Ind. Eng. Chem. Res. 2008, 47, 6275–6279
6275
Solubility Of Amino Acids In Water And Aqueous Solutions By the Statistical Associating Fluid Theory Peijun Ji† and Wei Feng*,‡ State Key Laboratory of Chemical Resource Engineering, College of Chemical Engineering and College of Life Science and Technology, Beijing UniVersity of Chemical Technology, Beijing, 100029, People’s Republic of China
The method previously developed is applied to determine the parameters of the statistical associating fluid theory (SAFT) equation of state for amino acids. With the parameters determined, the SAFT equation of state is applied to model the amino acid solubility in water with very good precision, including L-tyrosine, L-leucine, L-aspartic acid, L-tryptophan, L-glutamic acid, L-alanine, DL-alanine, DL-valine, L-phenylalanine, DL-serine, L-proline, L-serine, and glycine. On the basis of the results of binary systems of amino acid/water, the solubility of L-alanine and L-leucine in water at high pressures up to 3500 bar is simulated; the amino acid solubilities in an aqueous solution is modeled, including systems DL-alanine/DL-valine/H2O at 25 °C, DL-alanine/DL-serine/H2O at 25 °C, L-glutamic acid/L-aspartic acid/H2O at 25, 40, and 60 °C, L-serine/Laspartic acid/H2O at 25, 40, and 60 °C, and L-serine/L-glutamic acid/H2O at 25, 40, and 60 °C, and good agreement between modeling results and data are obtained. Introduction Amino acids are of importance in industrial processes, they can be utilized as food additives and constituents of pharmaceutical products. Amino acids are separated from mixtures that could be obtained from fermentation broths. The solubility of amino acids in water and in an aqueous solution is an important physical property for the design and scale-up of chemical processes for separation, concentration, and purification of amino acids. Solid-liquid equilibria in aqueous amino acid systems are phenomena that must be dealt with to obtain solubility data. For ternary or higher systems of amino acids, equilibrium is difficult to establish in solid-liquid systems, and equilibrium data are relatively lacking. Hence, it is necessary to develop a modeling approach to get knowledge of the phase equilibria for multicomponent systems. The thermodynamic models, such as Wilson, UNIFAC, and NRTL equations, have been applied to describe phase behavior for systems containing amino acids.1,2 However, these models can not be applied when the effect of pressure on the amino acid solubility and the density of aqueous amino acid solutions are taken into consideration. In this work, we are intended to use equations of state to model the solubility of amino acid in water and an aqueous solution. There are some predictive equations of state,3–5 which have been applied to a wide variety of fluids.6,7 In the previous work,8 by taking advantage of the method developed for determining the statistical associating fluid theory (SAFT) parameters for amino acids, the SAFT equation of state9 has been applied to describe the density of amino acid aqueous solutions. The objective of this work is to apply the SAFT equation of state to describe the amino acid solubility in water and an aqueous solution by extending the method10 previously developed to determine the SAFT parameters for amino acids. The results will be useful in the rational design and optimization of industrial processes. * To whom correspondence should be addressed. E-mail: fengwei@ mail.buct.edu.cn. † College of Chemical Engineering. ‡ State Key Laboratory of Chemical Resource Engineering.
