Ind. Eng. Chem. Res. 1996, 35, 4319-4327
4319
A Simplified Perturbed Hard-Sphere Model for the Activity Coefficients of Amino Acids and Peptides in Aqueous Solutions Mohammad K. Khoshkbarchi and Juan H. Vera* Department of Chemical Engineering, McGill University, Montreal, Quebec, Canada H3A 2A7
A simple perturbed hard-sphere model has been developed to correlate the activity coefficients of amino acids and peptides in aqueous solutions. The perturbation terms are those due to dispersion forces and dipole-dipole interactions. These interactions are represented by a Lennard-Jones (6-12) and a Keesom expression, respectively. Dipole moments of amino acids and peptides, calculated by a quantum mechanical approach using the Hyperchem molecular modeling software, are reported. The model can accurately correlate the activity coefficients of amino acids and peptides in aqueous solutions. The model was used to correlate and predict the solubilities of amino acids in aqueous solutions at different temperatures. New values of ∆g and ∆h of solution of amino acids are reported. Introduction The cost of separation and concentration of biomolecules from dilute aqueous media, in which they are usually produced, can be as high as 90% of their total cost of manufacturing (Eyal and Bressler, 1993). Separation processes based on precipitation and crystallization have been widely used for concentration and separation of biomolecules (Belter et al., 1988). For the design of equilibrium-based separation processes of biomolecules, the accurate prediction of their activity coefficients in solutions is essential. Since amino acids and peptides are the basic units of other biomolecules, such as proteins, in this study we focus our attention on the development of a simple model for their activity coefficients in aqueous solutions. The correlations of activity coefficients and solubilities of amino acids have been the subjects of many studies. Nass (1988) modeled the activity coefficients of a few amino acids by assuming that they were the product of a chemical reaction equilibrium term and a physical interactions term. The chemical reaction equilibrium term was justified using the fact that amino acids in aqueous solutions undergo equilibrium ionization reactions and form different ionic species. For the physical term, the Wilson equation (Wilson, 1964) was used. Chen et al. (1989) represented the activity coefficients of amino acids and peptides using their own version of the electrolyte nonrandom two liquid (NRTL) model (Chen et al., 1982). In this model, long-range interactions are represented by a Pitzer-Debye-Hu¨ckel form (Pitzer, 1980) and short-range interactions are given by a modified form of the NRTL equation (Renon and Prausnitz, 1968). Gupta and Heidemann (1990) used the universal functional activity coefficient (UNIFAC) model, as modified by Larsen et al. (1987), to model the activity coefficients of amino acids in water and used the same group parameters to predict the solubilities of antibiotics in water. Their model has the advantage of predicting solubilities of antibiotics in water, but suffers from the definition of too large groups and fails to correlate well the activity coefficients of some amino acids and peptides. Pinho et al. (1994) used the original UNIFAC model (Fredenslund et al., 1975) for the activity coefficients of amino acids in aqueous solutions. Their model, similar to the model proposed by Nass (1988), is a combination of a physical and a chemical * Author to whom correspondence should be addressed.
S0888-5885(96)00076-0 CCC: $12.00
equilibrium term with group sizes which are smaller than those used by Gupta and Heidemann (1990). In contrast to the work of Gupta and Heidemann (1990), Pinho et al. (1994) introduced a Debye-Hu¨ckel term to account for the activity coefficient of the different ionic species resulting from the ionization of amino acid molecules. This term seems to be unnecessary for aqueous solutions of amino acids due to the negligible number of amino acid molecules with a net charge in the absence of an electrolyte or of a proton donor or acceptor. In the present work, this term has been omitted. In all the above publications, the activity coefficients of amino acids in aqueous solutions were represented either by empirical local composition or by group contribution models. The perturbation theory is an attractive alternative for modeling the activity coefficients of amino acids and peptides in aqueous solutions. This theory has the advantage of having a sound thermodynamic basis, and it has proved to be able to represent Monte Carlo simulation data for many systems such as mixtures of Lennard-Jones potential molecules (Grundke et al., 1973) and mixtures of squarewell potential molecules (Bokis and Donohue, 1995). Another advantage associated with the perturbation theory, when compared to phenomenological local composition or group contribution methods, is the possibility of incorporating different forms of interactions arising from different structure and energy characteristics of the components in the system. This theory, on the other hand, has the disadvantage