NOTES
37 1
we find that both the Rh-H, and H-H stretches are shown to be infrared active species. The general result again is that whatever geometry is assumed, the nondegenerate stretching modes are infrared active, and consist of a single line per type of adsorbing face, unless split by the correlation fielda2 The infrared spectra listed in Table I contain a number of lines in the 3000cm-l region which have not been assigned. A possible assignment presents itself if these bands are considered to be the perturbed H-H stretches associated with a ,H-H Rh/ \Rh bridged structure. The 2000-~m-~ bands have essentially the same interpretation given at first, that of being Rh-H stretches, but now, due to the bridged structure are capable of vibrational coupling with the higher frequency H-H motions of the same symmetry species. The corresponding motions of the metal-hydrogen bond in these two extremes may be considered analogous ones in which the H-H interaction of neighboring protons on the surface is changed from a molecular bond to an atom-atom interaction term.
Quantum Yield of Photonitrosation of
Cyclohexane under a Flash Lamp by Kuya Fukuzawa and Hajime Miyama Basic Research Laboratories, Toyo Rayon Co., Ltd., Kamakura, Japan (Received August 8.2, 1967)
Baumgartner, et al.,’ reported the quantum yield of photonitrosation of cyclohexane as 1.48, while Shimokawa, et a1.,2 reported a rather contradictory value of 0.7 for the same reaction. To explain these data the former believes that the reaction proceeds via the following chain mechanism NOCl r i CsHio-CH2
Rh
/H-H\
‘\Rh+Rh
I
I
Rh
Again, since the metal-H frequency is so similar in both cases, it is clear that special sites must be responsible for dissociation, which then diffuse mobile hydrogen atoms onto the observed sites. In the case of the atomic mode of adsorption, the H . . . H motions will be included among the lattice modes predicted in the correlation mappings, Tables I1 and 111, and should be observable optically. In the 3000- and 2000-cm-l regions the large number of bands observed can be attributed to multiple crystallographic sites available on the polycrystalline films, each exhibiting as would be expected a different perturbation of the fundamentals discussed above. The discussion herein applies a group theoretical model to the discussion of spectra for H2 and D2 on polycrystalline rhodium substrates, and predicts the existence of certain fundamentals lying outside the region available to the investigators. A theoretical discussion of experimental problems in these regions has been given to indicate how these data may be obtaineda8 Acknowledgment. This work was supported in part by U. S. Atomic Energy Commission Contract No. AT(40-1)-2948.
(6) H. C. Eckstrom and W. H. Smith, J . Opt. SOC.Am., 57, 1132 (1967).
r+ C1. +C5Hio-CH* + HCl r+ NOCl --+ C5Hlo-CHN0 + C1, 1
1
C;Hlo-CH.
The latter, however, assumes the scheme NOCl
H . . .. H I I
+ hv + N O + C1.
7
+ hv +NO + C1.
1
+ C1. +C6Hlo-CH. + HC1 r-----l r----C5Hlo-CH. + NO +C5Hlo-CHNO 7
1
C5Hlo-CH2
Thus the quantum yield must necessarily be smaller than or at best equal to unity. There seems to be no compromise between these two observations and further investigations still to be awaited. We therefore performed this photoreaction under an intense flash lamp. Our conclusion is that the quantum yield is about 0.65 0.20 even under intense light.
Experimental Section The reaction vessel used in this work was of a triply walled, coaxial type with a flash lamp mounted at the center. The inner chamber was filled with a filter solution, and the outer chamber with a solution of cyclohexane. A photomultiplier was attached to the outer surface to serve as a monitor. The flash light was separated into peak wavelengths of 3600, 4100, and 5050 A by means of water solutions of cupric and cobaltous sulfate, cupric nitrate, and cupric and calcium chloride, respectively. A 0.006 M solution of potassium ferrioxalate3 was filled in the reaction chamber to measure the absolute number of incident light quanta at 3600 A, which together with the monitoring photomultiplier enabled us (1) P. Baumgartner, A. Desohamps, and C. Roux-Guerraz, Compt. Rend., 259, 4021 (1964). (2) Y. Shimokawa, et al., K o w o Kauaku Zasshi, 68, 937 (1965). (3) C. A. Parker, Proc. Roy. SOC.(London), A220, 104 (1953).
