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On interchange of relative stability of C4(g) isomers
Volume 161,number 3
CHEMICAL PHYSICS LETTERS
15 September I989
ON INTERCHANGE OF RELATIVE STABILITY OF C,(g) ISOMERS **** ZdenE?kSLANINA The J. Heyrovs@Instituteof PhysicalChemistryand ElectrochemistryCzechoslovakAcademyof Sciences, DolejSkova3, I82 23 Prague8 - Kobylisy,Czechoslovakia Received 12 May 1989; in final form 12 June 19539
The equilibrium has been studied between the rhombic and linear triplet structures of C,(g) on the basis of recent quantumchemical data. The existenceof interchange of their relative stability with increasing temperature is indicated for some approaches to the energetics. An approximate formula for estimating the equimolarity temperature has been suggested. A comparison has been carried out with existing thermodynamic data for the equilibrium between C,(g) and graphite.
1. Introduction
2. The C,(g) system
The interest in small carbon clusters C,(g) is due to a number of technological reasons leading to studies of the evaporation of graphite as well as to the presumed existence of these aggregates in interstellar space. As these species are sufficiently characterized experimentally only up to n=3, theoretical studies of these aggregates beyond this limit represent a valuable source of information about these clusters, see e.g. refs. [l-28]. The C4 cluster has recently been dealt with in a series of theoretical studies [8,13,14,16,18,19,22-24,26-281 which substantially extend the knowledge obtained from experiment [ 29-361. Thereby a sufficient amount of data from advanced quantum-chemical calculations has also been made available, which enables a theoretical evaluation of thermodynamic characteristics. The present paper provides an analysis of some interesting thermodynamic relations in this system and a comparison with the existing observed (but for testing of the theory, not yet employed) information [29,31,32].
theoretical calculations there exists a consensus that there are two energetically low-lying C4 structures, namely the rhombic one (DZh) and the linear triplet one ( Dccrh). The rhombic structure appears to lie mostly lower in terms of potential energy, but there is no unanimity with regard to the energy difference between the two structures (cf. table 1). According to a 6-31G*/MP2 study [ 191 the potential-energy difference AE is 64.86 kJ mol-I. The transition to higher orders of perturbation theory, viz. MP3 and MP4, leads [ 181 to values of 11.37 and 27.99 kJ mol-‘, respectively. Finally, the sophisticated approach reported in ref. [ 181, the so-called CCD+ST(CCD), provides a value of 21.71 kJ mol-‘. However, in the CCSD+T(CCSD) approach [27] the energy sequence is reversed: hE= -5.48 kJ mol-‘. The study [26] suggests the value of A@= 5 kcal mol-‘c20.92 kJ mol-’ for the ground-state energy difference between the linear triplet and the rhombic structures (i.e. the potential energy corrected for zero-point vibrations). For the purpose of studying the relative stability of the two isomers we will use three values of the AH: term,. namely the 6-31G*/MP2 value [ 191 Lwi ~65.15 kJ mol -I, the recommended estimate [ 26 ] AH: = 20.92 kJ mol-‘, and the negative value Mg= -5.19 kJ mol-’ derived from ref. [27].
* Dedicated to the late Academician V.P. Glushko. ** Part XL11 in the series Multimolecular Clusters and Their Isomerism; for part XLI, see.ref. [45 1.