of being in the form of an infinite series. This introduces truncation errors in the calculations for real fluid systems. The perturbation theory was first developed by Zwanzig (1954) and was later applied to canonical ensemble partition functions by Rowlinson (1964) and by McQuarrie and Katz (1966). The theory was extended to liquids by Henderson (1974). In the perturbation theory, the total interaction energy of the system is divided into a reference part and a perturbed part. The perturbed part is expressed in the form of a series in which the first-order term involves the two-body interactions and the higher order terms involve the three or higher body interactions. In this study we have developed a simplified perturbed hard-sphere model for the activity coefficients of amino acids and peptides in aqueous solutions. Since amino acids are compounds with very large dipole moments (Cohn and Edsall, 1965), the interaction © 1996 American Chemical Society
4320 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
energies for the perturbation term are considered to be due to dispersion forces, represented by a LennardJones (6-12) expression, and to dipole-dipole interactions, represented by an angle-averaged dipole-dipole interaction in the form of the Keesom equation (Maitland et al., 1981). As the dipole moments of amino acids are not experimentally measurable, these values were calculated by a quantum mechanical approach using the Hyperchem molecular modeling software. The activity coefficients of amino acids, calculated with the model, have been applied to correlate the solubilities of amino acids in aqueous solutions. Thermodynamic Framework In order to model the activity coefficients of amino acids in aqueous solutions, the perturbation theory in a primitive form has been employed. For the sake of simplicity and also due to the fact that the activity coefficient experimental data available in literature are only for single-component amino acid or peptide-water systems, in this section the derivation of the model is restricted to the single-component primitive systems. An extension of the model for multicomponent systems is presented in Appendix A. The primitivity of the model implies that the solvent in the system is treated as a dielectric continuum and the system solute/water is reduced to the treatment of a single-component system. The primitive nature of the model overcomes the shortcoming of the perturbation theory to represent systems with strong hydrogen bonding. In this theory, the chemical potential of a solute is considered to be composed of a contribution from a reference system and a perturbed term:
µi ) µiRef + µiPer
(1)
where the superscripts Ref and Per denote the contributions of the reference system and perturbation term, respectively. In this work, the contribution of the reference system to the chemical potential is represented by an equation of state for hard spheres proposed by Mansoori et al. (1971). This equation for the solute, considered as a single-component system, can be written as
µihg - µihg,ig 8ξ + 9ξ2 + 3ξ3 ) kT (1 - ξ)3
(2)
π ξ ) Fσ3 6
(3)
with
where σ is the size parameter, F is the number density of the component, T is the absolute temperature, k is the Boltzmann constant, and superscripts hs and ig denote the hard-sphere and ideal gas state, respectively. The contribution of the perturbed term to the chemical potential of the system is represented by an expression based on the Barker-Henderson perturbation theory (Barker and Henderson, 1967) truncated after the first perturbation term. Following Tiepel and Gubbins (1973), the chemical potential based on the first-order BarkerHenderson perturbation theory for a single-component primitive system can be written as
∫σ∞u(r) ghs(r) r2 dr
µiPer ) 4πF
(4)
where u(r) is the interaction energy as a function of intermolecular distance, r, and ghs(r) represents the radial distribution function of the reference term, i.e., of the hard-sphere system. The perturbation interaction energies include the effect of the dispersion, dipole, quadrupole, and induction forces. Due to the primitive nature of the model employed and the very large dipole moment of amino acids, only the contributions of dispersion and dipole-dipole interactions are considered here. The contributions of quadrupole and induction forces are neglected in comparison with the dipole-dipole forces. In this work the dispersion forces are represented by a Lennard-Jones (6-12) model as (Maitland et al., 1981)
(
uL-J(r) ) 4
)
σ12 σ6 r12 r6
(5)
where is the depth of the potential well according to the Lennard-Jones model. The dipole-dipole interactions are represented by an angle-averaged Keesom dipole-dipole interactions model as (Maitland et al., 1981)
uD-D(r) ) -
D4 3(4π0r)2kTr6
(6)
where D denotes the dipole moment of the component, 0 is the permittivity of the vacuum, and r is the relative dielectric constant of the medium. In principle, the radial distribution function used in the perturbed term should be the same as the one used in the hard-sphere term. Since this renders the analytical derivation of the model impossible, following the method proposed by Reed and Gubbins (1973), a uniform radial distribution function in the perturbed term is assumed as
