2102
Vol. 65
r\TOTF,S
followed by a rapid sequence
+
+ HCr04+ Cr04-
BC03 H20 --it A HCrOaB ---f A
+
seems to show a certain correlation between the basicity of the reactant and the rate constant. Such a correlation is to be expected in view of the existence of general free energy relationships." On the other hand, it is not surprising that any such correlation should be counteracted by the influence of the charge type, This influence can be inferred from the similarity between ~ I I o H and J G I I ~ H ~and , from the extremely low value for JGIIcop. kllCsHsO-
(7) (5a)
When i,he reactant is OH- step 7 is of course redundant. Since the choice of bases which can bring about the hydrolysis of bichromate is rather limited, no definite pronouncement can be made at present about the connection between ,WBand the nature of the reagent employed. However, a comparison between JCIIOH- and
(11) A . -4. Frost and R. G. Pearson, "Kinetics a n d Mechanism,' John Wiley and Sons, Inc., New York, h' Y. 1933, p. 214 ff.
NOTES MOLECULAR ORBITAL CALCULATIOSS FOR CYCLOOCTBTETRAEK'E' BY GUENTERAHLERSAND JAMES F. HORNIG Departmen! of Chemistry, Cniversity of California, Riverside, California Received December 2, 1960
Calculations for the molecule cyclooctatetraene indicate that a simple extension of Hiickel type molecular orbital theory2 may be used in this case to estimate the relative stability of geometric configurations of a molecule with a non-planar ?r-electron system. It is necessary only to modify the resonance integral p to take into account the relative twist of neighboring p orbitals. The effect on p of varying bond lengths is somewhat larger here than in planar ring systems, so we have taken it into account by means of an approximation due to Mulliken.3 The effect of bond length is smaller than that due to twist, so that t'he qualitative results do not depend critically on the approximation. Hiickel recognized that cyclooctatetraene probably would be puckered, so his calculat,ions for a planar model would not be appropriat'e. Lippincott and Lord4 demonstrated by analysis of Raman spectra that the molecule was indeed puckered. The detailed geometric configuration of the molecule became the subject of numerous investigations, and the boat, crown and chair configurations (Fig. 1) all were proposed. The most recent investigation is the electron diffraction study by Bastiansen, et al.,5 which establishes the boat configuration. The present calculations support this result. Modified Resonance Integral.-Experimental bond angles of cyclooctat'etraene indicat.e that the u bonds may deviate substantially from sp2hybridiza-
tion. For the degree of approximation intended here, we assume that the P electron orbitals still can be described adequately by linear combinations of hydrogen-like 2p functions. For neighboring hydrogen-like 2p orbitals twisted through an angle 4, the resonance integral is proportional to cos 4 64 = po cos The effect of variations in the bond length, p, on the resonance integral has been estimated by Mulliken.3 We incorporate his results in a factor
JCb) PP
Soc., 63, 4 1 (1941).
R Lippinoott and R. C. Lord, ibid., 78, 3889 (1951). ( 5 ) 0. Bimtienssn, L. Hedberg and K. Hedberp, J . Chem. Phvs., 31,
( 4 ) E.
1311 (1867).
k(f)Pl.39
As Mulliken points out, this estimate already includes a contribution from the compression energy of the u bonds. Nore refined estimates of k ( p ) would yield slightly different results, but since the contribution is smaller than that from twist, this approximation is satisfactory. If the effects of stretching and twisting are independent, the resonance integral may be written pp6 =
k ( f ) cos
p'1.39
Here po1.39 is the valug appropriate to an untwisted bond of length 1.39 A. The P electron energy of any particular configuration may be calculated by evaluating the secular equation containing the values of pp6 appropriate to the assumed geometry. Boat Configuration ( D a d ) .-In this configuration the molecule has four untlyisted bonds (type a, Fig. 1A) and four bonds (type b) with a relative twist, 4, given by COS
6
=
Cotze
where 6 is the C-C-C bond angle. The a and b bond types will have different lengths, so the secular equation will contain two different resonance integrals Lo = k(a) p'1.39
(1) Based on a thesis submitted b y Guenter Ahlers in partial ful-
fillment of the requirements for the degree of Bachelor of .Arts, University of California, Riverside, California. (2) E. HLickel, Z. P h y s i k , 7 0 , 204 (1931). (3) R . S. hluliiken, C. .A. Rieke and W.G . Brown, J. Am. Chem.
=
Pb6 = k(b)
Cot'
Ob01.39
The secular equation may be solved analytically, giving a ground state energy for the eight T electrons of Erboat = 8a:
+ 4(k(a) + [k(a)' + k(bY cot2 S ] ' / z ) p o ~as
Here a is the coulomb integral.
