appropriate reference HB free energy, the expansion was found to closely approximate the rigorous results. The applicability of the expansion method was illustrated for phase equilibria of pentane–pentanol mixture. The method was found to describe the behavior of the real fluid mixture accurately.
Appendix A We sum both sides of Eq. (11) once in i and once in j to obtain m m Nr exp(βAHB NijHB(R) = (Nd − NHB(R) ) Naj − NkjHB(R) (R) ) i=1 n
Nr exp(βAHB (R) )
NijHB(R)
=
(A.1)
k=1
Ndi
j =1
n HB(R) − (Na − NHB(R) ) Nik
(A.2)
k=1
HB(R) and nj=1 NijHB(R) in terms of known NHB(R) . Note that These equations readily give m i=1 Nij NijHB(R) is obtained explicitly using Eqs. (3) and (11) as represented by Eq. (17). This procedure also applies to obtain NijHB0 . To obtain the first order derivative with respect to AHB ij , we rewrite Eq. (7) Ndi − NijHB Nr exp(βAHB ij ) =
n
m j HB Nij Na − NijHB
j
(i = 1, 2, . . . , m,
j = 1, 2, . . . , n)
i
(A.3) Differentiating Eq. (A.3) with respect to AHB kl , we obtain
HB(R) NijHB Nr exp(βAHB exp(βAHB (R) ) + βNr Nij (R) )δik δjl n m m n
= − NijHB Naj − NijHB(R) − NijHB Ndi − NijHB(R) j
i
i
(A.4)
j
where F = (∂F /∂AHB kl )(R) . Summing Eq. (A.4) with respect to i and j and rearranging, we have n m
NijHB = − i
j
HB(R) βNr exp(βAHB (R) )Nkl m HB(R) n HB(R) Nr exp(βAHB + j N0j i Ni0 (R) ) +
(A.5)
C.S. Lee et al. / Fluid Phase Equilibria 187–188 (2001) 433–441
441
where HB(R) Ni0 = Ndi −
n NijHB(R) j
and
HB(R) N0j = Naj −
m NijHB(R)
(A.6)
i
Summing Eq. (A.4) once in i, once in j and then replacing the double summed derivative with Eq. (A.5), we have m
NijHB =
i
n
NijHB
j
βNr exp(βAHB (R) ) m HB(R) Nr exp(βAHB i Ni0 (R) ) + HB(R) HB(R) N0j Nkl HB(R) × δjl m HB(R) n HB(R) − Nkj Nr exp(βAHB ) + N + N i j 0j i0 (R)
βNr exp(βAHB (R) ) = n HB(R) Nr exp(βAHB j N0j (R) ) + HB(R) HB(R) Ni0 Nkl HB(R) × δik m HB(R) n HB(R) − Nil Nr exp(βAHB + j N0j i Ni0 (R) ) +
(A.7)
(A.8)
Putting Eqs. (A.5), (A.7) and (A.8) into Eq. (A.4) and rearranging, we obtain Eq. (16) for the expansion coefficients. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
B.A. Veytsman, J. Phys. Chem. 94 (1990) 8499. B.A. Veytsman, J. Phys. Chem. 97 (1993) 7144. C.G. Panayiotou, I.C. Sanchez, Macromolecules 24 (1991) 6231. I.C. Sanchez, C.G. Panayiotou, in: S.I. Sanlder (Ed.), Models for Thermodynamic and Phase Equilibria Calculations, Marcel Dekker, New York, 1993, p. 187. M.S. Yeom, K.-P. Yoo, B.H. Park, C.S. Lee, Fluid Phase Equilb. 158 (1999) 143. E.A. Guggenheim, Mixture, Clarendon, Oxford, 1952. S.S. You, K.-P. Yoo, C.S. Lee, Fluid Phase Equilb. 93 (1994) 193. S.S. You, K.-P. Yoo, C.S. Lee, Fluid Phase Equilb. 93 (1994) 215. M. Ronc, G.R. Ratcliff, Can. J. Chem. Eng. 54 (1976) 326.