On
the
basis
of
[ 8,13,14,16,18,19,22-24,281
0 009-2614/89/$ 03.50 Q Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
265
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CHEMICAL PHYSICS LETTERS
15 September 1989
Table 1 Survey of potential-energy differences AE and ground-state energy differences NJ: between linear triplet (Dmh) and rhombic (Dzh) isomers of C,(g) source of energy term
Level of approximation
AE(kJmol-I)
source of vibrational frequencies
Level of approximation
tiz
MCHS [ 191 RB [I81 VFZWE [ 261
6-31G*/MP2 CCD+ST(CCD) evaluation of data [8,13,14,16] CCSD+T(CCSD)
64.86 21.71 20.63 18.27 -5.48
MCHS 1191 MCHS [ 191 MCHS [ 191 RKY [ 141 MCHS [ 191
6-3lG*/MP2 6-3 lG*/MP2 6-3 lG*/MP2 DZP or 6-3 lG* 6-3 lG*/MP2
65.15 22.01 20.92 20.92 -5.19
BMB [27]
3. The temperaturedependenceof the relative stability of the two isomers If the temperature dependence of the relative stability of both structures is to be adjudged, it is necessary to carry out the calculation of the composition of equilibrium mixture of the isomers, e.g. in terms of their mole fractions wi [ 37-391. These quantities are determined by the energetics and by the partition functions of both structures. With respect to the extent of the molecular parameters available from quantum-chemical calculations the evaluation is to be limited to the usual approach of rigid rotor and harmonic oscillator (RRHO) [ 321. The structural data for the calculation of moments of inertia are taken from ref. [ 19 ] or ref. [ 27 ] (the BMB/MCHS approach). The construction of vibrational partition function makes use of the set of harmonic vibrational frequencies [ 191, and the estimates of sensitivity of the vibrational partition function to a change in the quantum-chemical approach also adopt the set of harmonic vibration frequencies from ref. [ 141. Table 2 presents the temperature dependence of the mole fraction WD2a of the rhombic structure evaluated in terms of four parameter sets. It is obvious that, for positive A# values, on increasing the temperature one always encounters a value T,, at which the equilibrium mixture will contain equimolar amounts of the two isomers. However, in the BMB/ MCHS approach [ 19,271 with the negative m8 term there is no stability exchange - the linear structure always prevails. The temperatures relevant for observations of the C,(g) formed by evaporation of graphite are very high [ 29,321. Then it is clear that under such conditions the two isomers should be 266
(kJ mol-‘)
Table 2 Temperature dependence of the mole fraction w,, of rhombic C,(g) in its equilibrium mixture with linear triplet C,(g) evaluated within various approximations a)
298.15 500 1000 1500
2000 2500 3000 3500 4000
MCHS/ MCHS
VFZWE/ MCHS
VFZWE / RKY
BMB/ MCHS
100.0 100.0 99.8 96.2 85.4 70.2 56.0 44.7 36.3
100.0 98.0 69.0 42.4 29.0 21.9 17.7 15.0 13.1
100.0 98.6 78.4 55.9 42.1 33.8 28.5 24.8 22.1
6.2 8.8 9.1 8.6 8.1 1.7 7.3 7.0 6.7
‘) The approximations are coded by X/Y where X and Y refer to the source of the AH: term and vibrational frequencies, respectively, according to table I.
presumed to coexist, i.e. to be present in amounts of the same order. The mole fractions WDz,, obtained from various approaches studied can differ considerably, which is not very surprising considering the differences of the A# terms used. The comparison of the VFZWE/ MCHS and VFZWE/RKY approaches reveals an interesting sensitivity of the results to the harmonic vibrational frequencies. It is connected with the fact that the C,,(g) aggregates exhibit a number of lowlying vibrational frequencies, and the vibrational partition function is very sensitive to even small shifts in frequencies in this region. The correction of the RRHO model for vibra-
tional anharmonicity
15September 1989
CHEMICALPHYSICSLETTERS
Volume 161.number 3
and rotational non-rigidity
should bring no significant
changes in the WD,, val-
ues, because this term is related to the ratio of the partition functions of both structures and, hence, these corrections will be extensively cancelled in the numerator and denominator of this ratio [ 381. Finally, it should be stressed that there is still no consensus on the A@ term and further progress is clearly possible. It is, inter alia, because we deal with the problem of calculating the energy difference between states of differing multiplicities, where the correlation effects can be quite different.
4. An approximaterule for TsoH Table 3 presents the TSO% temperature values for the first three approaches considered in table 2. Clearly, a decrease of the positive LW! value towards zero will cause a decrease of the temperature at which the mixture is equimolar. However, there arises a question whether it is possible to simply express the way in which the T,,,%varies in an isomeric mixture if the A@ term is (slightly) varied whereas all the other terms are fixed. More generally, we can consider the isomerization (1)
A(g) =B(g) 9
which exhibits a certain value of the equilibrium constant K for the value a,, at a temperature T, (in our particular case of the estimate of the TSO, temperature it is K= 1,but the considerations take the same way even if K# 1). Let us now change the M& term into the value A@$ and look for the
temperature T2 at which this value of the equilibrium constant K will be reached again. If qx,i denotes the partition function of the species X (X= A or B) at the temperature Ti(i=1or 2), then it is [37,38] K= (q&q*,,) =(qB,Zh,P)
exp( -A%,JRT1) exp(-~b/RTd.
