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J. Phys. Chem. 1986, 90, 1395-1396
STATISTICAL MECHANICS AND THERMODYNAMICS Application of a Hard-Sphere Perturbation Equation To Calculate the Heat Capacity of Argon in Water over a Wide Temperature Range Roberto Fernandez-Prini* and Maria L. Japas Departamento Quimica de Reactores, ComisiBn Nacional de Energfa AtBmica, 1429- Capital Federal, Argentina (Received: May 15, 1985; In Final Form: July 31, 1985}
Recently, direct experimental measurements of CPz0for Ar in water over a wide temperature range have been reported. These data show a steep rise of Cpzmabove 550 K which is not expected according to the usual structural models of aqueous solutions of nonpolar solutes. A perturbation theory employing a hard-sphere reference fluid with the packing fraction of water is capable of accounting very accurately for the observed behavior of CP2-over the entire temperature range. This evidence, added to the available information on gas solubility in water and its temperature dependence, suggests revision of models implying significant structural effects due to the introduction of nonpolar solutes in water.
The large partial molar heat capacity of nonpolar gases dissolved in water close to room temperature has frequently been singled out as a distinguishing feature of the behavior of aqueous solutions when compared to those in other solvent media.l This feature is considered related to the structure of the water solvent, Le., to the effect of the solute upon the extensive orientational correlation existing among neighboring water molecules. Experimental values for the heat capacity of dissolution of the gases, ACPzm, have usually been obtained from the temperature dependence of the solubility.2 The values of ACpzmobtained from measurements of the solubility of inert gases and methane in water over a wide temperature range3 showed a moderate decrease of this quantity with temperature. This trend appeared to agree with the accepted structural interpretation of the large magnitude of CP2-since the structure of water is known to decrease with temperat~re.~ Very recently Biggerstaff, White, and WoodS have reported the first direct determination of CP2-for solutions of Ar in water over a wide temperature range, thus providing a most important experimental evidence to understand the behavior of aqueous solutions. Their results appear at variance with the inferences from the temperature dependence of solubilities;as the temperature was observed to pass through a shallow minimum increases CPzm and after 550 K it increases steeply with temperature; the value of CpzOat 578 K was found to be greater than 900 J (K-mol)-’. As pointed out by Biggerstaff, White, and Wood,S their results are consistent within experimental uncertainty with the solubility data,3 the discrepancy between the CP2-values in both studies being due to the large uncertainty added by the double differentiation of the chemical potential to obtain the heat capacities from solubility data. They also noted that in spite of the fact that Cpzm is expected to go to plus infinity as the solvent’s critical point is approached, the steep increase found for Cpzmof Ar in water starts at a temperature which is lower than expected according to Wheeler’s theory which takes account of the effect of approaching the solvent’s critical point upon the solute’s partial molar properties.6 (1) Franks, F. In “WATER A Comprehensive Treatise”; Franks, F., Ed.; Plenum Press: New York 1973;Vol. 11, p 6. (2)Wilhelm, E.;Battino, R.; Wilcock, R. J. Chem. Rev. 1977,77,219. (3)Crovetto, R.; Fernindez-Prini, R.; Japas, M. L. J . Chem. Phys. 1982, 76, 1077. (4)Franck, E. U.; Roth, K. Discuss.Faraday SOC.1967,43,108. Heger, K.;Uematsu, M.; Franck, E. U. Ber. Bunsenges. Phys. Chem. 1980,84,758. ( 5 ) Biggerstaff, D.R.; White, D. E.; Wood, R. H. J . Phys. Chem. 1985, 89,4378.
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On account of the great interest in understanding the behavior of aqueous solutions it would be desirable to explain the values recently found for CP2-of Ar in water. W e have already used a perturbation theory to reproduce the chemical potential of nonpolar gases in water over a wide temperature range.’ Its success in dealing also with the Cp2-data is reported here. A hard-sphere fluid having the density of water and obeying the Percus-Yevick compressibility equation is employed as reference fluid; the perturbation solutesolvent intermolecular potential was taken to be of the Lennard-Jones type. The change of chemical potential of the solute when going from ideal gas at 0.1 MPa to infinitely dilute aqueous solution is
+
Apzo = Apzcav Ap2“
+
(1)
The first term gives the contribution due to the formation in the hard-sphere fluid of a cavity large enough to hold the solute particle. ApZatis the contribution of the attractive solute-solvent interaction and the last term accounts for the change of the solute’s standard state from ideal gas at 0.1 MPa to ideal gas at the density of the reference fluid. Percus-Yevick’s equation for the cavity terms is -In (1 - y )
+ -1(3Y-RY
+ R z ) + -9( 2
+
).’
yR 1-y
(1y- 2Y I 3
\
(2)
and for the second term in (1) we have Ap2at= 4?rp ~ m r % ’ l z ( r ) g o , 2 dr (r)
(3)
where u ’ is~the ~ Lennard-Jones solvent-solute interaction potential and g“ 12 is the pair distribution function for hard-sphere mixture^.^ The last term in (1) is Apid = k T In ( k T p )
(4)
Expressions 2-4 have as variables the packing fraction of the (6) Wheeler, J. C. Ber. Bunsenges. Phys. Chem. 1972,-76,308. (7) Fernindez-Prini, R.; Crovetto, R.; Japas, M. L.; Laria, D. Acc. Chem. Res. 1985, 18,207. ( 8 ) Reed, T. M.; Gubbins, K. E. ”Applied Statistical Mechanics”; McGraw-Hill Kogakusha: Tokyo, 1973. (9) Lebowitz, J. Phys. Rev. [Secr.] A 1964,133, 895.
