NOTES
metals15 although the existence of these species has been deduced by other means.) An ion current due to KOH was also observed, but this species resulted both from the formation of KOH on the amalgam during the loading process and from the reaction of vaporized potassium with residual HzO in the mass spectrometer. There was no evidence of K3+, HgKf, HgK2+, HgK3+, or of heavier ion species. At 20 ev, the relative ion intensities (in arbitrary units) of the + 78), three species, I K + (isotope 39), I K ~(isotope and IH,+ (isotope 202) were 400, 0.26, and 40, respectively; the ratios of IH,+/IK+and IK,+/IK+are 0.1 and 6.5 X respectively. The background ion intensities of masses 39 and 78 were nearly at the sensitivity limit of the instrument (0.001), while the background ion intensity of Hg202was 0.0432. (The obis directly proportional to served ratio of IK*+/IK+ the ratio P K J P K . From published data6$’for potassium at 200°, PK*/PK= 7.7 X 10-4.) In contrast, Roeder and Al~rawietz,~ to best fit their effusion data, calculated partial pressures for K, Hg, and HgK3 over a 77.5 at. % potassium amalgam at 200” of 6.00 X 2.60 X and 1.62 X torr, respectively; are the corresponding ratios of P H J P K and PHsK8/PR 0.43 and 0.27, respectively. Although HgK3 was not observed, the question arises as to whether the molecule exists in the vapor but has such a low dissociation energy that it cannot be observed with a mass spectrometer. This question can be qualitatively answered by considering the gaseous K2 and Hgz molecules. Even though the Kz molecule has a low dissociation energy of 0.51 ev,a it was easily observed with the mass spectrometer. If the HgK3 molecule had a dissociation energy of approximately 0.5 ev and were present in the vapor to the extent proposed by Roeder and Morawiets, it probably would have been detected. The Hgz molecule, with a dissociation energy of 0.06 ev19was not observed. The partial pressure expected for Hg2 under these experiatm. mental conditions was calculated to be 1.3 X Even though this pressure is very near the limit of detection of the mass spectrometer, it is felt that the Hgz+ ion, if present, would have been observed because of the very low background in the region of mass 400. If the dissociation energy of HgK3 is similarly low, one would not expect to observe HgK3even if it were present in the vapor to a considerable extent. However, one might reasonably expect to see one or more of the fragment species K3+, HgK+, or HgKz+-none of which was observed. Furthermore, these considerations indicate that the dissociation energy of an Hg-K bond in HgK3 would lie between 0.5 and 0.06 ev. Roeder and M ~ r a w i e t z ,however, ~ calculated an enthalpy of for-
3349
mation for gaseous HgK3 of 65 kcal/mole, which would correspond to a dissociation energy of 2.8 ev. A dissociation energy of this order of magnitude would certainly render HgK3 observable with the mass spectrometer unless, of course, its concentration was below the spectrometer’s limit of detection. The results of this experiment, therefore, indicate that the vapor phase over potassium amalgams consists of K, Kz, and Hg. The species HgK3 apparently does not exist or, at most, is a very minor component of the vapor.
Acknowledgment. Thanks are due to E. Veleckis for the preparation of the amalgam and for discussions concerning the experiment. H. M. Feder and M. Ader are also to be thanked for discussions concerning this note. (5) B. Siegle, Quart. Rev. (London), 19, No. 2, 77 (1965). (6) L. C. Lewis, 2.Physik, 69, 805 (1931). (7) F. D . Rossini, “Selected Values of Chemical Thermodynamic Properties,” Series 111, National Bureau of Standards, Washington, D . C., 1954. (8) G. Herrberg, “Spectra of Diatomic Molecules,” 2nd ed, D. Van Nostrand Co., Inc., Princeton, N. J., 1950, p 543. (9) A. G. Gaydon, “Dissociation Energies and Spectra of Diatomic Molecules,” 2nd ed, Chapman and Hall, London, 1953, p 226.
Spin Densities in Biphenylaminyl Network and Triphenylimidazolyl Network by Hisashi Ueda Department of Chemistry, The University of Tezas, Austin, Tezas 78711, and Department of Chemistry, Tokyo Institute of Technology, O h o k a y a m , Meguro, Tokyo, Japan (Received February 18, 1966)
Spin densities in the biphenylaminium anion, biphenylaminyl radical, 1-4 triphenylimidazoium anion, and triphenylimidazolyl radical’ have been reported. In those works calculations were made using the vaTable I : Spin Densities in Biphenylaminyl Ftadical and Aminium Anion Atomic positions
r
ortho
meta
para
C
N
Radical Experimental 0.084 -0.042 0.084 0.046 0.572 SCFMO 0.099 - 0 , 0 3 2 0 . 0 9 4 -0.011 0.587 Anion Experimental 0 . 0 9 3 0 . 2 5 4 - 0 . 0 9 3 -0.078 -0.046 SCFMO 0.172 0.172 -0.087 -0.078 -0.013
Volume 70, Number 10
October 1966
3350
NOTES
Table Il : Spin Densities in Triphenylimidaeolyl Radical and Imidaeoium Anion Atomic positions
Radical Experimental SCFMO Anion
c1
ortho
meta
... -0,042
0.156 0.131
-0.078 -0.076
...