(2)
Now it is necessary to specify somewhat the partition functions Qx,i.The species A and B will be nonlinear and linear molecules, respectively. As we are interested in the region of higher temperatures, we shall employ for the vibrational partition function of the harmonic mode of the frequency Oj its high-temperature limit (i.e. the term [ 37 ] kT/fiwj). Under these presumptions an extensive cancellation can be carried out in expression ( 2 ) , so that they are finally reduced to the equation +ln T,-AH&/RT, =f In T2-AH&/RT2,
(3)
which can be used as a transcendental equation for estimation of the Tso% temperature (this estimate will be denoted as TSoS(2/l ) ) for the enthalpy term ml,,, provided the T,% value is known for AWL. Table 3 presents the T,,(2/ 1) estimates carried out mutually between the first two situations in table 2. The approximate procedure appears to provide quite good estimates even at considerable differences in the values AE& and AJ&
5. The equilibriumconstant of formation of C,(g) above graphite Literature [29,31,32] gives the values of the equiconstant KP of formation of the C,(g) aggregate, librium
Table 3 Temperature positions of the equimolarity point T, in the equilibrium mixtures of rhombic and linear triplet isomers of C&I Approach @)
T,ow (K)
T50%(2/1) w1 b,
MCHS/MCHS VFZWE/MCHS VFZWE/RKY
3247 1321 1681
3313 1289
a) See tables 1 and 2. ‘) Approximate treatment described in the text.
4C(s)=G(g;
overall),
(4)
at two different temperatures (2400 and 3003 K). As the detection technique used was mass spectrometry, it was, of course, impossible to differentiate the presence of the two isomers, and they were in fact treated as a single pseudospecies (therefore, the Cq of eq. (4) is referred to as overall). In the light of theoretical findings about the isomerism of the Cd(g) cluster, two partial equilibria must be considered: 267
Volume 16 1, number 3
4’3s)
=G(g;
4C(s) =Cdg;
D2d
15 September 1989
CHEMICAL PHYSICS LETTERS
(5)
,
Dood ,
(6)
each of them having its own partial equilibrium constant Kp,i (i = 1 or 2). In the case of the equilibrium constant the relation between the partial and overall quantities is extremely simple, (7)
Kp =&,I +&,z >
which, however, need not necessarily be the case with the other thermodynamic quantities, e.g. the partial and overall heat capacities, cf. refs. [ 38,391. This isomerism in reaction (4), of course, must be properly respected when comparing the theoretical and the observed Kp values. Evaluating the equilibrium constant K,, we must include certain observed values which can hardly be determined from the quantum-chemical theory (which is a difference from the simpler situation of the evaluation of Wjterms). The quantum-chemical calculations provide primarily the total energy of the system. The treatment for determination of K,, however, needs not only the total energy of C,(g) clusters but also that of the free C(g) atom, and both these energies should be determined in the same approximation (the value for a single atom, however, was specified [ 181 only in some of the considered approaches), The transition from the binding energy of a cluster to its heat of formation at absolute zero necessitates the observed value [ 311 of the heat of formation of C(g). Finally, also the values of ther-
modynamical potential of graphite C (s ) are to be taken from observation [ 3 11. Table 4 presents a survey of the calculated values Kp,i and Kp for three selected approaches, and a comparison with experiment [ 29,3 1,321. The ap proaches MCHS/MCHS and MCHS/RKY appear to lead - at the level of the K,, term - to essentially the same, namely favourable, agreement with the observed values (the difference is less than one order of magnitude). In the case of the more sophisticated CCD + ST (CCD) energy, however, the agreement is worse by at least two orders of magnitude. Although a certain part of the incomplete agreement between theory and experiment is probably due to the RRHO approximation used, it is clear that the binding energy of the C,(g) clusters represents a critical point for further improvement here, yielding a final, reliable energy difference between them. Finally, it should be mentioned that when comparing the theoretical thermodynamic data with observation establishment of thermodynamical equilibrium is to be supposed in the latter situation. To clarify the question, kinetics should be analyzed, this being a considerably more difficult task in both theory as well as experiment. Nevertheless, such theoretical results can be useful not only for interpretation of carbon vapours [29,32] but even for understanding of populations of the vapour species frozen in matrices [ 40-441.
