J. Phys. Chem. 1988, 92, 5836-5839
5836
In this limit D I 2is given by Normalized values of D l l and D I 2are plotted against the total concentration of electrolyte MX in Figure 2. The calculations were made for the case of identical diffusivities of the anion and the free cation, Dx- = D M t . It is evident from Figure 2 that as the concentration of the electrolyte drops to zero, the electrolyte’s interdiffusion coefficient D l l approaches values that are significantly larger than the intradiffusion coefficient DMt*for ion M+. The M+ions are transported more rapidly during interdiffusion because they are pulled along by the electric field, which is not present during intradiffusion. In general, the limiting value of D l l will be less than or greater than the limiting value of DM+* depending on whether DMt is less than or greater than (1 + KC2)Dx-,respectively. In contrast to this behavior, the limiting intradiffusion coefficient and intradiffusion coefficient for nonelectrolyte solutes are identical,21cas illustrated in Figure 1.
Discussion In this paper the flow equations for diffusion of adsorbing solutes have been written in terms of total solute components. As a result, the terms for the rate of adsorption and desorption disappear from the continuity equation for the solute concentration. This mathematical simplification allows diffusion with adsorption to be treated as a pure diffusion problem. For example, one-dimensional transport of a solute into a uniform adsorbent is described by the time-dependent diffusion equation
where Dl I stands for the integral solute interdiffusion coefficient, the mean value of D , , along the diffusion path. It has been assumed in this paper that the adsorbed solute has zero mobility. If surface diffusion of the adsorbate is not negligible, however, it will be necessary to assign a nonzero value to the diffusivities of the adsorbed species, DM or D+Ms. In other cases, such as adsorption of low molecular weight solutes onto colloidal
Isomeric (CO,),-(g) Capacity+#“
particles or high polymer solutes,2a it will also be necessary to assign small but nonzero diffusivities to the adsorption sites since they will move slowly through the solutions. The approach taken in this paper has been to treat the adsorbent medium as a single homogeneous phase,’ analogous to an ordinary liquid solution. If the adsorbent consists of a liquid-filled porous solid, only a fraction of the cross-sectional area is available for transport of solute. In addition, the diffusion path is longer than in a liquid phase because only the pore networks can transmit solute. The homogeneous-phase diffusion coefficients used in this paper can be converted approximately to diffusion coefficients within the pore solution by multiplying the former by empirical “tortuosity” Typical values of the tortuosity factor range from 2 to 6.sb The diffusion coefficients for inert solutes in liquid solutions usually vary with temperature approximately as Tq-l, where q denotes the viscosity of the solvent.28 The effective diffusion coefficients for adsorbing solutes generally increase with temperature much more rapidly owing to the decrease in the extent of adsorption with temperature. According to eq 12 and 24 the intradiffusion and interdiffusion coefficients for a dilute adsorbing nonelectrolyte equal DA/KC2,provided KC2 >> 1. In this case In DA. or In D l l plotted against 1/T will have slope a In DA/a(11T), typical of inert solutes, plus the additional contribution A H o / R arising from the temperature dependence of the equilibrium constant. Here AHo refers to the standard enthalpy change for adsorption reaction I. Although there have been a number of careful experimental studies of diffusion with adsorption in connection with diffusion of charged dyes into interpretation of the results is complicated by ion exchange and the presence of at least two different kinds of adsorption sites. Also, the dye baths used in those studies contained supporting electrolytes, such as sodium chloride. To describe the diffusion behavior observed in these experiments, it will be necessary to extend the present theoretical treatment to multiple equilibria and quaternary diffusion. Acknowledgment is made to the Natural Sciences and Engineering Research Council for financial support of this research. (28) Wilke, C. R.;Chang, P. AIChE J . 1955, I , 264.
System: A New Record Isomerism Contribution into Heat
Zdengk Slanina The J . Heyrovsk? Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, Machova 7, CS-121 38 Prague 2, Vinohrady, Czechoslovakia’b (Received: January 22, 1988; In Final Form: March 25. 1988)
The results of a recent computational characterization of DZdand C, isomers of (C02)*-(g) have been used for evaluation of the isomeric interplay in relation with enhancement of the heat capacity of the system. It has been shown that the maximum value of the isomerism contribution into standard overall heat capacity can be 134 J/(K mol), which is a value 1 order of magnitude higher than the highest heat capacity contributions known for other systems.
