J . Phys. Chem. 1990, 94, 7913-7922
Conclusions
duced as the mass peak of this product overlaps with that of Nb,CO2. Isotopic substituted CD3CD0 would help to clarify this problem. The product formation channels can be summarized in the equations (s, m, and w indicate the relative product intensity being strong, medium, or weak, respectively) shown in Scheme I. The equations in Scheme I present only some of the possible processes that can account for the observed mass peaks. Indeed, all the stable molecules ( C H 3 0 H , CO, H,, HCHO, H 2 0 , CHJ are observed to be released from the reaction adduct, suggesting that the channels forming these products are energetically similar. Since no structural information is available, the product written as C H 3 0 H could be C O + 2H2 or H C H O H,.
The reactions of benzene derivatives and unsaturated nonaromatic hydrocarbons provide some insight into the dependence of the reaction product distributions and the possible reaction mechanisms on the reactant structure. The dehydrogenation probability is found to depend on the size of the niobium cluster as well as on the structure and stability of the organic molecule. Substituted benzenes such as chlorobenzene and benzonitrile form molecular addition type reaction products followed by dehydrogenation to different extents depending on the size of the niobium cluster. In the reaction of Nb, with pyridine, products due to the loss of C2H2 (C2D2) and Hz are important channels for the multiple molecular addition products, suggesting the opening of the aromatic ring. Due to the nonaromaticity and lack of resonance in 1,5-hexadiyne, cluster products due to the loss of C2Hz, C3H6,and H2 are also observed to be important reaction channels. In the reaction of Nb, with acetaldehyde, all of the reaction channels leading to the formation of stable molecules, e.g., H2, CO, CH30H, HCHO, HzO, and CHI, are observed. This suggests that, in this case, stable product formation drives the reaction. All of the above results suggest that the reactivity of the metal clusters is determined not only by the structure and stability of the metal clusters but also by those of the reagent molecules and the final product molecules.
+
SCHEME I Nb,
+ CH3CHO
------
Nb,O Nb,O2 Nb,C20 Nb,C
+
Nb,C20 (s) 2H2 Nb,C (s) + CHjOH Nb,CO (s) CH, Nb,CH4 (w) + C O Nb,C2 (s) + H2 HzO Nb,CH2 (m) + H C H O Nb,C2H2 (w) + H2O
-+
+ CH3CHO
+ CH3CHO + CH3CH0 + CH3CH0
7913
+
+
+ +
Nb,C02 (w) CHI Nb,C202 (m) 2H2 NbX02C20(w) 2H2 Nb,(C,O), (m) 2H, Nb,C,O (m) + 2H2
Acknowledgment. We thank the Office of Naval Research (Contract No. NOOO14-89-3-1350) for financial support.
+ +
Registry No. C6HSC1,108-90-7;C6H5CN,100-47-0;C,H4N2,28995-2;CSHSN, 110-86-1; HC+CH2CH,C-
3000
8
2000 6 3 0 .
0
'
"
0 10
1
1
'
1
1
(
~
/
1
0 40
0 20 0 30 MOLE FRACTION N p
Figure 17. Critical line of Nz-H20 in temperature-mole fraction coordinates. The data are from ref 5 0 the solid curve is the model prediction.
lustrated in Figures 14 and 15 and Table 111 for the Henry constant and Figures 16 and 17 for the critical-line data. We do not encounter the discrepancy noted in the Ar-H20 system. The model is not able to produce the shallow minimum in the T-x critical line. We believe that this is due to the problem repeatedly alluded to before, which forces the critical line to depart more or less parallel to the vapor pressure curve in p-T space (see Appendix). We conclude that the model shows the consistency of the Henry constant and critical line data to a reasonable extent.