gPer(r) )
{
0 1
rσ
(7)
where gPer(r) denotes the radial distribution function used in the perturbed term in eq 4. It should be emphasized that, while the assumption of the uniform radial distribution function is an approximation, and it may be considered to introduce an inconsistency in the model, it only implies that the perturbation is uniformly distributed and preserves the distribution of the reference system. Furthermore, it results in a simple algebraic equation for the activity coefficient. This, in turn, largely reduces the calculation time, which is a desirable condition for engineering applications. This advantage is expected to become more important in the case of multicomponent systems. The unsymmetric activity coefficient of a solute, γi, is related to its chemical potential by the following exact thermodynamic relation:
µi - µiid ) kT ln γi
(8)
The term µiid in eq 8 is the chemical potential of the component i in a hypothetical ideal solution at the same composition and temperature of the system. In the primitive model used here, since the solvent acts as a dielectric continuum, the ideal solution behavior corresponds to the hard-sphere ideal gas state. In this limit, all attractive energies between the solute molecules are zero and the contribution of the perturbation term to
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4321
the (ideal gas) chemical potential is null. Thus, the activity coefficient of the solute given by eq 8 is normalized to unity at the reference state of infinite dilution. Combining eqs 1-8, we obtain the unsymmetrically normalized activity coefficient of the solute in a singlecomponent primitive system as
8ξ + 9ξ2 + 3ξ3 ln γi ) (1 - ξ)3 D4 4πF 8 3 σ + kT 9 9(4π )2kTσ3
{
0 r
}
(9)
In eq 9, the number density F is directly related to the molality of the amino acid in the solution. However, the data for activity coefficients are usually reported in terms of molality of the amino acid. On the other hand, the density of the amino acid solutions is rarely available in the literature (Ellerton et al., 1964). Thus, as a first approximation, we have used the molality to evaluate the number density. It should be mentioned that, in addition to the interaction energies employed in the model proposed here, other forms of the interaction energies can be incorporated in the perturbed term. This provides flexibility for the application of the model to more complex systems such as those composed of mixtures of amino acids and amino acid-water-electrolyte systems. An extension of the present model pf the multicomponent systems is presented in the appendix. Evaluation of Parameters In order to correlate the activity coefficients of amino acids and peptides in aqueous solutions using the model proposed in this study, it is necessary to evaluate the size parameters, the depths of potential wells in the Lennard-Jones energy expression, and the dipole moments. The dipole moments of amino acids, due to their highly polar nature, which makes them insoluble in organic solvents, are not experimentally measurable (Greenstein and Winitz, 1961). The values reported in the literature, calculated from geometrical considerations, are scarce and rather old and inaccurate. Therefore, in this study, the dipole moments of the amino acids and peptides were calculated from a quantum mechanical approach using the Hyperchem molecular modeling software. The calculated values are presented in Table 1 and compared with literature values when available. The values reported in Table 1 are based on the quantum mechanical description of the dipole moment, in which the charge is a continuous distribution and the dipole moment is an average over the wave function of the dipole moment operator. To calculate the dipole moment, the geometry of the amino acid or peptide was first established by a geometry minimization procedure based on a Polak-Ribiere conjugate gradient method by minimizing the root mean square of the gradient to less than 0.1 kcal/Å mol on a MM+ force field, as proposed by Allinger (1977). The Hamiltonian function of the molecule was then solved through a geometry minimization procedure based on a PM3 semiempirical method, as proposed by Stewartt (1980a,b). It should be emphasized that the dipole moments used here are not adjustable parameters of the model. They are the best estimates of their values obtainable with up-to-date tools. The size parameter and the depth of potential well for each amino acid and peptide were considered to be
Table 1. Dipole Moments of Some Amino Acids and Simple Peptides in Debye Units amino acid alanine R-amino-n-butyric acid glycine hydroxyproline proline serine threonine valine β-alanine γ-amino-n-butyric acid -amino-n-caproic acid
dipole momenta
dipole momentb
9.53 10.87 11.50 9.34 10.56 10.34 9.80 10.68 17.55 11.91 10.72
12 12 12 12 15 22
peptide alanylalanine alanylglycine glycylalanine glycylglycine triglycine
21.07 22.13 21.72 23.09 37.54
21 21 21 21 26
a Calculated in this work. b Reported by Greenstein and Winitz (1961).