Table 4 Comparison of the observed and calculated values of the equilibrium constant Kp of formation of the Cd(g) clusters ‘) Process
2400
3003
log,& b’ MCHS/MCHS
MCHS/RKY
RB/MCHS
observed [29,31,32]
4C(s)=C&D2s) 4C(s)=Ca(g;Dmr.) 4C(s)=C,(g; overall)
-11.22 - 11.65 - Il.08
- 11.09 -11.79 - 11.01
- 14.37 - 13.87 - 13.75
- 10.23
4C(s) =G(g; D2d 4C(s)=C,(g; D-s) 4C( s) =C& overall)
-6.82 -6.92 -6.57
-6.70 - 7.07 -6.54
-9.34 -8.70 -8.61
-6.21
‘I Rhombic and linear triplet structures of Dri, and D mh symmetry, respectively; see tables 1 and 2 for a description of the approaches used. b, K, in atm; for the standard-state specification, see ref. [ 3 11,
268
Volume 161, number 3
CHEMICAL PHYSICS LETTERS
Note added Further data on the small C, systems have been made available in refs. [ 46,471,
References [ I ] KS. Pitzer and E. Clementi, J. Am. Chem. Sot. 81 (1959) 4471. [2] S.J. Strickler and KS. Pitzer, in: Molecular orbitals in chemistry, physics and biology, eds. B. Pullman and P.-O. Lijwdin (Academic Press, New York, 1964) p_28 1. [3] R. Hoffmann, Tetrahedron 22 (1966) 521. [ 41 D.H. Liskow, C.F. Bender and H.F. Schaefer III, J. Chem. Phys. 56 (1972) 5075. [5] 2. Slanina and R. Zahradnik, J. Phys. Chem. 81 (1977) 2252. [6] M. Kirby and 8. Liu, J. Chem. Phys. 70 (1979) 893. [ 7 ] P.J. Bruna, SD. Peyerimhoff and R.J. Buenker, J. Chem. Phys. 72 ( 1980) 5437. [S] R.A. Whiteside, R. Krishnan, D.J. DeFrees, J.A. Pople and P. von R. Schleyer, Chem. Phys. Letters 78 ( 1981) 538. [ 9 ] R.A. Whiteside, R. Krishnan, M.J. Frisch, J.A. Pople and P. von R. Schleyer, Chem. Phys. Letters 80 ( I98 1) 547. [lo] D.W. Ewing and G.V. Pfeiffer, Chem. Phys. Letters 86 (1982) 365. [ 111 P. Joyes and M. Leleyter, J. Phys. (Paris) 45 (1984) 1681. [ 121 J. Bemholcand J.C. Phillips, Phys. Rev. B 33 ( 1986) 7395. [ 13 ] D.H. Magers, R.J. Harrison and R.J. Bartlett, J. Chem. Phys. 84 (1986) 3284. [ 141 J.P. Ritchie, H.F. King and W.S. Young, J. Chem. Phys. 85 (1986) 5175. [ 15 ] K. Raghavachari, R.A. Whiteside and J.A. Pople, J. Chem. Phys. 85 (1986) 6623. [ 16) B.K. Rao, S.N. Khanna and P. Jena, Solid State Commun. 58 (1986) 53. [ 17 ] D.W. Ewing and G.V. Pfeiffer, Chem. Phys. Letters 134 (1987) 413. [ 18 ] K.Raghavachari and J.S. Binkley, J. Chem. Phys. 67 (1987) 2191. [ 191 D. Michalska, H. Chojnacki, B.A. Hess Jr. and L.J. Schaad, Chem. Phys. Letters 141 (1987) 376. [20] Z. Slanina, Chem. Phys. Letters 142 (1987) 512. [21] D.W. Ewing and G.V. Pfeiffer, in: Physics and chemistry of small clusters, eds. P. Jena, B.K. Rao and S.N. Khanna (Plenum Press, New York, 1987) p, 67, [22] L.S. Ott and AK. Ray, in: Physics and chemistry of small clusters, eds. P. Jena, 3.K. Rao and S.N. Khanna (Plenum Press, New York, 1987) p, 95.