Introduction Several recent research works have provided evidence2 of the existence of isomerism in molecular complexes formed by weak intermolecular interactions: the newest one concerns the specOf the hydrogen-bonded N2°.HF and ‘Dedicated to Professor Kenneth S. Pitzer on the occasion of his 75th birthday.
0022-3654/88/2092-5836$01.50/0
Thus experimental approaches are beginning to involve the concept Of cluster isomerism, which earlier was indicated by theoretical (1) (a) Multimolecular Clusters and Their Isomerism. Part 37. 36: Slanina. Z. J . Mol. Struct.. in Dress, (b) Reorint-reauest address: The J. Heyrovskf Institute of Physicfil Chemistry ind Electkchemistry, Czechoslovak Academy of Sciences, Dolejgkova 3, CS-182 23 Prague 8-Kobylisy, Czechoslovakia.
0 1988 American Chemical Society
Isomeric (CO,),-(g) System
The Journal of Physical Chemistry, Vol. 92, No. 20, 1988 5837
$00-
31
50-
200
400
660
do
260
0260
460
660
8bO
T (K)
Figure 1. Temperature dependences of weights w, of the Dzd and C, isomers of (C02)2-(g)in the MP2 (dashed lines) and MP3 (solid lines) approaches; 1 represents the Dzd structure.
predictions of manyfold local energy minima (see, e.g., ref 3). However, because of dependence on the rigidity of system, interference had to be expected with the fluxional behavior! which effectively made the isomerism uneasily distinguishable or indistinguishable and thus obliterates the simple picture of two or more well-defined, isolated structures. The isomerism of molecular complexes also makes itself felt in the values of thermodynamic functions of these systems. In this context it has recently been shown5 that isomerism could markedly affect the heat capacity term. So far the greatest effects have been f0und~9~ with the dimers (CO,), and CO-HF. Testing of other systems' revealed that the contribution of isomerization effects to the heat capacity in the ( C O z ) c anion is larger by 1 order of magnitude. The (C02)*- species are dealt with in the present communication.
Description of the System The C 0 2 clusters in both neutralsa and ionizedab states have (2) (a) Levy, D. H.; Haynam, C. A.; Brumbaugh, D. V. Faraday Discuss. Chem. Soc. 1982, No. 73, 137. Lisy, J. M. Zbid. 1982, 73, 174. Haynam, C. A.; Levy, D. H. J . Phys. Chem. 1983,87,2091. Hoffbauer, M. A.; Liu, K.; Giese, C. F.; Gentry, W. R. Zbid. 1983,87, 2096. Andrews, L.; Johnson, G. L. J . Chem. Phys. 1983, 79,3670. Stace, A. J. Chem. Phys. Lett. 1983, 99, 470. Engdahl, A,; Nelander, B. Zbid. 1983, 100, 129. Carrasquillo, E.; Zwier, T. S.;Levy, D. H. J . Chem. Phys. 1985,83,4990. (b) Joyner, C. H.; Dixon, T. A.; Baiocchi, F. A.; Klemperer, W. J. Chem. Phys. 1981, 74,6550. Lovejoy, C. M.; Nesbitt. D. J. Zbid. 1987, 87, 1450. (3) McGinty, D. J. Chem. Phys. Lett. 1972,13, 525. Burton, J. J. Zbid. 1972,17, 199. Burton, J. J. J . Chem. Soc., Faraday Trans. 2,1973,69,540. Slanina, Z. Collect. Czech. Chem. Commun. 1975,40, 1997. Bauer, S. H.; Frurip, D. J. J . Phys. Chem. 1977,81, 1015. Slanina, 2.Adv. Mol. Relax. Interact. Processes 1979, 14, 133. Slanina, Z. Adv. Quantum Chem. 1981, 13, 89. Slanina, 2.Znt. J . Quantum Chem. 1983, 23, 1563. Slanina, Z. Znt. Rev. Phys. Chem. 1987, 6, 251. Slanina, 2. Contemporary Theory of Chemical Isomerism; Academia: Prague, Reidel: Dordrecht, 1986. (4) Berry, R. S. In The Permutation Group in Physics and Chemistry; Hinze, J., Ed.; Springer-Verlag: Berlin, 1979. Berry, R. S. In Quantum Dynamics of Molecules; Woolley, R. G., Ed.; Plenum: New York, 1980. Amar, F. G.; Berry, R. S . J. Chem. Phys. 1986, 85, 5943. (5) Slanina, Z. J . Phys. Chem. 1986,90, 5908. (6) Slanina, 2. Thermochim. Acta 1986, 102, 287. (7) Slanina, Z., unpublished results, 1987. (8) (a) Brigot, N.;Odiot, S.; Walmsley, S. H.; Whitten, J. L. Chem. Phys. Lett. 1977, 49, 157. Brigot, N.; Odiot, S.; Walmsley, S. H. Zbid. 1982, 88, 543. Slanina, Z. J. Mol. Struct. 1983,94,401. van de Waal, B. W. J. Chem. Phys. 1983, 79, 3948. Slanina, 2.Surf. Sci. 1985,157, 371. van de Waal, B. W. J . Chem. Phys. 1987,86,5660. Barnes, J. A.; Gough, T. E. Zbid. 1987, 86.6012. Fraser. G. T.: Pine. A. S.: Leffertv. W. J.: Miller. R. E. Zbid. 1987. 87; 1502. (b) Engelking, P. C.J . Chem. Piys. 1987,87,936. Kondow, T: J . Phys. Chem. 1987, 91, 1307.