adjustable parameters and were evaluated from the curve fitting of the experimental data. For all cases, to evaluate the size parameter and the depth of potential well, the following objective function was minimized:
OF ) (γiexp - γicol)2
(10)
For each amino acid or peptide-water system, the model proposed here contains only two adjustable parameters: the depth of potential well and the size parameter. Results for the Correlation of Activity Coefficients The model was employed to correlate the activity coefficient data of several amino acids and peptides reported in the literature (Fasman, 1976). As the activity coefficient calculated from the model is normalized on the mole fraction base, the activity coefficients of amino acids reported in the literature were converted from the molality base to the mole fraction base using the following relation (Robinson and Stokes, 1970):
γ(x) ) γ(m)(1 + 0.001Msm)
(11)
where MS is the molecular weight of the solvent and the superscripts (m) and (x) represent the molality and the mole fraction base, respectively. The activity coefficients are normalized according to the unsymmetric convention. Figures 1 and 2 show the results of the correlation of the activity coefficients of several amino acids, at their natural pH, as a function of molality. Two different figures are presented due to the different range of molalities required by the data. Figure 3 shows the result of the correlation of the activity coefficients of four simple peptides as a function of their molalities. As shown in these figures, the model can accurately represent the activity coefficient experimental data of amino acids and peptides in aqueous solutions. The values of the regressed size parameter, depth of the potential well, and the root mean square deviation (rmsd) of the correlation of the experimental data for amino acids and peptides studied are presented in Tables 2 and 3, respectively. The values obtained for
4322 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
Figure 1. Activity coefficients of five amino acids at various molalities: b, experimental data (Fasman, 1976); s, Result of the modeling.
Figure 3. Activity coefficients of four peptides at various molalities: b, experimental data (Fasman, 1976); s, result of the modeling.
carboxyl group gains a proton and the amino acid molecule becomes positively charged, whereas in the presence of a proton acceptor agent, the amino group of the amino acid loses a proton and the amino acid molecule becomes negatively charged. The formation of different ionic forms of the amino acids can be shown by the following reactions:
Figure 2. Activity coefficients of three amino acids at various molalities: b, experimental data (Fasman, 1976); s, result of the modeling.
the size parameters and the depths of the potential wells of the amino acids are physically meaningful and are within the range of the size parameters and the depths of the potential wells of other chemicals reported in the literature (Reed and Gubbins, 1973; McQuarrie, 1976). The comparison of the rmsd of the correlation of the experimental data using the model proposed in this work and the results reported by Chen et al. (1989), Gupta and Heidemann (1990), and Pinho et al. (1994), when available, are also presented in Tables 2 and 3. As it can be seen from this comparison, the model proposed here produces a better correlation than any of the other models for all peptides and for many amino acids. It never gives unexpectedly high deviations as the other models do. For proline, the amino acid with large solubility in water, the deviation is probably due to the use of the density of water instead of the density of the solution. It is important to mention that the model proposed in this work has a simpler algebraic form than the other models, and it contains only two adjustable parameters per amino acid or peptide. An Application To Model the Solubility of Amino Acids Amino acids, when dissolved in aqueous solutions, form several ionic species due to the ionization of their carboxyl and amino groups. After dissolving in water, the carboxyl group of the amino acid loses a proton and becomes negatively charged, whereas the amino group gains a proton and becomes positively charged. Therefore, an amino acid molecule possesses, at the same time, a negative and a positive charge (zwitterion). In the presence of a proton donor agent, the amino acid
(a) NH2RCOOH / NH3+RCOO-
K1
(b) NH3+RCOOH / H+ + NH3+RCOO-
K2
(c) H+ + NH2RCOO- / NH3+RCOO-
K3
where K1, K2, and K3 are the equilibrium constants, defined as
K1 ) [NH3+RCOO-]/[NH2RCOOH]
(12)
K2 ) [H+][NH3+RCOO-]/[NH3+RCOOH] (13) K3 ) [NH3+RCOO-]/([H+][NH2RCOO-])
(14)
In aqueous solutions, almost all molecular amino acid molecules convert to the zwitterionic form according to reaction a, above. This, in turn, corresponds to a large value of K1, as it has been experimentally confirmed by Greenstein and Winitz (1961) and by Cohn and Edsall (1965). In the absence of a proton donor or acceptor, up to 99% of the amino acid molecules are in their zwitterionic form. This is reflected in the large values of the K2 and K3 (Cohn and Edsall, 1965). Therefore, it is reasonable to assume that, in the absence of a proton donor or acceptor, the physicochemical properties of different ionized forms of the amino acids in aqueous solutions are represented by those of the zwitterionic form. In other words, the activity coefficients of all ionic forms of amino acids in the absence of a proton donor or acceptor are considered to be equal. This is the reason why eqs 12-14 are written as the ratios of mole fractions rather than activities. For a system composed of an aqueous solution of an amino acid in equilibrium with a solid phase of pure amino acid, we can write
fiL ) fiS
(15)
Expressing the fugacity of the amino acid in the liquid phase, in terms of its unsymmetrically normalized activity coefficient, γi, the mole fraction, xi, and the
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4323 Table 2. Regressed Values of σ and E/k for Several Amino Acids and Root Mean Square Deviation of Correlation of Activity Coefficient Experimental Data amino acid
/k (K)
σ × 1010 (m)
rmsd × 100a
rmsd × 100b
rmsd × 100c
rmsd × 100d
alanine R-amino-n-butyric acid glycine hydroxyproline proline serine threonine valine β-alanine γ-amino-n-butyric acid -amino-n-caproic acid
86.40 83.00 165.40 95.50 28.00 189.33 122.40 8.30 121.10 124.50 129.70
4.09 4.73 4.76 4.08 3.79 4.83 4.50 4.22 5.36 5.44 5.75
0.33 0.37 0.86 0.09 4.12 1.16 0.15 0.05 0.42 0.38 1.12
0.04 0.32 2.07 0.36 1.30 2.80 0.70 4.47
8.97 17.34 4.20 0.06 3.01 3.32 13.78 12.15
0.19 17.84 1.67 0.39 1.21 0.24 26.12 16.43
a
This work. b Chen et al. (1989). c Gupta and Heidemann (1990).