Determination of the SAFT parameters for amino acids. The SAFT equation of state9 is applied to model the amino acid solubility in water and an aqueous solution. For describing amino acids, five parameters are required by the SAFT model, namely, the segment number m, segment volume υ00, and segment-segment interaction energy u0/k, association energy εAB/k, and association volume κAB. The method of parameter determination previously developed10 is extended to determine the SAFT parameters for amino acids. There are three steps for the parameter determination. First, both the parameters segment volume υ00 and association volume κAB are set to a value; the parameters of m, u0/k, and εAB/k are taken as the three variables in eqs 13 (the three equations are expressed by the SAFT model), which are the pressure at the critical temperature TC, the first derivative of pressure with respect to density at TC, PC, and VC, and the second derivative of pressure with respect to density at TC, PC, and VC, respectively. By making use of the critical properties of temperature, pressure, and volume, the three equations can be solved, and then, the parameters of m, u0/k, and εAB/k are determined. Second, with the five parameters (three of them are determined as mentioned above), the SAFT model is used to calculate normal boiling temperature. If the normal boiling temperature is calculated correctly, then we say the early value given to the parameter κAB is the correct one. Otherwise, κAB is adjusted to a new value, and the above procedures (fist and second steps) are repeated until the normal boiling temperature is calculated correctly. Third, with the above five parameters (four of them have been obtained through the determination steps), the SAFT model is used to calculate the density of amino acids at room temperature and atmospheric pressure. If the density is calculated with a small deviation from the crystal density of the amino acid at room temperature (the crystal density is taken as a reference density of the amino acid, to be in an assumed amorphous state at room temperature), then the early value given to the parameter υ00 is the correct one. Otherwise, the parameter υ00 is set to a new value, and the procedures of parameter determination mentioned above are repeated. In this way, the SAFT parameters are determined finally.
10.1021/ie800313h CCC: $40.75 2008 American Chemical Society Published on Web 06/25/2008
6276 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008
P|TC ) PC ∂P ∂F
[ ]
TC
[ ] ∂2P ∂F2
(1)
)0
(2)
)0
(3)
TC
During the process of parameter determination, the association number should be carefully chosen. On the one hand, the association number affects the solving of eqs 1–3, whereas on the other hand, the association number affects reproduction of the normal boiling temperature. For amino acids, the critical properties of temperature, pressure, and density, as well as the normal boiling temperature are estimated using Joback’s modification of Lydersen’s method,11 as shown in eqs 4–7.
∑ ∆T - (∑ ∆T) ] ) [0.113 + 0.0032n - ∑ ∆P] V ) 17.5 + ∑ ∆V T ) 198 + ∑ ∆b
2 -1
TC ) TB[0.584 + 0.965 PC
-2
A
(4) (5)
C
(6)
B
(7)
How to calculate ∆T, ∆P, ∆V, and ∆b has been described in detail by Reid et al.11 On the basis of the group contribution method and using molecular structure information, the critical properties of temperature, pressure, and volume as well as normal boiling temperatures have been estimated. The values are listed in Table 1. Using the method of parameter determination described above, the SAFT parameters have been determined and are listed in Table 2. Calculation of Amino Acid Solubility. Solubility of amino acid in water or in an aqueous solution can be calculated using eq 812
( )
ln
xiφli φl0i
(
)
(
)
∆CPsli Tmi ∆CPsli ∆Hsli T T -1 + ln )1+ RT Tmi R T R Tmi (8)