0 1986 American Chemical Society
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J. Phys. Chem. 1986, 90, 1396-1403
y I
I
I
II
I
i'i
Figure 1. Cpzm against temperature for Ar in water. Experimental values from ref 5, @. Curves calculated with perturbation theory: (-) 17.2 MPa, ( - - - ) 10.0 MPa.
solvent y = (rpdI3)/6 ( p is the number density of water and d , = 2.70 h; is the hard-sphere equivalent diameter of the solvent) and the ratio of solute to solvent sizes R = (d2/dI). Equations 1-4 fit the experimental Ap? over all the temperature range if d2 decreases with temperature as predicted by theory. For Ar in water it was found dd2/dT = -2.0 X h ; / K . Differenriating eq 1-4 twice with respect to temperature, it is possible to calculate the heat capacity of dissolution at constant pressure, AC,,-, from which CP2-may be obtained by addition of the contribution of gaseous Ar. The term containing the second derivative of g o l 2 with temperature has been neglected in this calculation.
The molecular parameters describing the cross interaction which are necessary to write u'12were calculated by simple combination rules from those of the pure substances.'" The depth of the Lennard-Jones attractive well was obtained by using for water ( t , / k ) = 220 K and for Ar ( t 2 / k ) = 121.8 K.I0 The (PVT) properties of water were taken from the Steam Tables" and the calculation was made up to 623 K. Figure 1 compares the experimental data of Biggerstaff, White, and Wood5 with the results of the present calculation for pressures of 10.0 and 17.2 MPa. The agreement between the calculated and the experimental values is considered remarkable. In the low-temperature range, the calculated values are quite close to the experimental ones (experimental uncertainty5 is 20 J (mol. K)-') and the steep increase of Cp2m is very well predicted. It must be emphasized that the maximum temperature considered in this calculation is far from the region where the critical singularity of liquid water is clearly appreciable. The simple theory employed in this note is able to predict the chemical potentials as well as the heat capacities for Ar dissolved in water, the latter based on the parameters found to represent best the solubility data. It is notable that a theory which uses a hard-sphere reference fluid may be able to explain the properties of aqueous solutions over such a wide temperature range where the solvent structure changes significantly; Le., the orientational correlation among neighboring molecules is strongly affected by temperature. This new evidence supports our contention that the introduction of small nonpolar solutes in water may be described by employing a hard-sphere reference fluid having the density of water; the observed behavior reflects essentially the properties of the pure solvent. It is as if for the solute particle the medium were constituted by hard spheres having the experimental packing fraction: in other words, as if the introduction of small nonpolar solutes in water did not entail a significant reorientation of the water molecules surrounding it. Registry No. Ar, 7440-37-1. (IO) Alvarez, J.; Crovetto, R.; Fernindez-Prini,R. Z. Phys. Chem. (N.F.) 1983, 136, 135. ( I 1) Schmidt, E.; Grigull, U. "Properties of Water and Steam in SI Units"; Springer-VerIag: West Berlin, i 9 s i
Tensorial Formulation of Extended Nonequillbrium Thermodynamics of Lumped Systems S. Sieniutyczt and P. Salamon* Department of Mathematical Sciences, San Diego State University, San Diego, California 92182 (Received: June 13, 1985; In Final Form: November 5, 1985)
In the space of thermodynamic variables a tensor formulation of extended irreversible thermodynamics is proposed for lumped systems with the entropy depending on the rate of the process as well as on the state of the system. The requirement of covariance under a change of thermodynamic variables leads to two new functions of state: a thermodynamic Lagrangian A which replaces the entropy of a closed system or the second differential of entropy, bZS,in the dynamical equations of nonlinear processes and a function Q which plays the role of an integral of the reversible thermodynamic motion for an open system and whose time derivative equals the rate of entropy production in the general irreversiblecase. R simplifies to (1/2)SzS for a closed system in the vicinity of equilibrium. The derivations also show that it is the deviations from an instantaneous equilibrium state which provide the natural variables for describing the dynamics of a thermodynamic system.
I. Introduction We treat a covariant nonmetric formulation of dynamics for lumped thermodynamic systems neglecting relativistic effects. Our use of covariance refers to the space of thermodynamic variables rather than the variables of physical space or spacetime. The 'Permanent address: Department of Chemical Engineering, Warsaw Technical University, Warsaw, Poland.
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desirability of such a formultion is spurred by the recently recognized1"V2Riemannian structure on the space of thermodynamic variables defined by the metric whose matrix coincides with the matrix of ~ c o n dderivatives of entropy3 computed in the f m r ~ e (1) (a) Weinhold, F. J . Chem. Phys. 1975, 63, 2479. (b) 1976, 65, 559. (2) Salamon, P.; Andersen, B.; Gait, P. D.; Berry, R. S. J . Chem. Phys. 1980, 73, 1001-1002.
0 1986 American Chemical Society