Table 3. Regressed Values of σ and E/k for Several Simple Peptides and Root Mean Square Deviation of Correlation of Activity Coefficient Experimental Data peptide
/k (K)
σ × 1010 (m)
rmsd × 100a
rmsd × 100b
rmsd × 100c
alanylalanine alanylglycine glycylalanine glycylglycine triglycine
114.50 149.65 147.90 170.20 157.80
6.50 7.03 6.83 6.50 8.70
0.43 0.60 0.57 1.22 0.00
2.04 2.92 2.44 3.25 0.67
0.52 28.22 0.59 17.06 10.00
a
This work. b Chen et al. (1989). c Pinho et al. (1994).
fugacity at its standard state, fiθL, yields:
xiγifiθL ) fiS
(16)
where fiθL is the standard state fugacity for the case when the activity coefficient approaches unity as the amino acid approaches infinite dilution. From the definition of fugacity, the ratio fiS/fiθL can be related to the temperature of the system through the following relation.
fiS fiθL
∆h (∆sR - RT )
) exp
(17)
where ∆s and ∆h are the change in the molar entropy and enthalpy of the amino acid from the standard state to the solid state and T is the absolute temperature. Combining eqs 16 and 17, we write
∆h (∆sR - RT )
xiγi ) exp
(18)
Once the activity coefficient of the amino acid is known in terms of the mole fraction of the amino acid in the liquid phase, the temperature dependence of the solubility data can be obtained using eq 18. The influence of pH on the solubility of amino acids can be implemented using the relation between the concentration of the hydrogen ions in the solution obtained from the ionization reactions and eq 18. As discussed before, at high or low pH values, the presence of different forms of the amino acid molecules should be considered. Following Gupta and Heidemann (1990), the mole fraction of the amino acid at different pH values can be written in terms of the mole fraction of different ionic forms of the amino acid by
[
xi ) xi( + xi+ + xi- ) xi( 1 +
]
K3 [H+] + + K2 [H ]
Combining eqs 18 and 19 yields
(19)
d
Pinho et al. (1994).
(
xiγi ) exp
)[
K3 [H+] ∆s ∆h 1+ + + R RT K2 [H ]
]
(20)
where [H+] denotes the molality of hydrogen ion in the solution and K2 and K3 are the equilibrium constants defined by eqs 13 and 14. It should be noted that the above procedure to account for the effect of pH is not exact. This is because the data for the activity coefficients of amino acids have been obtained at their isoelectric pH values. At this condition, the predominant ionic species of an amino acid is its zwitterionic form and the concentration of other ionic forms is negligible. On the other hand, experimental data for the solubility of amino acids at high or low pH values can be obtained only by addition of an acid or a base, which in turn introduces the counterion of the acid or base to the solution. The introduction of the counterion of the acid or base results in the interaction between the ions and amino acid molecules, which most probably will affect the activity coefficient of the amino acid. In order to correlate the solubility of an amino acid in aqueous solution, the values of ∆h, ∆s, and of the activity coefficients are required. The values of the activity coefficients can be directly obtained from the model for the activity coefficients of amino acids and peptides without adjusting any new parameters. The model proposed in eq 18 can be used to predict the solubility of amino acids in aqueous solutions, provided that values of ∆h and ∆s are available. In the absence of reported values for ∆h and ∆s, they can be evaluated from the curve fitting of the experimental solubilities data. In such case, to correlate the solubility of the amino acids over the whole range of temperatures, only two adjustable parameters are required. Results for the Solubility of Amino Acids The activity coefficient model with parameters presented in Tables 1 and 2, together with eq 18, was used to correlate the solubility data of amino acids in aqueous solutions at various temperatures. The values of the dielectric constant of water at various temperatures were calculated from the relation proposed by Malmberg and Maryott (1956). The values of ∆s/R and ∆h/R were initially treated as two adjustable parameters. The activity coefficients of the optical isomers of amino acids are assumed to be equal. Experimental measurements of the activity coefficients of optical isomers of some amino acids in aqueous solutions at low to moderate concentrations also confirm the validity of this assumption (Robinson et al., 1942; Schrier and Robinson, 1974). Although this assumption might not be accurate at concentrations near the saturation point, in the absence
4324 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 Table 4. Regressed Values of the Parameters of Eq 18 for Correlating the Solubility Experimental Data of Several Amino Acids and Peptides in Aqueous Solutions Reported by Fasman (1976) amino acid
∆h/R (K)a
∆s/Ra
988.37 1115.07 1654.55 1054.60 959.77 2265.35 2663.05 1264.96 498.01
-0.2174 0.2127 2.4184 0.7170 2.6235 2.7173 5.7438 -0.1305 -2.8980
L-alanine DL-alanine
glycine hydroxyproline L-proline DL-serine L-serine DL-valine L-valine
∆h/R (K)b
∆s/Rc
920.9 1107.2 1711.1 704.6 654.2 2717.7
-0.30
754.9 452.9
2.61 -0.59 1.96
-3.27
a
Figure 4. Solubility of seven amino acids at various temperatures: b, experimental data (Fasman, 1976); s, result of the modeling.