where xi is the mole fraction of component i in the solution, φil is the fugacity coefficient of component i in the solution, and l l φ0i is the fugacity coefficient of the pure substance i. φil and φ0i 9 are calculated by the SAFT equation of state. Tmi is the melting temperature of species i, ∆Hsli is the enthalpy change of melting, and ∆CslPi is the heat capacity change of melting. For most amino acids, due to decomposition before reaching melting temperature, ∆Hsli , Tmi, and ∆CslPi are not measurable and are unavailable. ∆Hisl, Tmi, and ∆CPsli are taken as constants; they are determined by solubility data. Results and Discussion Amino Acid Solubility in Water. The solubility constants ∆Hsli , Tmi, and ∆CslPi of eq 8 are obtained by fitting the solubility data;13 they are listed in Table 3. As presented in Figures 1–3, amino acid solubilities in water are modeled, including Ltyrosine, L-leucine, L-aspartic acid, L-tryptophan, L-glutamic acid, L-alanine, DL-alanine, DL-valine, L-phenylalanine, DL-serine, L-proline, L-serine, and glycine. Lines represent the results calculated by the SAFT model. The relative deviation of the modeling is presented in Table 4. The results in Figures 1–3 and Table 4 show that the SAFT model can describe the amino
Table 1. Critical Properties and Normal Boiling Points substance
TC/K
PC/bar
VC/cm3/mol
TB/K
aspartic acid leucine glutamic acid tyrosine phenylalanine tryptophan
875.84 761.52 894.92 939.50 850.64 966.03
84.15 37.55 47.24 45.65 37.68 38.20
330.5 412.5 386.5 441.5 434.5 568.5
654.03 553.84 676.91 707.16 626.54 736.39
Table 2. Parameters of the SAFT Equation of State substance
m
a
L-alanine
3.044 2.582 a L-serine 2.926 glycinea 2.865 a proline 1.325 a DL-valine 2.203 waterb 1.165 aspartic acid 2.637 glutamic acid 2.264 leucine 2.866 DL-serine 3.034 tyrosine 1.702 phenylalanine 1.250 tryptophan 2.145 a
DL-alanine
u/k/K
υ00/cm3/mol
εAB/K
κAB
Nass c
170.411 176.335 203.42 161.162 233.773 197.446 194.29 225.049 237.451 218.789 227.788 275.165 249.81 152.33
10.0 12.0 12.0 8.0 40.0 22.0 8.0 16.0 23.0 19.0 12.0 40.0 55.0 40.0
4249.11 4147.86 6783.33 4055.48 4815.41 4806.18 3229 7054.38 7967.80 5053.87 7134.12 8367.10 6566.94 5501.39
0.050 0.056 0.0025 0.065 0.011 0.022 0.052 0.0056 0.002 0.057 0.0035 0.0012 0.0038 0.017
3 3 3 3 3 3 3 3 3 2 2 3 3 4
a From ref 8. b From ref 10. c Nass is the number of association sites. For the amino acids, the association sites are distributed on -COOH groups. For L-alanine, DL-alanine, glycine, aspartic acid, L-serine, proline, glutamic acid, tyrosine, phenylalanine, and DL-valine, two sites on oxygen and one site on hydrogen; for leucine and DL-serine, one site on oxygen and one site on hydrogen; for tryptophan, two sites on oxygen and two sites on hydrogens.
Table 3. Solubility Constants of Amino Acids in Water amino acid
∆Hisl/J/mol
Tmi/K
∆CPsli/J/mol K
DL-valine glycine L-alanine DL-alanine L-serine DL-serine L-aspartic acid L-leucine L-glutamic acid L-proline L-tyrosine L-phenylalanine L-tryptophan
71684.60 13708.99 28019.71 30702.21 27986.52 26873.17 76709.89 76586.02 97576.61 26727.68 160194.8 113385.4 115862.0
567.15a 688.28 704.07 618.29 592.11 586.70 597.57 610.15a 592.95 495.15a 699.42 759.94 620.21
212.71 0.000 45.78 60.44 0.000 34.12 129.31 224.26 187.74 23.074 275.56 158.26 318.98
a
From ref .
acid solubility in water quite well. Figure 4 presents the solubility of L-alanine and L-leucine at different pressures up to 3500 bar. The predicted amino acid solubilities at different pressures are in good agreement with the experimental data. Amino Acid Solubility in an Aqueous Solution. On the basis of the binary system results as mentioned above, ternary systems are described. The solubilities of an amino acid in an aqueous solution containing another amino acid are modeled as presented in Figures 5–9. Figures 5 and 6 present the results of DL-alanine/DL-serine/H2O and DL-alanine/DL-valine/H2O; the solubility of DL-serine increases, while that of DL-valine decreases slightly with the DL-alanine molality. The lines are predicted by SAFT on the basis of the calculation of solubility of individual amino acids in water. The prediction results are in good agreement with the data. In Figures 7–9, the solubility of an amino acid is enhanced by the presence of another amino acid. Figure 7 presents the solubilities of L-glutamic acid and L-aspartic acid in an aqueous solution at 25, 40, and 60 °C. The lines represent the results by the SAFT model. The dashed line represents the results of SAFT correlation with a binary
Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 6277 Table 4. Relative Deviation for Amino Acid Solubility in Water system DL-valine
glycine L-alanine DL-alanine L-serine DL-serine L-aspartic acid L-leucine L-glutamic acid L-proline L-tyrosine L-phenylalanine L-tryptophan
temperature/°C
relative deviation/%a
0–100 0–70 0–100 0–100 10–60 0–100 0–100 0–100 0–60 0–100 0–100 0–100 0–90
1.50 4.72 0.07 0.14 5.69 1.83 2.96 1.09 1.41 2.95 6.29 0.43 4.17
a Relative deviation ) [solubility(cal) - solubility(exp)]/solubility(exp) × 100.