COZ-HZO The data sources for this system are two sets of critical-line data-those of Tiidheide and Francks2 and of Takenouchi and Kennedys3 -and one set of supercritical apparent molar volume data by Crovetto et al.s4 The critical-line data are mutually consistent only at mole fractions of 0.1 or less. We were not able ~~~~~
~~~
( 5 2 ) TWheide, K.;Franck, E. U. 2.Phys. Chem. (Munich) 1%3,37, 387. (53) Takenouchi, S.; Kennedy, G . C. A m . J . Sci. 1964, 262, 1055. (54) Crovetto, R.: Wood, R. H.; Majer. V . J . Chem. Thermodyn. 1990, 22, 23 1,
to00
640
660
680
700
720
TEMPERATURE, K
Figure 20. Apparent molar volume of CO, in H 2 0 on three supercritical isobars,s4at approximately 38, 33, and 28 MPa. The solid curves are
model predictions for the nominal pressure values and for the mole fraction 0.005. The experimental mole fractions vary from 0.001 to 0.014. At 38- and 33-MPa isobars, the apparent molar volumes do not depend on m. At the 28-MPa isobar, the experimental apparent molar volume for the high mole fractions, x 2 0.005 (filled symbols), is much higher than for the low mole fractions, x < 0.005 (open symbols). The model cannot reproduce this feature. See, however, note added in proof. to find a set of model parameters that simultaneously represents the critical-line and apparent molar volume data. The best we could do is the set given in Table I. The description of the critical line is shown in Figures 18 and 19. The model is not capable of reproducing the initial downturn of the T-x diagram, as was noted before (see also Appendix). In Figure 20, we show the prediction of the apparent molar volumes. At the higher isobars, of 33 and 38 MPa, we represent the data rather well. At 28 MPa, the model passes through the low-concentration data well, but it misses an important feature noted at the higher concentrations, where the experimental apparent molar volumes actually are much higher than those closer to infinite dilution. This is the opposite trend from that seen in all the other cases, and our model is incapable of producing this feature.
High-Temperature Aqueous Solutions
The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 7921
Apparent Molar Heat Capacity, NaCl in H 2 0
Apparent Molar Heat Capacity, Ar in H,O
::::I nn
-2.0
-
-4.0 -
-6.0 -
Y
-
-8.0-
2
-10.0 -
_Y 4000.0
E
3
*8
"i.12.0 8
2000.0
-14.0 -
1
-16.0 -
0 0
.1*.0t
8
10-5
10-4
10-3
10-2
2
-2000.0 I 0.5
10'
1 104
10-3
MOLE FRACTION NaCl
Figure 21. Model prediction (solid curve) of the approach to infinite dilution of the apparent molar heat capacity of NaCl in water at 668.19 K and 32.28 MPa as a function of the mole fraction of salt. The experimental data, from ref 39, appear to be effectively at infinite dilution.
It is possible to obtain a substantially better fit of the critical line, but at the expense of a poorer fit to the apparent molar volumes. Except for the recent phase equilibrium calculations of Heilig and F r a n ~ kthis , ~ ~is the first attempt to represent a substantial region of thermodynamic data near the high-temperature end of the critical line of C 0 2 - H 2 0 by means of a mixture Helmholtz free energy. Our attempt leaves much to be desired. Not only are we unable to simultaneously represent the critical-line and apparent molar volume data, we also fail to represent the composition dependence of the apparent molar volume at 28 MPa. We have no explanation for this result. Nore Added in Proof. While this paper was in press, it was brought to our attention by Dr. R. Crovetto that a mistake was found in the evaluation of the supercritical apparent molar volume data of ref 54. The reevaluated data, which will be published, do no longer show the strong composition dependence that we could not explain. According to our model, the reevaluated data appear to be fully consistent with the critical-line data and with a new set of Henry constants, to be published by Dr. Crovetto.
I
Summary
We have used a generalized principle of corresponding states for formulating a variety of thermodynamic property data for four different aqueous electrolyte and nonelectrolyte solutions in the general vicinity of the critical point of steam. This is a Helmholtz free energy formulation, and pure water is used as the reference fluid. A total of only four to seven adjustable parameters are used. Results are given for infinite-dilution properties as well as for properties at finite compositions up to roughly I O mol %, in ranges of 100 K or more around the steam critical point. In several of our applications, this is the first time a number of disparate properties, such as phase boundaries, Henry constants, and apparent molar properties, are simultaneously correlated by a model that is very sparse in adjustable parameters. It is to be expected that inconsistenciesbetween data sets will be encountered. In NaCI-H20, the case were many previously evaluated data from different sources are available, we obtain a reasonable but not perfect description of the various data sets. Further progress should be possible but will require additional adjustable parameters. In most other cases, we make sense of several but not all properties. In the case of nonelectrolytes, we cannot reproduce the initial decline of the T-x critical line, for reasons that we believe we understand (Appendix). The model should thus be considered a rough first cut that will require refinement. The model has been very useful for estimating at what dilution a mixture approaches the condition of infinite dilution to within a preset tolerance. Near the solvent's critical point, a nonlinear dependence of the partial molar properties on the composition must (55) Heilig, M.; Franck, E. U.Ber. Bunsen-Ges. Phys. Chem. 1989, 93,
898.