I5 September I989
[ 231 K. Raghavachari and J.S. Binkley, in: Physics and chemistry of small clusters, eds. P. Jena, B.K. Rao and S.N. Khanna (Plenum Press, New York, 1987) p. 3 17. [24] G. Pacchioni and J. Koutecky, in: Physics and chemistry of small clusters, eds. P. Jena, B.K. Rao and S.N. Khanna (Plenum Press, New York, 1987) p. 439. [25] Z. Slanina, Thermochim. Acta 127 (1988) 237. [26] R.J. Van Zee, R.F. Ferrante, K.J. Zeringue, W. Weltner Jr. and D.W. Ewing, J. Chem. Phys. 88 (1988) 3465. [ 271 D.E, Bemholdt, D.H. Magers and R.J. Bartlett, J. Chem. Phys. 89 (1988) 3612. [ 281 J.M.L. Martin, J.P. Franpis and R. Gijbels, J. Chem. Phys. 90 (1989) 3404. [ 291 J. Drowart, R.P. Bums, G. DeMaria and M.G. I&ram, J. Chem.Phys.31 (1959) 1131. [30 ] K.R. Thompson, R.L. DeKock and W. Weltner Jr., J. Am. Chem. Sot. 93 (197 1) 4688. [31] D.R. Stull and H. Prophet, eds., JANAF thermochemical tables, NSRDS-NBS 37 (US Dept. of Commerce, Washington, 1971). [32] P.D. Zavitsanos and G.A. Carlson, J. Chem. Phys. 59 (1973 1 2966. [33] W.R.M. Graham, ICI. Dismuke and W. Weltner Jr., Astrophys. J. 204 (1976) 301. [34] R.J. Van Zee, R.F. Ferrante, K.J. Zeringue and W. Weltner Jr., J. Chem. Phys. 86 (1987) 5212. [35]S. Yang, K.J. Taylor, M.J. Craycraft, J. Conceicao, CL. Pettiette, 0. Cheshnovsky and R.E. Smalley, Chem. Phys. Letters 144 (1988) 431. [ 36 ] M. Algranati, H. Feldman, D. Kella, E. Malkin, E. Miklazky, R. Nasman, Z Vager and I. Zajfman, J. Chem. Phys. 90 (1989) 4617. [ 371 G.J. Jtmz, Thermodynamic properties of organic compounds (Academic Press, New York, 1967). [38] Z. Slanina, Contemporary theory of chemical isomerism (Academia/Reidel, Prague/Dordrecht, 1986). [38] Z. Slanina, Thermochim. Aeta 128 (1988) 157. 1401 V.E. Bondybey and J.W. Nibler, J. Chem. Phys. 56 (1972) 4719. [41] L.E. Brus and V.E. Bondybey, J. Chem. Phys. 63 (1975) 3123. [42] V.E. Bondybey, J. Chem. Phys. 65 (1976) 2296. [43] V.E. Bondybey and J.H. English, J. Chem. Phys. 68 (1978) 4641. [44]M. Rasanen, J.Y. Liu, T.P. Dzugan and V.E. Bondybey, Chem. Phys. Letters 142 (1987) 308. [45] Z. Slanina, Thermochim. Acta, 142 (1989) 343. [46] DC. Parent and S.W. McElvany, J. Am. Chem. Sot. I 11 ( 1989) 2393. [ 471 P. Jensen, Collection Czech. Chem. Commun. 54 (1989) 1209.
269
CHEMICAL PHYSICS LETTERS
15 September I989
ON INTERCHANGE OF RELATIVE STABILITY OF C,(g) ISOMERS **** ZdenE?kSLANINA The J. Heyrovs@Instituteof PhysicalChemistryand ElectrochemistryCzechoslovakAcademyof Sciences, DolejSkova3, I82 23 Prague8 - Kobylisy,Czechoslovakia Received 12 May 1989; in final form 12 June 19539
The equilibrium has been studied between the rhombic and linear triplet structures of C,(g) on the basis of recent quantumchemical data. The existenceof interchange of their relative stability with increasing temperature is indicated for some approaches to the energetics. An approximate formula for estimating the equimolarity temperature has been suggested. A comparison has been carried out with existing thermodynamic data for the equilibrium between C,(g) and graphite.