0.052
para
0
N
Ca
...
0 -0,066
0
...
0.156 0.140
0.275
0
...
either Experimental
c2
C4
ortho'
meta'
...
...
0.190
-0.101
0.130 0.057
-0.055
...
or
0 +lS
SCFMO'
0 0.004
0 0.022
0.052 0 -0.024
0 -0.072
0
--0.126
0 0.079
0
0
0.358
0.077
0
either 0.260 or 0.260 0.208 0.250 0.250 0.226 -0.009 0.208
para'
0 -0,032
0
0 0.265
Twenty-five electrons are assumed with the parameters hN = 2.0, kcN = 1.0, and X = 1.2.
lence bond method* and the Huckel MO method.' The purpose of the present note is to improve the results by employing McLachlan's SCFMO (to be abbreviated simply as SCFMO) method.s The networks and the name of the positions are illustrated in the previous paper,' and the results obtained are listed in Tables I and 11. The results for the biphenylaminyl radical (Table I) have been improved by the SCFRIO method in which the parameters hn = 1.0, ~ C N= 0.75, and X = l.05 are used. The nmr experiments by Gutowsky, et aL13indicate that at the meta positions the sign of the spin densities is negative. Accordingly, the present calculation has also indicated that negative spin densities are to be placed on the meta positions. Therefore, it seems that the SCFRIO calculation can give the correct signs of spin densities in this particular network. Accordingly, in the case of the biphenylaminium anion, the sign of the experimental spin densities at the para positions is assumed to be negative as shown in Table I. With the parameters hN = 2.0, ~CCN = 0.75, and X = 1.2, the calculated result fits the experimental result considerably better than that by the Huckel MO method, except the spin densities for the ortho and the meta positions. In the framework of the present calculation, there is no way to place different spin densities at the ortho and meta positions in biphenylaminium anion, while the difference between the observed values for these two positions is far beyond experimental error. As pointed out by Karplus, et aLj6 if there is an effect of some out-of-plane vibrations of the ortho and meta carbon atoms on the Huckel resonance integral parameters, /3, and if this value is not equal for the ortho and meta positions, spin densities will be distributed unequally between the ortho and meta positions, especially in the case of negative ions. Actually, the sum of the The Journal of Physical chemistry
spin densities at the ortho and meta positions is 1.388 by experiment and 1.376 by this calculation. I n case of triphenylimidazolyl radical, the combination of parameters, h N = 1.0, ~CCN = 1.0, and X = 1.8, yields fairly good spin densities except for a 50% lower estimate for the ortho' positions (Table 11). Finally, the result for the triphenylimidazoium anion is not improved at all if 25 electrons are assumed on the network (Table 11). 3Iost of the spin densities are accounted for by the theoretical values obtained from the 15th lowest Huckel orbital, $151 in which the unpaired electron is localized on the phenyl ring including C4 (see ref 1). The remaining unaccounted spin densities, 0.051 for each of four equivalent positions, are not explained by this orbital. It might be explained if one assumes that there is small admixture of other wave functions with the $15, and as a result, there are small spin densities on the rings, including C1 and C1', and that as a result of the out-of-plane vibration, the unpaired electron spin densities are concentrated on either of the ortho or meta position. If the unpaired electron is in the $15 orbital, a total of 29 electrons must be assumed on the network. This, however, means that the molecule has six negative charges. As there are four rings in the molecule, it might be possible to place a maximum of four negative charges on the network. This suggests that the energy relation between the $14 and the $15 is reversed. The (1) H.Ueda, J . Phys. C h a . , 68, 1304 (1964). (2) H.Ueda, 2. Kuri, and S. Shida, J . Chem. Phys., 36, 1676 (1962). (3) H.S. Gutowsky, H. Kusumoto, T. H. Brown, and D. H. Anderson, ibid., 30, 860 (1959). (4) T. H.Brown, D. H. Anderson, and H. S. Gutowsky, ibid., 33, 720 (1960). (5) A. D.RIcLachlan, Mol. Phys., 3, 233 (1960). (6) M.Karplus, R. G . Lawler, and G . K. Fraenkel, J . Am. Chem. SOL, 87, 5260 (1965).