460
600
-
860
T (K)
T (K)
_4
Figure 2. Temperature dependences of the standard heat capacities at constant pressure Cop (left) and the corresponding isomerism contributions SC, (right). Left side: solid lines with the indexes 1 and 2 representing the partial COP,,(i = 1 or 2) terms for the Dzd and C, structures of (C02)c(g),respectively; the solid and dashed line between the solid lines with the indexes 1 and 2 represent the standard overall isofractional terms CO,, in the MP3 and MP2 approaches, respectively; the solid and dashed line with a maximum represent the standard overall heat capacity Cop in the MP3 and MP2 approaches, respectively. Right side: the sequence of three curves of the same printing (either the solid lines (the MP3 approach) or the dashed lines (the MP2 approach)) represents in the order from the highest to the lowest curve the SC,, SC,,H, and SCp,w isomerism contributions.
already been studied many times; recently Fleischman and Jordan9 published a computational characterization of two structural forms of (C02),- in terms of the a b initio calculations with a flexible basis set and including the effects of electron correlation. These two local energy minima belong to the DZdand c, point groups of symmetry, the D2d structure lying about 30 and 20 kJ/mol lower9 in the MP2 and MP3 approaches to the contribution of electron correlation, respectively. The paperg contains full information enabling construction of the partition functions of the two isomeric species in the usual (see, e.g., ref 10) approximation of the rigid rotor and harmonic oscillator.
Temperature Interplay of Dw and C, Isomers of (C02)2-(g) The composition of the two-component isomeric system can be described5 by mole fractions w1 and w2 of the two isomers in their equilibrium mixture in terms of their partition functions q1and q2 and the enthalpy terms at absolute zero temperature AHoo,l and Wo,2 (wl + w2 = 1; subscripts 1 and 2 denote the structures D2d and c,,respectively): WI
=
41
41 + 42 e x p [ - ( ~ O 0 , 2- ~ o o , l ) / R T l
(1)
Figure 1 presents the temperature dependence of the weight factors wi in the MP2 and MP3 approximations and shows that in the
two cases a rapid mutual approach and subsequent interchange of relative stabilities of both isomers take place. This relative stability interchange takes place at about 469.1 and 3 17.9 K in the MP2 and MP3 approximations, respectively. Each of the structures 1 and 2 has its own heat capacity term at constant pressure (Cop,1and If the components are present in their equilibrium mixture, the experimentally observed will generally be different from overall heat capacity value (COP) (9) Fleischman, S. H.; Jordan, K. D. J . Phys. Chem. 1987, 91, 1300. (10) Hoare, M. R. Ado. Chem. Phys. 1979,40,49. For other theoretical studies of thermodynamics of molecular complexes (though without isomerism treatment), see: Yamabe, S.; Minato, T.; Hirao, K. J. Chem. Phys. 1984,80, 1576. Yamabe, S.;Minato, T.; Sakamoto, M.; Hirao, K. Can. J. Chem. 1985, 63,2571. Pullman, A.; Claverie, P.; Cluzan, M.-C. Chem. Phys. Lett. 1985, 117, 419. Nguyen, M. T. Zbid. 1985, 117, 571. Yamabe, S.; Furumiya, Y.; Hiraoka, K.; Morise, K. Zbid. 1986,131,261. Hiraoka, K.; Shoda, T.; Morise, K.; Yamabe, S.;Kawai, E.; Hirao, K. J . Chem. Phys. 1986,84, 2091. Hiraoka. K.: Takimoto. H.: Yamabe. S. J. Phvs. Chem. 1986. 90. 5910. Pal. P.; Hoarel M. R. Zbid. 1%7,91, 2474. Ngiyen, M. T.; Hegarty, A. F.; Ha; T.-K. J. Mol. Struct. 1987, 150, 319.