Regressed from eq 18. b Reported by Fasman (1976). c Calculated from values of ∆g and ∆h of Fasman (1976), using eq 25. Table 5. Calculated and Experimental Values of ∆g/RT0, T0 ) 298.15 K amino acid L-alanine DL-alanine
glycine hydroxyproline L-proline DL-serine L-serine DL-valine L-valine
∆g/RT0a
∆g/RT0b
3.532 3.527 3.131 2.820 0.595 4.881 3.188 4.373 4.555
3.431 3.165 2.990 0.271 3.168 4.833
a
Calculated from the left hand side of eq 24 and values reported in Table 4. b Calculated from the values reported by Fasman (1976) according to the right hand side of eq 24. Figure 5. Solubility of proline at various temperatures: b, experimental data (Fasman, 1976); s, result of the modeling.
transferring 1 mol of solute from the saturated solution to a hypothetical aqueous solution at an activity of 1 m and at T0 ) 298.15 K, ∆gF°. From the definition of fugacity, we write
∆gF° fiid(1 m) ) ln RT0 fs
(21)
i
However, as shown in Appendix B, fiid(1 m) and fiθL are directly related by
fiid(1 m) ) 0.001MwfiθL
(22)
where Mw is the molecular weight of water. Combining eqs 21 and 22 and substituting the numerical value of Mw ) 18.02, for water, yields Figure 6. Solubility of glycine at various pH values: b, experimental data (Needham et al., 1971); s, result of the modeling.
of the experimental data it can be accepted. Figure 4 shows the correlation of the solubility of some amino acids in aqueous solutions as a function of temperature. Figure 5 shows the correlation of solubility for proline, which is much more soluble in water than the other amino acids. As shown in these figures, the model developed in this study can accurately correlate the solubility of amino acids in aqueous solutions over a wide range of temperatures. The numerical values of the parameters of eq 18 are presented in Table 4. Unlike the models proposed by Chen et al. (1989) and Pinho et al. (1994), which require the regression of three parameters to correlate the solubility experimental data, the model proposed here requires only two parameters. It is interesting to compare the values of ∆h/R and ∆s/R obtained here with those that can be obtained from independent measurements. Fasman (1976) reports the value of the free energy change of the solute, for
ln
fiS fi
θL
)-
∆gF° - 4.016 RT0
(23)
Comparison of eqs 17, at T0 ) 298.15 K, and 23 gives
∆s ∆gF° ∆h ) + 4.016 RT0 R RT0
(24)
where the left hand side of eq 24 represents the value of ∆g/RT0 obtained in this work and the right-hand side represents the equivalent value which can be obtained from Fasman (1976) tables. These two values are compared in Table 5. As it can be seen from Table 5, the agreement between both values is quite satisfactory. One possible reason for the slight disagreement is that the values of ∆h/R and ∆s/R were adjusted over a wide temperature range, while the values from Fasman (1976) were obtained from measurements at around 298.15 K.