NOTES
3351
+
+
energy eigenvalues for these are CY 0.6287p and CY 1 .Ob,respectively. The actual energy eigenvalues are, therefore, either higher for the g14 or lower for the 4,s than the calculated values. Acknowledgment. The author thanks the Robert A. Welch Foundation for financial support.
Volume-Energy Relations in Liquids a t
OOK from Equations of State
by S. T . Hadden Gulf Research & Development Company, Pittsburgh, Pennsylvania 16260 (Received March 16, 1966)
A variety of ways for evaluating the volume-energy relations in the hypothetical liquid state of 0°K for CCl, and some hydrocarbons was recently presented by Another method of evaluating these relations from equations of state for liquid hydrocarbons is presented here. Several equations of state have been derived in close parallel with the cell equation of state of Prigogine and c o - w o r k e r ~ ,and ~ ~ ~some of these and one not previously discussed in the literature will be mentioned. Comments on the Prigogine equation of state have been publi~hed.~ Modifications of the Prigogine equation of state must be made if a model different from Prigogine's is assumed. Hence, it is helpful to review the assumptions for Prigogine's model2 (beyond those in the usual quasicrystalline model, e.g., Guggenheim6). These are summarized as follows. (A) The planar, open-chain, zigzag molecular model consists of s identical units (ignoring end effects) which are subject to a Lennard-Jones 6-12 potential when centered on their respective lattice sites. (B) When the units oscillate off their lattice sites, the change of the intermolecular field can be represented adequately by a harmonic oscillator field. (C) Any differences between intra- and intermolecular distances can largely be removed in the model by choosing CH, as the repeating unit (thus propane is a dimer, pentane a trimer, etc.). Also, the effect of temperature and pressure on the departure from a regular lattice can be ignored for volume changes of only a few per cent. The molecular degrees of freedom can be subdivided into (a) intramolecular degrees of freedom (bond stretching and bond angle vibrations) which are independent of volume and (b) intermolecular degrees of freedom.
(D) The isolated s-mer has free rotation about the bonds linking the s monomer units. The validity of assumption D for n-alkanes has been commented on4 and has been confirmed6 for s 2 2 with the modified equation of state discussed below. Using the foregoing assumptions, a partition function can be evaluated from which the equation of state is derived using P = kT(b In Z / ~ V ) Twhere , Z is the partition function. If changes in the assumptions for the model are made, a new equation of state can be obtained by redefining the partition function. Changes in the Prigogine model are necessary to represent adequately other hydrocarbon molecules. For example, for aromatic and alkylated aromatic hydrocarbons, change assumption C so that each aromatic ring is represented as two units (Le., benzene is two units; biphenyl is four units) making the repeating unit of the ring approximately C3H3,and change assumption D to assume that free rotation occurs only between pairs of units representing separate rings."' As a second example, for oligomers having end effects, change assumption A so that the planar, open-chain, zigzag molecular model consists of s identical units plus two end units, etc., and change assumption C so that there are s CH4units (roughly -C2H4- in the molecule) and two end units. For n-alkanes (CnH2*+2),if n is an odd number, each end unit is a CH4 unit and its characteristics differ from the central units; and, if n is an even number, each end unit corresponds with the -CHI group in the molecule. These definitions for end units preserve the uniform lattice.6 The equations of state for the various types of hydrocarbons have the general form
6M1(BQ2 - C )(BQ2 - 2C)-'
+
2 -M2Q2(BQ2 IcT
- C)
(1)
The contribution to the potential field of neighbors beyond the first shell are accounted for by the constants B and C which, for a face-centered-cubic lattice, are (1) A. A. Miller, J . Phys. Chem., 6 9 , 3190 (1965). (2)I. Prigogine, N. Trappeniers, and V. Mathot, Discussions Faraday SOC.,15, 93 (1953). (3) I. Prigogine, "The Molecular Theory of Solutions," Interscience Publishers, Inc., New York, N. Y.,1957. (4) R.Simha and S. T. Hadden, J . Chem. Phys., 25, 702 (1956); 26, 425 (1957). (5) E.A.Guggenheim, "Mixtures," Oxford University Press, London, 1962. (6)S. T. Hadden, Doctoral Thesis, New York University, Jan 1965. (7) 9. T. Hadden and R. Simha, J . Chem. Phys., 36, 1104 (1962).
Volume 70.Number 10 October 1966
