Ind. Eng. Chem. Res., Vol. 49, No. 3, 2010 1397 Table 3. PC-SAFT Parameters for Water24 m
σ
1.2047
see eq 8
/ k
N
AB hbi j / k
353.95
2
2425.67
Ai B j
k hb
0.0451
tures, the binary parameter is assumed to be constant (i.e., temperature independent) and was fitted to the experimental data of the solute with the lower solubility. For cross-association systems in this study, the strength of cross-associating interactions between two associating substances can be described by applying simple mixing and combining rules, as suggested by Wolbach and Sandler. 23 AiB j hb )
AiB j κ hb )
√
1 AiBi ( 2 hb
AiBi A jB j hb hb
[
+
A jB j hb )
(6)
√σ iiσ jj (1/2)(σ ii
+
σ jj)
]
3
(7)
Thus, no adjustable correction parameters have to be introduced in the association term. The PC-SAFT parameters used for the modeling are given in Tables 3 and 4, where N denotes the number of the association sites acting as proton donators and as proton acceptors. Most amino acids are modeled with two association sites, with the amino group acting as a proton acceptor and the acid group acting as a donor. As amino acids are zwitterionic molecules, the overall charge is mostly neutral and the ionic character is not regarded in modeling. We used a temperature-dependent segment diameter for water as described by Cameretti and Sadowski.24 The calculation for the segment diameter is given in eq 8, where σ represents the segment diameter and T is the temperature (in kelvin): σ (T )
)
2.7927
+
10.11 exp(-0.01775T ) 1.417 exp(- 0.01146T ) (8)
Parameter Estimation
The PC-SAFT parameters of glycine, DL-alanine, and L-valine have already been determined in previous works by Fuchs et al.9 and Cameretti and Sadowski. 24 In this work, the parameters for glycine, L -alanine, L-valine, L -aspartic acid, L-glutamic acid ( β-form), L-leucine, and L-tyrosine were fitted to different
experimental data, such as solubilities, binary mixture densities, and amino acid activity coefficients (see Table 4). With the obtained parameters, not only solubilities but also solution densities and (water and amino acid) activity coefficients can be described (see Table 4). L -Aspartic acid and L -glutamic acid were assumed to exist in a neutrally charged form in aqueous solution. On the one hand, this differs from the real solutions as both amino acids are not only present as neutral zwitterions but also present as anions and cations (e.g., L -glutamic acid, p I 5 ) 3.217 ). On the other hand, we describe this by applying more than one association site acting as acidic groups. Furthermore, some of the former parameters were readjusted as the values of the melting properties were unphysically high (e.g., 24 T SL L-valine ) 1800 K ). Now the values of the melting temperatures are more reasonable (for L-alanine, T SL ) 692.4 K, and for L-valine, T SL ) 680.0 K). The parameters used for the modeling are listed in Tables 3 and 4, as are the average absolute deviation (AAD, see eq 9) and the average relative deviation (ARD, see eq 10) of the calculated binary data to the experimental data. NP
AAD )
1 exp |( ycalc - yk )| ∑ k NP k )1 NP
ARD )
|(
calc
yk 1 1 - exp ∑ NP k )1 yk
(9)
)|
(10)
Considered Ternary and Quaternary Mixtures
We considered ternary and quaternary mixtures of amino acids in water with one pure amino acid in the solid phase. Thus, mixtures forming a solid solution, such as L-leucine12 L-isoleucine-water, are not included. Furthermore, the solubility of amino acids in water is pH-dependent (e.g., refs 9, 28, and 34). This dependency is expressed by an increased solubility at pH values near the p K a values of the amino acid. We first do not take into account the pH influence; hence, the considered amino acids shall possess similar isoelectric points and p K a
Table 4. Pure-Component and Binary PC-SAFT Parameters for Amino Acids, Calculated 10 and Adjusted Melting Properties, and Deviations between Correlated and Experimental Data
parameter m
σ
/ k [K] N A Bi k [K] hbi /
AB κhbi i
T SL [K] SL R [K] ∆h / SL T calc [K] SL R [K] ∆hcalc / k ij,25 °C(H2O) k ij,T (H2O)
solution density T range [K] ARD [%] AAD [kg/m 3] solubility T range [K] ARD [%] AAD [mol/kg] amino acid activity coeff T range [K] ARD [%] AAD
L-Ala
L-Asp
L-Glu
Gly
L-Leu
L-Tyr
L-Val
5.4647 2.5222 287.59 2 3176.60 0.0819 692.4 2543.7 580.58 2749.4 -2 -6.12 × 10 -4 2.91 × 10 ref 25 298 0.20 0.23 this work 288-346 1.05 0.02 ref 30 298 0.07 <0.01
2.9998 3.3668 207.74 3 3265.67 0.0436 619.0 2802.7 595.43 3241.3 -4 -1.92 × 10
3.0248 3.4781 164.54 3 2536.56 0.0160 586.8 3022.6 596.0 3558.8 -1 -1.29 × 10
4.8495 2.3270 216.96 2 2598.06 0.0393 714.3 2109.3 462.50 3415.7 -2 -6.12 × 10
ref 26 298 0.03 0.25 ref 5 273-373 5.68 0.01 n.a.
this work 298.43 0.01 0.1 ref 29 278-342 2.23 <0.01 ref 31 310 0.14 <0.01
ref 25 298 0.09 0.96 ref 5 298-373 2.88 0.13 ref 32 298 1.64 0.01
8.3037 2.7000 330.00 2 3600.00 0.0200 620.9 4499.8 582.55 3330.3 -2 -6.30 × 10 -4 4.09 × 10 ref 27 298 0.03 0.25 this work 288-346 1.98 <0.01 n.a.
8.1390 2.2798 289.37 3 2500.00 0.0400 542.5 5000.3 601.67 4764.0 -4 -2.77 × 10 -4 2.90 × 10 ref 28 298 -318 0.01 0.02 ref 5 273-373 5.68 0.01 n.a.
6.5370 2.7211 397.07 2 3332.49 0.0386 680.0 3197.2 581.83 3012.8 -2 -6.15 × 10 -4 3.85 × 10 ref 27 298 0.02 0.17 this work 303-346 2.30 0.02 ref 33 298 0.18 <0.01