5838 The Journal of Physical Chemistry, Vol. 92, No. 20, 1988
the above-mentioned individual terms. It will be useful to introduce the term of the isomerism contribution to heat capacity (K,)as a quantity that must be added to the term to obtain the above-mentioned overall term:
COP =
COPJ
+ SC,
(2)
In a two-isomer system it iss
SC, = w2(ACoP,2- A C p , l ) + WIWZ
( A H o ,- AH'z)~ RT2
(3)
where the enthalpy terms AHoI and AHo2 now apply to the measurement temperature considered. The second term of eq 3 is connected with the temperature dependences of w1 and w2; if it is neglected, the relation is reduced to the so-called5isofractional contribution to heat capacity: SCp,. = WZ(AC0,,2
- A.C',,I)
(4)
and thereto corresponds the overall term of isofractional heat capacity: cop,w
= C0p,1 + 6Cp,w
(5)
On the other hand, the first term of eq 3 is connectedSwith the temperature dependence of enthalpy, and its omission leads to another methodologically interesting quantity: (AH'1 - AH'2)' Scp,H
=
wlw2
Slanina
TABLE I: Partiala and Overall StandardbMolar Changes of Heat
Capacity at Constant Pressure for Gas-Phase ReactionsC2C02(g) + e - k ) = (C02)&) 2 C 0 2 e- = 2 c o * + e- = 2 c 0 2 + e- = (co2)2- (overall) MP3 (co2)2-(C8) MP2 T, K (C02MD2,A 100 -26.8 -12.1 -26.8 -26.8 200 -18.6 -8.6 -18.5 -4.6 298.15 -16.1 -9.6 -1.8 111.6 300 -16.0 -0.8 111.8 -9.6 400 -14.2 -10.2 98.4 33.7 500 -12.7 -10.3 93.3 0.7 600 -1 1.6 -10.3 30.2 -6.6
+
'Associated with production of (CO2)T(g)of Du or C, symmetry. bIdeal gas state, standard pressure is irrelevant; in J/(K mol). cData for the components in the left side taken from ref 11.
TABLE 11: Illustrationa of Parameter Behavior of the Simple Phenomenological Model w , = exD(-AF) at Tn = 500 K n
A.
1 2 3 4 5
0.319 0.637 0.127 0.255 0.510
K-" X X X X X
lo-' lo-"
2"
0.648 2.59 5.83 10.4 16.2
I
.
5.38 21.5 48.5 86.2 135
"The maximum position preselected at To = 500 K. bSee eq 9 for T
= To and eq 14.