1. Introduction
2. The C,(g) system
The interest in small carbon clusters C,(g) is due to a number of technological reasons leading to studies of the evaporation of graphite as well as to the presumed existence of these aggregates in interstellar space. As these species are sufficiently characterized experimentally only up to n=3, theoretical studies of these aggregates beyond this limit represent a valuable source of information about these clusters, see e.g. refs. [l-28]. The C4 cluster has recently been dealt with in a series of theoretical studies [8,13,14,16,18,19,22-24,26-281 which substantially extend the knowledge obtained from experiment [ 29-361. Thereby a sufficient amount of data from advanced quantum-chemical calculations has also been made available, which enables a theoretical evaluation of thermodynamic characteristics. The present paper provides an analysis of some interesting thermodynamic relations in this system and a comparison with the existing observed (but for testing of the theory, not yet employed) information [29,31,32].
theoretical calculations there exists a consensus that there are two energetically low-lying C4 structures, namely the rhombic one (DZh) and the linear triplet one ( Dccrh). The rhombic structure appears to lie mostly lower in terms of potential energy, but there is no unanimity with regard to the energy difference between the two structures (cf. table 1). According to a 6-31G*/MP2 study [ 191 the potential-energy difference AE is 64.86 kJ mol-I. The transition to higher orders of perturbation theory, viz. MP3 and MP4, leads [ 181 to values of 11.37 and 27.99 kJ mol-‘, respectively. Finally, the sophisticated approach reported in ref. [ 181, the so-called CCD+ST(CCD), provides a value of 21.71 kJ mol-‘. However, in the CCSD+T(CCSD) approach [27] the energy sequence is reversed: hE= -5.48 kJ mol-‘. The study [26] suggests the value of A@= 5 kcal mol-‘c20.92 kJ mol-’ for the ground-state energy difference between the linear triplet and the rhombic structures (i.e. the potential energy corrected for zero-point vibrations). For the purpose of studying the relative stability of the two isomers we will use three values of the AH: term,. namely the 6-31G*/MP2 value [ 191 Lwi ~65.15 kJ mol -I, the recommended estimate [ 26 ] AH: = 20.92 kJ mol-‘, and the negative value Mg= -5.19 kJ mol-’ derived from ref. [27].
* Dedicated to the late Academician V.P. Glushko. ** Part XL11 in the series Multimolecular Clusters and Their Isomerism; for part XLI, see.ref. [45 1.
On
the
basis
of
[ 8,13,14,16,18,19,22-24,281
0 009-2614/89/$ 03.50 Q Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
265
Volume 16 1, number 3
CHEMICAL PHYSICS LETTERS
15 September 1989
Table 1 Survey of potential-energy differences AE and ground-state energy differences NJ: between linear triplet (Dmh) and rhombic (Dzh) isomers of C,(g) source of energy term
Level of approximation
AE(kJmol-I)
source of vibrational frequencies
Level of approximation
tiz
MCHS [ 191 RB [I81 VFZWE [ 261
6-31G*/MP2 CCD+ST(CCD) evaluation of data [8,13,14,16] CCSD+T(CCSD)
64.86 21.71 20.63 18.27 -5.48
MCHS 1191 MCHS [ 191 MCHS [ 191 RKY [ 141 MCHS [ 191
6-3lG*/MP2 6-3 lG*/MP2 6-3 lG*/MP2 DZP or 6-3 lG* 6-3 lG*/MP2
65.15 22.01 20.92 20.92 -5.19
BMB [27]
3. The temperaturedependenceof the relative stability of the two isomers If the temperature dependence of the relative stability of both structures is to be adjudged, it is necessary to carry out the calculation of the composition of equilibrium mixture of the isomers, e.g. in terms of their mole fractions wi [ 37-391. These quantities are determined by the energetics and by the partition functions of both structures. With respect to the extent of the molecular parameters available from quantum-chemical calculations the evaluation is to be limited to the usual approach of rigid rotor and harmonic oscillator (RRHO) [ 321. The structural data for the calculation of moments of inertia are taken from ref. [ 19 ] or ref. [ 27 ] (the BMB/MCHS approach). The construction of vibrational partition function makes use of the set of harmonic vibrational frequencies [ 191, and the estimates of sensitivity of the vibrational partition function to a change in the quantum-chemical approach also adopt the set of harmonic vibration frequencies from ref. [ 141. Table 2 presents the temperature dependence of the mole fraction WD2a of the rhombic structure evaluated in terms of four parameter sets. It is obvious that, for positive A# values, on increasing the temperature one always encounters a value T,, at which the equilibrium mixture will contain equimolar amounts of the two isomers. However, in the BMB/ MCHS approach [ 19,271 with the negative m8 term there is no stability exchange - the linear structure always prevails. The temperatures relevant for observations of the C,(g) formed by evaporation of graphite are very high [ 29,321. Then it is clear that under such conditions the two isomers should be 266
(kJ mol-‘)
Table 2 Temperature dependence of the mole fraction w,, of rhombic C,(g) in its equilibrium mixture with linear triplet C,(g) evaluated within various approximations a)
298.15 500 1000 1500
2000 2500 3000 3500 4000
MCHS/ MCHS
VFZWE/ MCHS
VFZWE / RKY
BMB/ MCHS
100.0 100.0 99.8 96.2 85.4 70.2 56.0 44.7 36.3
100.0 98.0 69.0 42.4 29.0 21.9 17.7 15.0 13.1
100.0 98.6 78.4 55.9 42.1 33.8 28.5 24.8 22.1
6.2 8.8 9.1 8.6 8.1 1.7 7.3 7.0 6.7
‘) The approximations are coded by X/Y where X and Y refer to the source of the AH: term and vibrational frequencies, respectively, according to table I.