Figure 3. Solubility of L-proline, L-serine, and glycine in water. Data from ref 13.
Figure 1. Solubility of L-tyrosine, L-leucine, L-aspartic acid, L-tryptophan, and L-glutamic acid in water. Data from ref . Figure 4. Solubility of L-alanine and L-leucine at different pressures. Data from ref 14.
solubility of L-aspartic acid in the aqueous solution containing L-serine is significant. The lines represent the results by the ass SAFT model; a binary interaction parameter function k12 ) -4 -6 2 -0.09729 + 8.78571 × 10 T - 4.28571 × 10 T is required. This function is for the cross association attraction, and it is obtained in the temperature range of 25-60 °C. Table 5 presents
Figure 2. Solubility of L-alanine, DL-alanine, and DL-serine in water. Data from ref 13.
DL-valine, L-phenylalanine,
interaction parameter of -0.6 between L-glutamic acid and L-aspartic acid. With the binary parameter, the solubilities of L-glutamic acid and L-aspartic acid in aqueous solutions at 25 and 40 °C are predicted as presented by the solid lines. In Figure 8, for describing the great enhancement of L-glutamic acid solubility in the solution of L-serine, the SAFT model needs a dis function of binary interaction parameter, k12 ) -0.68 + -5 2 0.01167T - 6.66667 × 10 T , between L-glutamic acid and dis L-serine. k12 is for the cross van der Waals attraction; its function is obtained in the temperature range of 25-60 °C. Figure 9 presents the solubilities of L-serine and L-aspartic acid in an aqueous solution at 25, 40, and 60 °C. The enhancement of the
Figure 5. Solubilities of DL-alanine and at 25 °C. Data from ref 2.
DL-serine
in an aqueous solution
6278 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008
Figure 6. Solubilities of DL-alanine and at 25 °C. Data from ref 2.
DL-valine
in an aqueous solution Figure 9. Solubilities of L-aspartic acid and L-serine in an aqueous solution at 25, 40, and 60 °C. Data from ref 15. A function of binary interaction ass parameter k12 ) -0.09729 + 8.78571 × 10-4T - 4.28571 × 10-6T2 between L-serine and L-glutamic acid is used. Table 5. Relative Deviation of Calculations for Ternary Systems relative deviationa (RDi)/% system
T/°C
RD1
RD2
+ DL-serine(2) + water(3) DL-alanine(1) + DL-valine(2) + water (3) L-glutamic acid(1) + L-aspartic acid(2) + water(3) L-glutamic acid(1) + L-aspartic acid(2) + water(3) L-glutamic acid(1) + L-aspartic acid(2) + water(3) L-glutamic acid(1) + L-serine(2) + water(3) L-glutamic acid(1) + L-serine(2) + water(3) L-glutamic acid(1) + L-serine(2) + water(3) L-aspartic acid (1) + L-serine(2) + water(3) L-aspartic acid (1) + L-serine(2) + water(3) L-aspartic acid (1) + L-serine(2) + water(3)
25
0.32
3.55
25
1.17
3.00
60
1.27
6.29
40
1.56
0.91
25
1.99
4.64
60
2.05
7.38
40
4.36
6.09
25
3.46
6.10
60
7.16
8.21
40
7.75
6.97
25
7.40
6.52
DL-alanine(1)
Figure 7. Solubilities of L-glutamic acid and L-aspartic acid in an aqueous dis solution at 25, 40, and 60 °C. Data from ref 15. A binary parameter k12 )-0.6 between L-glutamic acid and L-aspartic acid is used.
a
RD ) [solubility(cal) - solubility(exp)]/solubility(exp) × 100.