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4325
above procedure for prediction of the solubility of amino acids at different pH values is not exact. As mentioned before, this is due to the fact that the activity coefficients of amino acids available in the literature were measured at their isoelectric points. The presence of an acid or a base will produce net charges in the amino acids and also the counterions of the acid or base which will affect the activity coefficient of the amino acid. However, the prediction shown in Figure 6 shows that, despite these shortcomings, this model can, as other models previously did, represent the effect of pH on the solubility of amino acids with a reasonable accuracy. Conclusions Figure 7. Solubility of glycine and L-alanine at various temperatures: b, experimental data (Fasman, 1976); s, result of the two-parameter correlation of model; ‚‚‚ ) result of prediction of model.
A different comparison can be performed with the individual values of ∆h/R and ∆s/R. Fasman (1976) reports the enthalpy change of solution of the solid crystalline amino acid into an infinitely dilute solution. Since the enthalpy of the solute at its standard state is equal to the enthalpy of the solute at its reference state at infinite dilution, this value corresponds exactly to ∆h of this work. Thus, it is possible to back calculate a value of (∆s/R)F from Fasman’s tables as
( ) ( ) ( ) ∆s R
)
F
∆h RT
-
F
∆gF° - 4.016 RT0
(25)
This value, together with (∆h/R)F obtained directly from tables of Fasman (1976), has been added to Table 4. It is interesting to note that both experimental and regressed values show the possibility of both positive and negative values for ∆s/R. The difference between the regressed and back calculated values of ∆s/R is most probably due to some questionable simplifying assumptions used in the estimation of the experimental values of ∆h/R, mentioned by Fasman (1976). If independent values of ∆g/R and ∆h/R were available, eqs 18 and 20 could be used in a purely predictive manner. Figure 7 compares the results of the prediction and correlation of the solubility for glycine and L-alanine at various temperatures using both sets of parameters reported in Table 4. The model for activity coefficients of amino acids developed in this work can accurately predict the solubility data when the values of ∆g/R and ∆g/R are available and reliable. No adjustable parameter is required in this case. This verification indicates that the perturbation model reflects well the physical reality of the amino acids in aqueous solutions. It should be mentioned, however, that experimental values available for ∆g/R and ∆h/R are rather old and should be used with caution. Thus, the values regressed in this work are additions to the information available for these systems. As it has been previously done by other researchers (Nass, 1988; Chen et al., 1989; Gupta and Heidemann, 1990; Pinho et al., 1994), eq 20, together with the experimental values of the dissociation constants (Fasman, 1976), can be used to correlate the solubility data of amino acids at different pH values without any additional adjustable parameters. Figure 6 shows the result of the prediction of eq 20 for the solubility of glycine at different pH values. We emphasize that the
A two-parameter theoretical model based on a simplified perturbation theory of a hard-sphere reference system has been developed to correlate the activity coefficients of several amino acids and peptides in aqueous solutions. The two-parameter simple model can accurately correlate the activity coefficients and the solubilities of amino acids over a wide range of concentrations and temperatures. The values of the dipole moments of several amino acids and simple peptides, required for the Keesom expression in the perturbed term of the model, were calculated by a quantum mechanical approach using the Hyperchem molecular modeling software. The results of the model indicate that, due to the high dipole moments of amino acids and peptides, the dipole-dipole interactions play a central role in the chemical potential of amino acids. Using the proposed activity coefficient model, a model was developed to predict and correlate the solubilities of several amino acids in aqueous solutions. The results show that the model can accurately correlate the solubility experimental data over a wide range of temperatures with two adjustable parameters. It was found that, upon availability of independently evaluated experimental data for ∆h/R and ∆g/R, the model can be used to accurately predict the solubility of amino acids in aqueous solutions without any adjustable parameters. Appendix A The perturbed hard-sphere model proposed in this study can be generalized to model the activity coefficients of amino acids in multicomponent systems. Following Mansoori et al. (1971), eq 2 of the text can be written as