102
101
MOLE FRACTION Ar
Figure 22. Model prediction (solid curve) of the approach to infinite dilution of the apparent molar heat capacity of argon in water at 668.6 K and 32.29 MPa as a function of the mole fraction of the gas. The experimental data, from ref 44, appear to be effectively at infinite dilution.
Apparent Molar Volume, COP in 10 0
-
E" .
0
E
I
8'ol
H20
le
6.0
>
,
0.0
10-5
104
10-3
102
101
MOLE FRACTION CO2
Figure 23. Model prediction (solid curve) of the approach to infinite dilution of the apparent molar volume of C02 in water at 665.4 K and 28.0 MPa, as a function of the mole fraction of the gas. Although it is possible for the model to show a slight increase with composition, it cannot follow the steep rise of the experimental data, ref 54. See, however, note added in proof.
be e ~ p e c t e d ; l ~ it, ~is~ induced *~' by the high compressibility of the solvent. Several examples of very nonlinear dependence of partial molar properties on composition are given in Figures 21-23. Note that the composition scale is logarithmic. For compositions larger than in mole fraction, the partial molar properties vary strongly with composition. If the mole fraction is decreased below the property values level off and become effectively constant, which means that the infinite-dilution value has been reached. The model in its present state may not give an accurate value for this infinite-dilution property, but it should give the right order of magnitude for the composition range where the infinite-dilution value can be observed. In both electrolyte and nonelectrolyte solutions, effectively infinite-dilution apparent molar heat capacity values in supercritical steam at 32 MPa are obtained at mole fractions lower than The composition dependence of the apparent molar volume of C 0 2 in water at 28 MPa, however, cannot be described by the model (Figure 23) for reasons explained in the note added in proof. On the other hand, the composition dependence of the apparent molar heat capacity of NaCl in water is quite reasonably represented (Figure 5). The model has also been used for reducing finite-concentration solubility data to Henry constants. It apparently predicts the nonideality corrections of the gaseous and liquid phases well and guides the extrapolation to infinite dilution properly, compared to alternative schemes that have been used.42.48,49 (56)Chang, R. F.; Morrison, G.; Levelt Sengers, J. M . H. J. Pbys. Chem. 1984, 88, 3389.
Levelt Sengers and Gallagher
7922 The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 I n principle, the model can also be used to calculate activity and osmotic coefficients. It will be very interesting to find out what the range is where these coefficients are better predicted from a Helmholtz free energy than from the usual Gibbs free energy approach. Finally, the Debye-Hiickel limiting law can be incorporated into the model straightforwardly and without the difficulty associated with the traditional approach.I5 It is probably not reasonable to do this without simultaneously incorporating the association of the electrolyte in the low-dielectric-constant medium. Acknowledgment. This work received support from the Office of Standard Reference Data. Prepublication results were obtained from Profs. K . S. Pitzer and R. H. Wood. We profited from conversations with Drs. J. Tanger, IV, and A. H . Harvey. We are indebted to Prof. R . H . Wood, Dr. R. Crovetto, and the reviewers of this paper for constructive criticism.