estimated for amino acids using group contribution method. These properties are used to determine the SAFT parameters for amino acids. With the parameters determined, the SAFT equation of state is applied to model the amino acid solubility in water and an aqueous solution containing another amino acid, and good agreement between modeling results and experimental data are obtained. Acknowledgment Figure 8. Solubilities of L-glutamic acid and L-serine in an aqueous solution at 25, 40, and 60 °C. Data from ref 15. A function of binary interaction -5 2 parameter kdis 12 ) -0.68 + 0.01167T - 6.66667 × 10 T between L-serine and L-glutamic acid is used.
the relative deviation of calculations by the SAFT model for the ternary systems. Conclusions The hypothetical critical properties of temperature, pressure, and volume as well as normal boiling point temperature are
This work was supported by the National Science Foundation of China (Grant Nos. 20676009 and 20676014) and the National Basic Research Program of China (2007CB714302) Appendix: Statistical Associating Fluid Theory (SAFT) Equation of State The SAFT equation of state we have applied is the version of Huang and Radosz.9 The general expression for the Helmholtz energy is given by
Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 6279
a )a res
+a
seg
(
a 4η - 3η )m + RT (1 - η)2 seg
+a
chain
2
assoc
∑ ∑ D [ kTu ] [ ητ ] i
ij
i
j
)
j
1 - 0.5η achain ) (1 - m)ln RT (1 - η)3 aassoc ) RT
∑ [ln X
A
]
A
(A2) (A3)
XA 1 + M 2 2
-
(A1)
(A4)
The mole fraction of molecules not bonded at site A is expressed as XA ) [1 + NAV
∑ FX ∆ B
AB -1
]
(A5)
B
where ∆AB, the association strength, is expressed as (A6) ∆AB ) g(d)hs[exp(εAB/kT) - 1](σ3κAB) In the above expressions, the reduced density is calculated by
[
[ ]]
-3u0 3 (A7) kT and the temperature-dependent dispersion energy of interaction between segments, u/k, is expressed as η ) τFmυ00 1 - C exp
e u u0 ) 1+ (A8) k k kT Cand e/k are constants set to 0.12 and 10.0, respectively. For oxygen and carbon dioxide, e/k are set to 0 and 40, respectively. For each fluid, 3 parameters are needed for a nonassociating component, segment number, m, segment volume, υ00, and segment-segment interaction energy, u0/k. And, two additional parameters are needed for an associating component, the association energy εAB/k and volume κAB. For mixtures, the following mixing rule9 is used
[
]
∑ ∑ X X m m [ kT ](υ ) uij
u ) k
i j
i
i
0
j
ij
j
∑ ∑ X X m m (υ ) 0
i j
i
i
j
(A9)
ij
j
where (υ0)ij )
[ 21 [(υ )
0 1⁄3 0 1⁄3 3 i + (υ )j ]
( )
ui uj uij ) (1 - kdis ij ) k k k
]
(A10)
1⁄2
List of Symbols a ) Helmholtz energy Dij ) square-well energy constants g(d)hs ) hard sphere distribution function k ) Boltzmann constant m ) SAFT parameter, segment number NAV ) Avogadro’s number Nass ) number of association sites on a molecule P ) pressure R ) gas constant T ) temperature TB ) boiling point temperature TC ) critical temperature PC ) critical pressure VC ) critical volume
(A11)
u/k ) temperature-dependent dispersion energy of interaction between segments (K) u0/k ) SAFT parameter, temperature-independent dispersion energy of interaction between segments (K) υ00 ) SAFT parameter, temperature-independent segment volume (cm3/mol) X ) mole fraction XA ) mole fraction of molecules not bonded at site A Greek Symbols κAB ) SAFT parameter, volume of interaction between sites A and B εAB/k ) SAFT parameter, association energy interaction between sites A and B ∆AB ) strength interaction between sites A and B η ) reduced density F ) density τ ) constant, 2π/6 kijdis ) binary interaction parameter for u/k between i and j kijass ) binary interaction parameter for εAB/k between i and j
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ReceiVed for reView February 25, 2008 ReVised manuscript receiVed April 25, 2008 Accepted May 2, 2008 IE800313H