NOTES
Nov., 1961
2103
stable than the boat, though comparable to the crown form. For the calculation we chose param130°, eters relatigely favorable for stability: 6 a = 1.33 A., b = 1.46 A., and c = 1.50 A. The energy in this configuration was found to be El,chlr
=
8a!
+ 6.5 8’1
39
Conclusions The boat configuration is favored energetically B. C R O W N ( D 4 d ) over the crown and chair configurations by about 3.5 Po139 in these calculations, thus agreeing with Bastiansen’s experimental results favoring the boat configuration. It is interesting to note that although the calculation is intended primarily to give relative results for several possible configurations, the predicted stabilization energy of the boat configuration is comparable to the experimental resonance energy. The energy of four C . CHAIR ( C 2 h ) isolated double bonds would be 8k(1.33)$139 or 9.5P0139, so our calculations predict a stabilization energy of 0.3/P01391, or about 11 kcal./mole, using -37 kcal./mole as a reasonable estimate of Po1 39.697 The experimental resonance energy is 4 kcal. /mole. Fig. 1.-Ground state pi electron energies calculated for It may be possible to estimate the relative stathree configurations of cgclooctatetraene. bility of various configurations of other non-planar Using Rastiansep’s molecular parameters, e = T electron systems in this way. Reliable quantita126.5’’ a = 1.33 A., b = 1.46 A., and Mulliken’s tive calculations, or comparisons of configurations differing widely in C-C-C bond angles would redata for k ( p ) , one obtains quire a more careful treatment which included E,,bost -= 8a: 9.8fia1.39 change in hybridization and an explicit evaluation Crown Configuration (Da).-All eight bonds are of the energy of the u system. equivalent in this configuration (Fig. lB), with a (6) G. Glockler, J Chem P h y s , 21, 1242 (1953); R. S. Muiliken s n d R G Parr,t b d , 19, 1271 (1951) relative twist, 4, given by
+
cos I$ =
‘COS
1
6
+ 2 COS 7r/4 - 1 1+ e
I
COS
Solution of the secular equation gives a ground state energy of = 801
E,mown
For p
=
1.42A., 6
+ 8.668p4
125O, thisgives
=
EI,crown
= 8a
+ 3.3P01.a9
Since ,BO1 39 is negative, this configuration is unstable with reference to the boat configuration by 5.5 f1°1.391. Assuming even the most favorable bond length, that appropriate to benzene, c = 1.39 A., and a relatively flattened molecule, e = 130°, gives E X ~ r o m n=
801
+ 6.W1
39
which is still 3.6 lPo1.39 less stable than the boat form. Chair Configuration (Czh).-The chair configuration has three kinds of bonds (Fig. 1). The four type b bonds are twisted by an angle $ J b given by COS 4 b
= cotze
just as in the boat configuration. The two type a bonds are untwisted, but the remaining two type c bonds are twisted by an angle cpc given by COS
,
I
= 12Cot2
e-
11
The twist in the c bond is considerable, so that neighboring p orbitals are eclipsed completely in the vicinity of 6 = 125’. A numerical solution of the secular equation involving three different resonance integrals showed that the chair configuration is considerably less
(7) H. D. Springall, T. R. White and R. C. Cass, Trans. Faraday Soe., PO, 815 (1954).
A NEW ME:THOD FOR STUDYING PORE SIZES BY THE USE OF DYE LUMINESCENCE BY JEROME L. ROSENBERG A N D DOVALD J. SHOMBERT Contnbutaon X o . 108P f r o m the Department 0.1’ Chemzstry, Unzverszty of Pzttsburgh, Pzttsburgh 1 3 , Pennsylaanza Recezaed January 10, I Q f i l
I n connection with some recent work on the mechanism of the reaction of oxygen with photoexcited adsorbed dyesY2 we found that the penetration of oxygen through the pores of the adsorbent was rate-limiting in some cases. I t occurred to us that photochemical observations under these conditions might be used to study the pore characteristics of the adsorbent. This note summarizes the experimental basis and outlines the possibilities of application for such a porosimeter. Procedure.-The method of study has been described in detail previously.2 Briefly, a suitable phosphoreecent dye was adsorbed on the porous substance. The principal criteria in the selection of the dye wrre high adsorbability (1) Presented before the 138th National Meeting of the American Chemical Society, New York, September 16, 1960. This work was supported by the National Science Foundation under Grant NSFG6271. The material presented here is abstracted from a dissertation presented t o the University of Pittsburgh by Donald J. Shombert in partial fulfillment of the requirements for the Ph.D. degree in January, 1959. (2) J. L. Rosenberg and D. J. Shombert, J . Am. Chsm. Soc,, 89,