RT2
Figure 2 presents the temperature dependencies of the Cop,l, co~.zv Co~9wr and S c ~ wsc~J+ P" terms* The ac~.H term appears to be a c h i v e component of the isomerism contribution to heat capacity in the given particular case. This term is reand a c ~ sponsible for the marked temperature maxima Of the functions, this extreme course being in an obvious contrast to the monotonous course Of the and C O functions. ~ The values of the isomerism contribution to heat capacity are reached at 442.4 and 300.2 K in the MP2 and MP3 approximations, respectively. The maximum values of this enhancement are 133.7 and 127.8 J/(K mol), the contribution of the isofractional term SCP,.being only about 0.93 and 1.9876, respectively. These values of isomerism contribution to heat capacity are the highest ones of those reported so far for a particular system. The (C02),-(g) anion represents-from the thermodynamic point of view-a markedly exotic system obviously lying outside the possibility of usual measurements. It is of course impossible to prepare (C02),-(g) as a defined isolated (two-component) species, it being possible only to prepare complex mixtures including these species as a minor component. This circumstance, however, does not change the fact that-in our context-this anion represents useful material evidence that the isomerism contribution to heat capacity can attain considerable values. These high values are due to a certain convenient interplay of all the parameters involved. It is not very important that the partition functions used have the nature of rigid rotor and harmonic oscillator. More important is especially the relation between the enthalpy terms AHo, and the rate of temperature change of the weight factors w,, In principle, such a convenient interplay of these values can also take place in more rigorous approaches to the partition functions. On the other hand, however, it is obvious thatespecially for distinctly nonrigid systems (which usually also are distinctly anharmonic)-it would be desirable to examine also the problems of the barrier height separating the two isomers (in this respect no data are available in the particular case of (COz)zJ and thus move to the partition functions of a single fluxional, unified supersystem that is not further separated into the individual isomers. At any rate, the results presented are important for the special class of isomeric systems characterized by a fast change of relative stability of the structures involved, the behavior of local energy minima in the sense of rigid rotor and harmonic oscillator being reasonably fulfilled, the separating potential barrier(s) is
(are) sufficiently high, but in spite of that, a sufficiently rapid possibility of establishing of thermodynamic equilibrium is maintained (these last two requirements, of course, can be contradictory). The isomerism contributions to heat capacity SC, are advantageous also from another point of view: they are transferable to any chemical reaction. With their help, e.g., it is possible to compute the terms of the overall standard heat capacity change A ~ for~ the, reaction
In Table I they are compared with the partial quantities. The heat capacity values at the left-hand side are taken from ref 11. (Strictly speaking, for any chemical reaction we could also introducel2 two types of heat capacity: the usual standard terms and those corrected for the contribution to heat capacity due to the temperature changes of mutual populations of all the reaction components. This more general problem, however, exceeds the aims of the present work; hence Table I gives only the usual conventional terms.) A Simple Model It can be useful to neglect the relatively complex interplay of parameters at the molecular level by postulating a certain functional dependence of thermodynamic quantities and thus carrying out a rather phenomenological analysis. Let us focus attention on the SC,,H term, which appeared dominant in the above discussion and has its origin5 in one of the two components resulting from temperature differentiation of products of molar enthalpies and weight factors wi:
where it is immaterial what reaction is connected with AHoi, (11) Thermodynamical Properties of Individual Species; Glushko, V. P., Ed.; Nauka: Moscow, 1978, 1979; Vol. I and 11. (12) Woolley, E. M.; Hepler, L. G. Can. J. Chem. 1977, 55, 158. Allred, G. C.; Larson, J. W.; Hepler, L. G. Ibid. 1981,59, 1068. Larson, J. W.; Zeeb, K. G.; Hepler, L. G. Ibid. 1982, 60, 2141. Jolicoeur, C.; Lemelin, L. L.; Lapalme, R. J. Phys. Chem. 1979, 83, 2806. Peiper, J. C.; Pitzer, K. S. J. Chem. Thermodyn. 1982, 14, 613. Mains, G. J.; Larson, J. W.; Hepler, L. G. J. Phys. Chem. 1984,88, 1257.
J. Phys. Chem. 1988,92, 5839-5842 2K(1
5839
+ K)-dK - T(l + 3K) dT
(E)'+ -
2TK(1
+ K ) -d2K =0 d P
(11) Now we must introduce some presumption about the temperature dependence of K; this can be done by postulating the temperature dependence of w1 as
100.
w1 = exp(-AP)
(12) where A and n are certain free parameters. Then eq 11 for the temperature position To of the extreme of the z function leads to the transcendental equation AT," exp(ATo") - 2 exp(ATo")
+2 =0
(13) which has the solution uo = ATo" = 1.593624..., the height of the corresponding maximum being
50.