Volume 78, Number 1
January 1968
XOTES
372 to determine the number of light quanta at the longer wavelengths. Nitrosyl chloride was bubbled into a cyclohexane solution saturated with gaseous hydrochloric acid. The concentration of nitrosyl chloride was ca. 1%. I n order to determine the extent of the photoreaction, resulting oxime hydrochloride was treated with chlorine to give chloronitrosocyclohexane, the absorption of which was measured at 6500 A. The experimental error involved in the whole procedure was estimated to be within *30%. Results summarized in Table I show that the quantum yield as measured a t the three wavelengths was
accurate way of estimating the number of relatively short self-avoiding walks. The relative positions of the monomer units in the molecule are represented by the successive states of a Markov chain. The Markov chain contains an absorbing state, and transitions into this state correspond to configurations of infinite energy (normally where two monomer units occupy the same lattice point). Suppose that the states of the Markov chain are labeled (1,2, . . . , m) where 1and m are the absorbing and initial state, respectively. Let p15(n)be the n-step transition probability from state i to state j and let pi5(') be written p i 5 . The (mzm) transition probability matrix P = ( p v ) may be partitioned as
Table I Wavelength, A
3600 3600 4100 4100 5050 5050
Oxime produced,
M
0.83 x 0.76 x 1.39 x 1.18 x 0.86 x 0.87 x
10-4 10-4 10-4 10-4 10-4 10-4
Light quanta
absorbed, M
1.22 x 1.22 x 2.02 x 2.00 x I . 36 x 1.30 x
Quantum yield
10-4 10-4 10-4 10-4 10-4 10-4
0.68 0.62 0.69 0.59 0.63 0.67
invariable within experimental error. The fact that the values obtained are less than unity does not necessarily exclude the chain mechanism. Our observation, however, seems to support Shimokawa's mechanism.
where Q is the transition probability matrix between all nonabsorbing states, R is the column vector of transition probabilities into the absorbing state, and 0 is a zero-row vector. A particular configuration of the molecule will then be represented by a realization of the Markov chain, i.e., by a vector of the form (m, il, i ~ ,. . . , is). If, at the rth step the absorbing state is reached, the corresponding vector is (m, il,iz, . . . , &.-I, 1, 1, . . ,, 1). Suppose that the coordination number of the lattice is c and that walks with adjacent steps in opposite directions are forbiden; then the number of otherwise unrestricted walks is c(c
The Treatment of End Effects in a Markov Chain Model of the Configurational Properties
p* = 1 - pml(")
The Journal of Physical Chemistry
(2)
(3)
so that the number of walks, f(n), not reaching the absorbing state is given by f(n) = p*(c
Unilever Research LaboTdoTy, The Frythe, Welwyn,, Hertshire, England (Received August 80,1067)
The majority of treatments of the configurational properties of long-chain molecules have begun with the assumption that the properties of the molecule are sufficiently well represented by the properties of a selfavoiding random walk on a regular lattice. The mathematical difficulties introduced by the self-avoiding condition can be avoided by approximating the self-avoiding walk by a first-order homogeneous Markov chain. I n Mazur's treatment, the self-avoiding condition is introduced by including an absorbing state in the Markov chain and by averaging only over the transient states of the chain. The approach can be adapted to include end effects and then proves to be a reasonably
= (c - 1)"
The probability ( p * ) that a walk, starting in state m will not have reached the absorbing state 1 in n steps is
of Long-chain Molecules
by S. G. Whittington and P. J. Williams
- 1)-
- 1)"
(4)
To the level of approximation of this treatment, this is the number of self-avoiding walks. pml(")can readily be expressed2 in terms of the eigenvalues and eigenvectors of P,leading to the expression (5)
where X, is the j t h eigenvalue of P and x5 and y5 are the corresponding right and left eigenvectors. am5is the mth element of xj and p5, is the first element of yj. If the eigenvalues are numbered in decreasing order J. Mazur, J. Chem. Phys., 41, 2556 (1964). (2) L. Takacs, "Stochastic Processes," Methuen and Go. Ltd., London, 1962, p 9. (1)