presumed to coexist, i.e. to be present in amounts of the same order. The mole fractions WDz,, obtained from various approaches studied can differ considerably, which is not very surprising considering the differences of the A# terms used. The comparison of the VFZWE/ MCHS and VFZWE/RKY approaches reveals an interesting sensitivity of the results to the harmonic vibrational frequencies. It is connected with the fact that the C,,(g) aggregates exhibit a number of lowlying vibrational frequencies, and the vibrational partition function is very sensitive to even small shifts in frequencies in this region. The correction of the RRHO model for vibra-
tional anharmonicity
15September 1989
CHEMICALPHYSICSLETTERS
Volume 161.number 3
and rotational non-rigidity
should bring no significant
changes in the WD,, val-
ues, because this term is related to the ratio of the partition functions of both structures and, hence, these corrections will be extensively cancelled in the numerator and denominator of this ratio [ 381. Finally, it should be stressed that there is still no consensus on the A@ term and further progress is clearly possible. It is, inter alia, because we deal with the problem of calculating the energy difference between states of differing multiplicities, where the correlation effects can be quite different.
4. An approximaterule for TsoH Table 3 presents the TSO% temperature values for the first three approaches considered in table 2. Clearly, a decrease of the positive LW! value towards zero will cause a decrease of the temperature at which the mixture is equimolar. However, there arises a question whether it is possible to simply express the way in which the T,,,%varies in an isomeric mixture if the A@ term is (slightly) varied whereas all the other terms are fixed. More generally, we can consider the isomerization (1)
A(g) =B(g) 9
which exhibits a certain value of the equilibrium constant K for the value a,, at a temperature T, (in our particular case of the estimate of the TSO, temperature it is K= 1,but the considerations take the same way even if K# 1). Let us now change the M& term into the value A@$ and look for the
temperature T2 at which this value of the equilibrium constant K will be reached again. If qx,i denotes the partition function of the species X (X= A or B) at the temperature Ti(i=1or 2), then it is [37,38] K= (q&q*,,) =(qB,Zh,P)
exp( -A%,JRT1) exp(-~b/RTd.
(2)
Now it is necessary to specify somewhat the partition functions Qx,i.The species A and B will be nonlinear and linear molecules, respectively. As we are interested in the region of higher temperatures, we shall employ for the vibrational partition function of the harmonic mode of the frequency Oj its high-temperature limit (i.e. the term [ 37 ] kT/fiwj). Under these presumptions an extensive cancellation can be carried out in expression ( 2 ) , so that they are finally reduced to the equation +ln T,-AH&/RT, =f In T2-AH&/RT2,
(3)
which can be used as a transcendental equation for estimation of the Tso% temperature (this estimate will be denoted as TSoS(2/l ) ) for the enthalpy term ml,,, provided the T,% value is known for AWL. Table 3 presents the T,,(2/ 1) estimates carried out mutually between the first two situations in table 2. The approximate procedure appears to provide quite good estimates even at considerable differences in the values AE& and AJ&
5. The equilibriumconstant of formation of C,(g) above graphite Literature [29,31,32] gives the values of the equiconstant KP of formation of the C,(g) aggregate, librium
Table 3 Temperature positions of the equimolarity point T, in the equilibrium mixtures of rhombic and linear triplet isomers of C&I Approach @)
T,ow (K)
T50%(2/1) w1 b,
MCHS/MCHS VFZWE/MCHS VFZWE/RKY
3247 1321 1681
3313 1289
a) See tables 1 and 2. ‘) Approximate treatment described in the text.