µihg - µihs,ig πPhsσi3 ) -ln(1-ξ3) + + E + 3F kT 6kT
(A1)
where
ξn )
E)
π
n
∑ Fkσkn
6k)1
3ξ2σi + 3ξ1σi2 (1 - ξ3)
(n ) 0, 1, 2, 3)
(A2)
9ξ22σi2 +
[
ξ23σi3 ξ33
2(1 - ξ3)2
-
2 ln(1-ξ3) +
]
ξ3(2 - ξ3) (1 - ξ3)
(A3)
4326 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
F)
[ [
ξ22σi2 ξ32
ln(1-ξ3) +
ξ32
ξ3 (1 - ξ3)
-
ξ23(3 - ξ3)
3ξ1ξ2
ξ0
6kT + + P ) π (1 - ξ3) (1 - ξ )2 (1 - ξ3)3 3 hs
] ]
2(1 - ξ3)2
(A4)
µi
n
∂ ) ∂Ni
n
FjFl∫σ ujl(r) gjlhs(r) r2 dr] ∑ ∑ j)1 l)1 ∞
[2πV
jl
(A5)
(A6)
In eq A6, N is the number of the molecules of component i in the system, V is the total volume of the system, r is the intermolecular distance, ujl(r) is the interaction energy as a function of intermolecular distance, and gjlhs(r) represents the radial distribution function of the hard-sphere reference. The sums run over all components. As discussed before, since the incorporation of the hard-sphere radial distribution function in eq A6 renders the analytical solution of the model impossible, following the method proposed by Reed and Gubbins (1973), the radial distribution function in the perturbed term is approximated by a uniform radial distribution function as
gijPer(r) )
{
r < σij r > σij
0 1
(A7)
where gijPer(r) denotes the radial distribution function used in the perturbed term in eq A6. The interaction energies in the perturbation term can include the effect of dispersion forces, dipole-dipole, and dipole-induced dipole terms, etc. The dispersion forces are represented by a LennardJones (6-12) model as (Maitland et al., 1981)
(
uijL-J(r) ) 4ij
σij12 r12
r12
)
(A8)
where mixing rules are required for σij and ij. The dipole-dipole interactions are represented by an angle-averaged Keesom dipole-dipole interaction model as (Maitland et al., 1981):
uijD-D(r) ) -
Di2Dj2
(A9)
3(4π0r)2kTr6
If necessary, dipole-induced dipole interactions can be directly added. The activity coefficient of a component is related to its chemical potential by eq 8 of the text. The combination of above equations provides the thermodynamic framework for the activity coefficients of the components using a perturbed hard-sphere model as:
ln γi )
µihg - µihg,ig
{
-
kT
4πFj 8 Di2Dj2 3 ijσij + j)1 kT 9 9(4π )2kTσ N
∑
0 r
dfi dm
(B1)
ni)0
where the subscript ni)0 indicates that the derivative is evaluated in the limit ni ) 0 at constant number of moles of solvent, nW. Thus, at an activity of 1 m,
fiid(1 m) )
( ) dfi dm
(B2)
ni)0
Applying the chain rule to the right-hand side of eq B2, we write
fiid(1 m) )
( ) ( ) dfi dxi
ni)0
dxi dm
(B3)
ni)0
From the definition of the standard-state fugacity, when the activity coefficient approaches unity as the solute approaches infinite dilution, in the mole fraction scale, we have
fiθL )
( ) dfi dxi
(B4)
ni)0
In addition, from the relation between mole fraction and molality, for a single solute in a solvent of molecular weight Mw,
( ) dxi dm
ni)0
) 0.001Mw
(B5)
Thus, combining eqs B3, B4, and B5, we obtain
fiid(1 m) ) 0.001MwfiθL
(B6)
Acknowledgment
σij6 -
( )
fiid ) m
The contribution of the perturbed term to chemical potential can be represented by the first-order BarkerHenderson perturbation theory (Barker and Henderson, 1967). Following Tiepel and Gubbins (1973), this equation for a multicomponent system can be written as Per
the solute approaches infinite dilution, in the molality scale, is given by
}
3 ij
(A10)
Appendix B The fugacity of a solute i in an ideal solution, for the case when the activity coefficient approaches unity as
The authors are grateful to the Natural Sciences and Engineering Research Council of Canada for financial support and to Professor M. A. Whitehead for his help in using the Hyperchem molecular modeling software. Nomenclature D ) dipole moment f ) fugacity g(r) ) radial distribution function h ) enthalpy K ) equilibrium constant k ) Boltzmann constant (1.381 × 10-23 J K-1) MS ) molecular weight of the solvent N ) number of molecules OF ) objective function P ) pressure R ) universal gas constant r ) intermolecular distance s ) entropy T ) absolute temperature u ) interaction energy V ) molar volume x ) mole fraction Greek Letters γ ) activity coefficient ) depth of potential well 0 ) permittivity of vacuum (8.854 × 10-12 C V-1 m-1)
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4327 r ) relative dielectric constant µ ) chemical potential ξ ) packing factor F ) number density σ ) size parameter Superscripts cal ) calculated D-D ) dipole-dipole interaction term exp ) experimental hs ) hard sphere id ) ideal solution ig ) ideal gas L ) liquid L-J ) Lennard-Jones Per ) perturbed term Ref ) reference term S ) solid θ ) reference state (m) ) molality base normalized (x) ) mole fraction base normalized
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Resubmitted for review April 29, 1996 Revised manuscript received July 23, 1996 Accepted July 30, 1996X IE960076X
Abstract published in Advance ACS Abstracts, October 1, 1996. X
