Ind, Eng. Chem. Fundam. 1983, 22, 249-258
solubility in the aqueous solution p* = H[C02]
(AW
The equilibrium vapor pressure of C02can be calculated from eq 3,4, A5-A8, and All-A12 as
Nomenclature AMP = 2-amino-2-methyl-1-propanol DEA = diethanolamine DIPA = diisopropanolamine MDEA = N-methyldiethanolamine MEA = monoethanolamine PE = 2-piperidine ethanol ai = concentration of dissolved COz at the interphase (pi/H) a. = concentration of dissolved COz in the liquid bulk (p,/H) fb)= (S- [SZ - 4y (1 - y)]'/2)/2 H = Henry's law constant for COP Ka = ([RZ"I[H+IJ/([R~NH~+I) k , = rate constant for reaction of C 0 2 with amine K c = [RzNC00-]/{ [RZNH] [HCOS-]) Kc, = W+I[HCO,-II/ [COP1 Kcz = W+l[co32-ll/ W O 3 - I k L = N / ( q - ao) k , = rate constant for zwitterion formation from amine and COZ m = total molar concentration of amine N = COz absorption flux per unit area p* = equilibrium vapor pressure of COz p e , pi = partial pressure of COz in the bulk and at the interphase, respectively s = 1 + (l/Kcrn)
249
y = moles of COP absorbed per mole of amine Registry No. COz, 124-38-9;KzC03, 584-08-7;HOCHzCHzNHCOZH, 6998-39-6;( H O C H ~ C H ~ ) ~ N C O 84836-10-2; ~H, HOCHZC(CH3)2NHC02H,84836-11-3; 2-amino-2-methy1-1-propano1, 124-68-5;2-piperidine ethanol, 1484-84-0. Literature Cited Astarita, 0. "Mass Transfer with Chemical Reaction"; Elsevier, 1967. Capiow. M. J. Am. Chem. SOC. 1868, 90, 6795. Chan, H. M.; Danckwerts. P. V. Chem. Eng. Sci. 1881, 36, 119. Danckwerts, P. V.; McNeil, K. M. Trans. Inst. Chem. Eng. 1967, 45,T32. Frahn, J. L.; Mliis, J. A. Aust. J. Chem. 1864, 17, 256. Isaacs, E. E.; Otto, F. D.; Mather, A. E. Can. J. Chem. Eng. 1874, 52, 125. Jensen, M. B.; Jorgensen, E.; Faurholt, C. Acta Chem. Scand. 1954, 8 , 1137. Jensen, M. B. Acta Chem. Scand. 1857, 1 1 , 499. Jou, F. Y.; Lal, D.; Otto, F. D.; Mather, A. E. "The Solubllity of H2S and COP in Aqueous Methyldiethanolamine Solutions", Paper 20e, presented at AIChE Meeting, Houston, Apr 5-9, 1981. Jones, J. H.; Fronlng, H. R.; Clayton, E. E., Jr. J. Chem. Eng. Data 1858, 4 , 85. Kent, R. L.; Eisenberg, 8. "Proceedings, Gas Conditioning Conference"; University of Okalahoma, 1975; Paper E. Kohl, F.; Reisenfeld, F. C. "Gas Purification", 3rd ed., Gulf Publishing: Houston, 1979. Lee, J. I.; Otto, F. D.; Mather, A. E. Can. J. Chem. Eng. 1878, 54,214. Melchior, M. T., Exxon Research and Engineering Company, Linden, NJ, private communication, 1977a. Melchior, M. T., private communlcatlon, 1975. Melchior, M. T., private communication, 1977b. Savage, D. W.; Astarita, G.; Joshi, S. Chem. Eng. Sci. 1880, 358 1513. Sharma, M. M. Trans. Faraday SOC. 1865, 61, 661. Tennyson, R. N.; Schaaf, R. P. Oil Gas J . 1877, 75(2), 78. Tosh, J. S.;Field, J. H.; Benson, H. E.; Haynes, W. P. "Equilibrium Study of the System Potassium Carbonate, Potassium Bicarbonate, Carbon Dioxide, and Water"; US. Bureau of Mines Report of Investigations, No. 5484. 1959.
Received for review August 31, 1981 Revised manuscript received January 3, 1983 Accepted January 21, 1983
Thermodynamics of Highly Solvated Liquid Metal Solutions Montgomery M. Alger and Charles A. Eckert' Department of Chemical Engineering, University of Illinois, Urbana, Illinois 6 180 1
Chemical theory has been found to be well-suited for representing abrupt changes that are often observed in the activity coefficients and partial molar enthalpies of the components in systems that are known to form compounds in the solid phase. A chemical model is presented in which the Gibbs energy and the enthalpy are coupled through a simple analytical expression. The parameters that are obtained from the fitting of experimental data can be interpreted physically because of the rigorous thermodynamic foundation of the model. The model provides a rational method for extending limited thermodynamic data for a given system to a wider range of temperatures and compositions.