4C(s)=G(g;
overall),
(4)
at two different temperatures (2400 and 3003 K). As the detection technique used was mass spectrometry, it was, of course, impossible to differentiate the presence of the two isomers, and they were in fact treated as a single pseudospecies (therefore, the Cq of eq. (4) is referred to as overall). In the light of theoretical findings about the isomerism of the Cd(g) cluster, two partial equilibria must be considered: 267
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4’3s)
=G(g;
4C(s) =Cdg;
D2d
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CHEMICAL PHYSICS LETTERS
(5)
,
Dood ,
(6)
each of them having its own partial equilibrium constant Kp,i (i = 1 or 2). In the case of the equilibrium constant the relation between the partial and overall quantities is extremely simple, (7)
Kp =&,I +&,z >
which, however, need not necessarily be the case with the other thermodynamic quantities, e.g. the partial and overall heat capacities, cf. refs. [ 38,391. This isomerism in reaction (4), of course, must be properly respected when comparing the theoretical and the observed Kp values. Evaluating the equilibrium constant K,, we must include certain observed values which can hardly be determined from the quantum-chemical theory (which is a difference from the simpler situation of the evaluation of Wjterms). The quantum-chemical calculations provide primarily the total energy of the system. The treatment for determination of K,, however, needs not only the total energy of C,(g) clusters but also that of the free C(g) atom, and both these energies should be determined in the same approximation (the value for a single atom, however, was specified [ 181 only in some of the considered approaches), The transition from the binding energy of a cluster to its heat of formation at absolute zero necessitates the observed value [ 311 of the heat of formation of C(g). Finally, also the values of ther-
modynamical potential of graphite C (s ) are to be taken from observation [ 3 11. Table 4 presents a survey of the calculated values Kp,i and Kp for three selected approaches, and a comparison with experiment [ 29,3 1,321. The ap proaches MCHS/MCHS and MCHS/RKY appear to lead - at the level of the K,, term - to essentially the same, namely favourable, agreement with the observed values (the difference is less than one order of magnitude). In the case of the more sophisticated CCD + ST (CCD) energy, however, the agreement is worse by at least two orders of magnitude. Although a certain part of the incomplete agreement between theory and experiment is probably due to the RRHO approximation used, it is clear that the binding energy of the C,(g) clusters represents a critical point for further improvement here, yielding a final, reliable energy difference between them. Finally, it should be mentioned that when comparing the theoretical thermodynamic data with observation establishment of thermodynamical equilibrium is to be supposed in the latter situation. To clarify the question, kinetics should be analyzed, this being a considerably more difficult task in both theory as well as experiment. Nevertheless, such theoretical results can be useful not only for interpretation of carbon vapours [29,32] but even for understanding of populations of the vapour species frozen in matrices [ 40-441.
Table 4 Comparison of the observed and calculated values of the equilibrium constant Kp of formation of the Cd(g) clusters ‘) Process
2400
3003
log,& b’ MCHS/MCHS
MCHS/RKY
RB/MCHS
observed [29,31,32]
4C(s)=C&D2s) 4C(s)=Ca(g;Dmr.) 4C(s)=C,(g; overall)
-11.22 - 11.65 - Il.08
- 11.09 -11.79 - 11.01
- 14.37 - 13.87 - 13.75
- 10.23
4C(s) =G(g; D2d 4C(s)=C,(g; D-s) 4C( s) =C& overall)
-6.82 -6.92 -6.57
-6.70 - 7.07 -6.54
-9.34 -8.70 -8.61
-6.21
‘I Rhombic and linear triplet structures of Dri, and D mh symmetry, respectively; see tables 1 and 2 for a description of the approaches used. b, K, in atm; for the standard-state specification, see ref. [ 3 11,
268
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Note added Further data on the small C, systems have been made available in refs. [ 46,471,
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