zo = n2u02/(exp(uo)- 1) = 0.647610 ...n2
0 200
400
600
800
T (K) __. Figure 3. Temperature course of the SCP,,= zR terms (see eq 9) within the model based on the assumption of (12) for n = 1-5 (in the circles).
because only the difference in the enthalpy content of the two isomers is relevant. Let us introduce the equilibrium constant K of the isomerization of 1 to 2 (where 1 denotes the isomer that is dominant a t low temperatures). Then w I = 1/(1 K ) , and hence
+
(9) d In K A H Q 2 - AH'I = R P dT For the position of the temperature extreme of the z function we can obtain the following from the usual condition dz/dT = 0:
(14)
and hence it increases with the n parameter above any limit (see Table 11), the value n = 5 being necessary in this model for reaching the values actually found with the (C02)z-(g) system. Figure 3 presents the course of the z quantity in a wider temperature interval. In this simple model it is possible to derive the formulas for the further quantities, e.g., SC,,, which in this model provide substantially higher values than those observed for the (C02)2-(g) system.
Concluding Remarks Using the system of two isomers of (C02)2-(g),we could describe the (so far) largest effects of isomerism in overall heat capacity. By means of a simple model it was shown that, in principle, there exists no limitation for the increase of magnitude of this isomerism contribution. It remains an open question as to what are the highest values attainable in a real isomeric system or whether there exists such an isomeric system whose heat capacity enhancement can be useful in practical applications (e.g., the heat transfer) or for an explanation of the substantial anomalies in the behavior of a real thermodynamic or biothermodynamic system. Registry No. (CO,),,26796-46-3.
Klnetlcs and Thermochernlstry of the Equlllbrium P(Acety1ene) = Vlnylacetylene. Direct Evidence Against a Chain Mechanism E. Ghibaudi and A. J. Colussi* Department of Chemistry, University of Mar del Plata, 7600 Mar del Plata, Argentina (Received: September 8, 1987) It is shown that recent real-time measurements of the initial rates of decomposition of vinylacetylene behind incident shock waves together with earlier kinetic data for acetylene dimerization closely satisfy detailed balance, i.e., k1/2k-l = Kl (Hidaka, Y.; Tanaka, K.; Suga, M. Chem. Phys. Lett. 1986, 130, 195; Skinner, G. B.; Sokoloski, E. M. J. Phys. Chem. 1960,64, 1952). The short times involved ( t < 20 p s ) allow molecular species, or their isomers, to take part in reactions 1 and -1 but specifically exclude free monoradical chain mechanisms. Since a one-step concerted dimerization is otherwise untenable on kinetic grounds, the intermediacy of vinylidene (H2C=C:) in reaction 1 is proposed on the basis of ab initio calculations.
Introduction Acetylene, a major component of flames and the direct precursor of soot, is a molecule that is extraordinarily stable toward dis~ociation.~*~ However, it readily dimerizes to vinyla~etylene:~ 2CZHz = C4H4 (1) (1) Hidaka, Y.; Tanaka, K.; Suga, M. Chem. Phys. Lerr. 1986,130,195. (2) Skinner, G. B.: Sokoloski, E. M. J. Phys. Chem. 1960, 64, 1952. (3) Harris, S.J.; Weiner, A. M. Annu. Reu. Phys. Chem. 1985, 36, 31. (4) Wodtke, A. M.; Lee, Y. T. J . Phys. Chem. 1985,89, 4744. (5) Cullis, C. F.; Franklin, N. H. Proc. R . Soc. London,A 280, 1964, 139.
0022-3654/88/2092-5839$01.50/0
Now, in principle, the adverse kinetic factors associated with H-atom fission reactions such as those required in the sequence from C,H, species to soot (mainly CsH) suddenly Compare reactions 2 and 3: C4H4 = C4H3 H AH2 = 98 kcal/mol (2) M + C2H2 = C2H + H + M AH, = 130 kcal/mol (3)
+
(6) DurSn, R. P.; Amorebieta,V. T.; Colussi, A. J. J . Phys. Chem. 1988, 92,636. (7) Colussi, A. J. In Chemical Kinetics of Small Organic Radicals; Alfassi, Z., Ed.;CRC: Boca Raton, 1988.
0 1988 American Chemical Society