Introduction Many liquid metal solutions that form compounds in the solid phase do not conform to the regular solution theory requirement of zero excess entropy of mixing. Experimental entropies of mixing (Hultgren et al., 1973) often indicate the presence of unusual amounts of ordering in the liquid phase at compositions where compounds are known to exist in the solid phase. Jordan (1979) and Predel(l979) have presented evidence that suggests that compounds present in the solid phase also exist in the liquid phase when the alloy melts. Any attempt to model deviations from ideal mixing behavior in these systems should account for compound formation in the liquid phase. 0196-43 13/83/1022-0249$01.50/0
Eckert et al. (1981) have recently presented a general treatment of chemical theory as applied to strongly solvating (compound-forming) liquid metal systems. A comparison made between the Van Laar, Wilson, and three suffix Redlich-Kister equations and chemical theory showed that chemical theory gave a better fit of -y@ in the Mg-Bi system with the same number of parameters. In this paper a unified chemical theory model is presented. The model is capable of representing the Gibbs energy and the temperature dependence of the Gibbs energy in compound-forming systems with a minimum number of parameters. All the model parameters are based on well-defined thermodynamic functions. Because of the simple analytical form of the model, mathematical oper0 1983 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
ations are readily performed which permit the derivation and representation of various thermodynamic functions. Chemical Theory The underlying assumption of chemical theory is that molecular complexes or compounds exist in solution. Prigogine and Defay (1954) have shown that by assuming compound formation in the melt the chemical potential of the bulk species A is related to that of its true solution monomer by yAb*
=
@AtrUe
(1)
Expressed in terms of activity, eq 1 may be written (2) where zA is the true mole fraction of A and c ~ Ais the true activity coefficient of A. The true activity coefficients characterize the forces existing in solution after the monomers have equilibrated with the compounds. The true mole fraction of species i is defined by eq 3 XAYA = Z A ~ A
zi
ni
=
N ~.
nA + nB + E n j
The solution to eq 9 and 10 occurs where F1 = F2 = 0. Equation 9 is the normalization condition for the true mole fractions, and eq 10 ensures that the bulk moles of A and B are distributed between all the chemical compounds including the monomers. So that the true mole fractions of the monomers will be recovered from eq 8 for i = 1 , 2 , the following definitions are made: K1 = K,, = 1, K2 = Ka2= 1,al = b2 = 1, and a2 = bl = 0. With these definitions z1 = Z A and z2 = ZB for 1 = 1,2. By defining the first two terms in this manner notation is greatly simplified and the resulting equations are more easily programmed on a computer. By combining eq 8,9, and 10, eq 11and 12 are obtained, which can be solved for ZA and ZB given xA, all Ki's and a physical theory model relating the true activity coefficients ai to the true mole fractions .ti
(3)
where
j=3
where the moles of species i, ni, is a function of the extent of compound formation. Although chemical theory converts a binary problem into a multicomponent problem, the observed activity coefficient of A, yA,can be related to the true activity of the monomer A by rearranging eq 2 to the following form
pi
= (ai
+ bi)XA - ai
(13)
Cox (1979) has shown that by solving eq 12 for zA/xA and taking the limit as X A goes to zero, the infinite dilution activity coefficient -yAm may be expressed in terms of the equilibrium constants and the infinite dilution physical theory activity coefficient by
(4) In order to obtain the true activities of the components A and B from chemical theory, the functional relationship between xA, zi, ai, and the equilibrium constants Ki must be established. The chemical compounds that are assumed to form in the liquid phase are in equilibrium with the monomers through the expression given by aiA + biB = A,Bb, (i = 3, N) (5) for which the equilibrium constant, Ki, is defined
and the true activity coefficient equilibrium constant, K,, is defined (7)
Rearranging eq 6 gives an expression for the true mole fractions zi in terms of the two monomer true mole fractions Z A and Z B
where N is the total number of compounds including the monomers. Since there are only N - 2 equilibrium expressions, the two additional equations required to specify uniquely all the true mole fractions given x A and all Ki's are N
F1 = C Z-~1 = 0 i=l
N
N
F2 = X A - Caizi/(C(ai i=l
i=l
+ bi)zi) = 0
(9) (10)
where the summation is over all compound equilibrium constants containing only 1 atom of A. A similar expression is obtained for y B m
The infinite dilution activity coefficients are related to the compounds containing only one atom of the solute since compounds containing only one atom of the solute are the only ones capable of forming near infinite dilution. Compound Selection To use chemical theory, the compounds that are assumed to form in the liquid phase must be specified a priori. Cox (1979) has presented a method for selecting compounds based upon the chemical similarity of elements in the same column of the periodic table. In Cox's method the intermetallic compounds that exist in the solid phase of the system are listed along with the intermetallic compounds that exist in the solid phase of chemically similar systems. This list provides a set of candidate compounds that may be used to model the system. Examples, from Cox, of the compounds which exist in systems which are chemically similar to the MgBi system are: Mg-Bi: Mg3Bi2;Mg-Sb: Mg3Sb2;Ca-Bi: Ca3Bi2,CaBi,, (CaBi), Ca,Bi; Sr-Sb: Sr3Sb2,SrSb3, SrSb, Sr,Sb; Ba-Bi: Ba3Bi2,BaBi,, BaBi, (Ba2Bi). Sometimes there are a number of chemically similar compounds, all of which cannot be used. In this case the candidate compounds must be screened so that only the most important ones are retained. Final compound selection should be based on the following considerations: (1) the infinite dilution activity coefficients, related to
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983 251
compounds containing only a single atom of the solute, (2) the shape of the liquidus curve, and (3) experimental measurements which suggest unusual behavior in the melt at a particular composition. It is important to include compounds that satisfy the infinite dilution activity coefficient requirement. If compounds are not chosen to fulfill the infiiite dilution activity coefficient requirement, then the infinite dilution activity coefficient will be given by the contribution from the physical theory only. In the case of the Mg-Bi system for example, if only the strongest compound Mg3Bi2were assumed, then the theory would give a value of yMgm = 1, as a hook in the curve of log y vs. x . In fact, yMm< and in no system studied have we encountered a system where the curve “hooks”. To represent the ymdata, compounds must exist which, in eq 15, can mathematically yield a value of yMgm