Appendix I n this Appendix, we gather relations pertaining to the thermodynamics of dilute near-critical mixtures and discuss the difficulties resulting from combining a nonclassical (scaled) reference with the classical principle of corresponding states for mixtures. The classical Helmholtz free energy A”(V,T,x) of a dilute mixture can be expanded around the solvent’s critical point. The symbol ” implies that the ideal-mixing term RT[x In x (1 x ) In ( 1 - x)] is subtracted from the Helmholtz free energy A and treated separately. To leading orders, this expansion is of the following form’5.56*57
+
+
A”(V.T,x) = A(V,T,x) - RT[x In x (1 - x) In (1 - x)] = A‘’‘ + ACV(6V) + Af’C,(6T) A”:x + AcW(6V)(6T) A$,(Glr)x ... + ACW7(6V)2(6T)/2+ Abx(6V)2x/2 + ... + Ac&6V)4/24 ... ( A l )
+ +
+
+
where the subscripts V, T, and x denote (repeated) differentiation with respect to the pertaining variables volume V, temperature T, and mole fraction of solute x. The symbol 6 denotes the difference between the value of the variable that follows it, and its value at the solvent’s critical point. The superscript c denotes that the pertaining derivative is evaluated at the solvent’s critical point. Note that -A; equals the critical pressure of the solvent, -AFT equals the limiting slope of its vapor pressure curve and critical isochore at the critical point, and Abequals (-dp/dx)&. The symbol ” has been omitted in all cases where it made no difference. Asymptotic expansions for the pressure p and isothermal compressibility Kj- are readily obtained from ( A l ) : p = -AV = -ACv - ACYT( 6 T ) - AC,x - ACwr(6V)(6T) AC,,(6V)x - AC&6V)’/6 + ... (A2) (VKj-)-l = ACw= Ac&T
+ AC,,X + A C 4 6 V ) * / 2+ ... (A31
The initial slope of the critical line (CRL) is obtained from this expansion by imposing the first criticality condition26 A,A,
-A,~ =
o
(A4)
(57) Rozen, A . M. Russ. J . Phys. Chem. (Engl. Transl.) 1976, SO, 837.
The results are55i56
The derivative value A,; plays a crucial role in dilute-mixture thermodynamics. Its sign determines whether the initial slope of the critical line in the p-T plane is higher or lower than the slope of the vapor pressure curve (A7). A higher slope is indicative of a volatile (Ab, < 0) and a lower slope of a nonvolatile solute (Ab, > 0). This derivative, coupled with properties of the pure solvent, also determines the sign and size of the infinite-dilution and enthalpy f l of the solute asymppartial molar volume totically near the solvent’s critical point, through the relation^'^^^^-^^
E
6 = Vi HT = HI
+ A“:
AE,VKj-,
- TA‘‘?,
(A81
+ TAG,AGj-VKj-1
(A9) where the subscript 1 refers to properties of the pure solvent. Since Kj-, diverges strongly at the solvent’s critical point, the dominant terms in (A8) and (A9) are the ones which contain AC, as the multiplier; this explains the importance of this coefficient. The derivative A;, also determines the variation of the Henry constant k, with temperature along the saturation curve u near the critical point of the solvent, through the r e l a t i ~ n ~ ~ . ~ ’ R T dk,/dq,
AC, dV,I/dT16
where the subscript 1,l refers to the saturated liquid volume of the pure solvent. Our model, which has an accurate equation of state for water and steam as a reference, should therefore demonstrate consistency of the critical-line slopes, apparent molar properties, and Henry constants of a mixture (solution) near the solvent’s critical point through the value it ascribes to the constant A;,. Although a nonclassical (scaled) Helmholtz free energy cannot be expanded at the solvent’s critical point, and displays subtle anomalies at the mixture’s critical line,5Eseveral of the asymptotic relations given above remain valid; in particular, this is true for eqs A6 and A8. It is readily seen that a problem arises with eq AS. If the solvent is described by a scaled equation, then the derivative AcwT equals zero, because the temperature divergence of the compressibility is characterized by a critical exponent y which is larger than 1. In our case, the reference Helmholtz free energy of pure water, although classical, mimics closely the behavior of the real fluid, which is nonclassical.59@ Thus, one must expect an “abnormally” low value of Acwj- compared to classical equations such as that of van der Waals. As a consequence, the value of dT/dxIERL will be abnormally high, which results in forcing the critical line to remain close to the (extended) vapor pressure curve, eq A7. This may be the reason why the model has no difficulty representing the critical line of NaCI-H20, which in fact runs almost parallel to the vapor pressure curve of water in the p-T plane,I5 while it is not able to yield a negative initial value of dT/dXI&, observed in aqueous solutions of volatiles (Figures 12, 17, and 19). (58) Griffiths, R. B.; Wheeler, J. C . Phys. Reu. A 1970, 2, 1047. (59) Levelt Sengers, J. M. H.; Greer, S. C . Inf. J . Heat Mass Transfer 1972, 15, 1865. (60) Levelt Sengers, J. M . H.; Kamgar-Parsi, B.; Sengers, J. V . J . Phys. Chem. Ref Data 1983, 12, 1.
