J. Phys.
524
Chem. 1983, 87, 524-527
layer concept as an approximation. Let us introduce a frame of reference where the two transformed concentrations are equal at x = 0 Cl(0) = 4 0 ) = co (A4)
where "a" can be expressed from (A9) and (A12) as
Now according to the reaction layer concept
The last step of the present derivation is to show that most of the reaction takes place in the domain -a < x < +a. The reaction density Rc (the rate of the consumption of C per unit area of the electrode) between x = -a and x = +a
clc2 = co2= constant
(A5)
holds within the layer. The first derivative of (A5) a t x = 0 gives the relation
R, =
1'"rc
dx
-a
(A141
where See Figure 6 for the geometric meaning of the characteristic distance a. The second derivative of (A5) at x = 0 gives the following relation co($)
x=o
+co(a>
x=o
+2(%)x=(z)
x=o
k,
rc = -[X-][C] With (A2) and (A5) DC DX-
r, = -k,c1c2
=
0 (A71
(A15)
z
DC
= -kcco2
Dx-
and R,
Based on (A3)-(A7)
DC R, = --ltCc,2(2a)
DXIt can be shown by using (A12) and (A13) that and regarding (A3), (A4), and (A@,we obtain Dx-/a2 = kcco
(A9)
We need another equation to calculate a or co. (A9) was derived by using the reaction layer concept. Another relationship can be found by applying the diffusion layer concept as the reaction is fed by a diffusion current. As we have a symmetrical problem, half of the reaction takes place where x < 0 and the other half in the region of x > 0. Consequently, the diffusion current density of component C entering into the diffusion layer of thickness 6 (Jc)x=a
= -D,(dcz/dx)x=,
-Dc([c]b/s)
(A10)
= -DC(dc,/dx),=o = l/z(JJx=a
(All)
and regarding (A17) and (AB), we obtain Rc = Dc([c]b/6) =
I(Jc)x=aI
W9)
That is, the whole current density of "C" entering into the diffusion layer is used up by the reaction in the region -a < x < +a. In other words, the above region is the reaction layer the thickness of which 1.1 = 2a (A20) and with (A12)
will be halved at x = 0 (Jc)x=o
Regarding (AlO), ( A l l ) and (A6), we obtain c o l a = '/2([cIb/S)
(A12)
Registry No. Br-, 24959-67-9;I-, 20461-54-5;HBr02,3769127-3; HIOp,30770-97-9;HBr03,7789-31-3;HIO,, 7782-68-5; H#04, 7664-93-9.
Tricopper. A Fluxional Molecule D. P. DILella, K. V. Taylor, and M. Moskovits' Lash Miller Chemical Laboratories, Department of Chemistry, University of Toronto, Toronto, Ontario, Canada (Received July 30, 1982)
The resonance Raman spectrum of matrix-isolated Cu3is reported. On the basis of the irregularity of the observed progression and the unusual isotopic structure shown by the vibrational spectral components, Cu3is proposed to be a fluxional Jahn-Teller molecule. A symmetric stretching frequency of 354 cm-' is found for the molecule. The UV-visible absorption spectroscopy copper cluster species isolated in rare gas matrices has been the subject of numerous papers.' Although spectral features ascrib(1) H. Huber, E. P. Kiindig, M. Moskovits, and G. A. Ozin, J . Am. Chem. Soc., 97, 2097 (1975); M. Moskovits and J. E. Hulse, J. Chem. Phys., 67, 4271 (1977).
able to aggregates consistingof as many as four atoms have been reported, the lack of vibrational structure in these has made speculation into the geometries of these species difficult. In any case such conclusions, being drawn from absorption spectra, would have been pertinent only to the excited state* Molecular orbital calculations on Cu, species have been
0022-3654/83/2087-0524$01.50/00 1983 American Chemical Society
The Journal of Physical Chem/stfy, Vol. 87, No. 3, 1983 525
Tricopper 1
16810.7
l
0
o
,
I
I
500
I
I
I
l
l
IO00
,
I
1
1
1
I
I
X
J
I500
c m-‘
Flguro 1. Resonance Raman spectrum of Cwontaining argon matrix excited wRh 16 820-cm-’ R6G laser radiation.
i\\
undertaken by several groups2and, taking for example the species Cu3, there is no general agreement regarding its shape. In this paper we report a resonance Raman study of Cu3 isolated in an Ar matrix. As regards other triatomics, the Raman (or perhaps the preresonance Raman) spectrum of Ag33and the resonance Raman spectrum of Ni; have been reported. Extensive calculations have been performed on Li3,5which was predicted to be a pseudorotating, Jahn-Teller molecule. Recent matrix ESR work suggests that K3 is, in fact, pseudorotating even at matrix temperatures (-15 K).*
Experimental Section Matrices containing Cu were prepared by evaporating the metal from a copper wire-wound tantalum filament. The metal and argon streams were cocondensed onto an aluminum paddle cooled to about 12-15 K by means of a Displex closed-cycle refrigerator. The paddle was situated so that different parts of it received different rates of metal flux, while its entire surface received approximately constant argon flow. Raman emissions were excited by means of an Ar+, Kr+, or argon-pumped R6G dye laser radiation and recorded by using a Spex double monochromator equipped with photon counting, in some cases interfaced to a Tektronix 4051 computer. Results and Discussion The resonance Raman spectrum obtained when a copper-containing argon matrix was excited with 16820-cm-’, R6G dye laser radiation is shown in Figure 1. It consists of a main progression consisting of four lines at 355, 710, 1063, and 1430 cm-l and a second, weaker progression at 404, 760, and 1115 cm-’. The irregularity of the main progression is at once noticeable; it cannot be characterized by a single anharmonicty constant a”&“,. These spectral emissions could only be excited in a narrow range (approximately 50 cm-’ in width) of frequencies centered (2) (a) P. Joyes and M. Leleyter, J.Phys. E. 6,150 (1973);(b) R. C. Baetzold and R. E. Mack, J. Chem. Phys., 62, 1513 (1975); (c) A. B. Anderson, ibid.,64,4046 (1976);(d) S. C.Richtameier, J. L. Gole, and D. A. Dixon, Proc. Natl. Acad. Sci. U.S.A.,77,5611 (1980); (e) C. W. Bauschlicher, Jr., S. P. Walch, and P. E. M. Siegbahm, J. Chem. Phys., 76,6015 (1982). (3)W. Schulze and H. V. Becker, Chem. Phys. Lett., 35,177 (1978). (4)M.Moskovita and D. P. DiLella, J. Chem. Phys., 72,2267(1980). (5)W.H.Gerber and E. Schumacher, J. Chem. Phys., 69,1692(1978); W. H. Gerber, Ph.D. Thesis, University of Bern, Bem, Switzerland, 1980.
16809.8
16809.6
16809. I
I , 16807.6 4216805.8
I 689.6
1
I
I
701.6
I
I
713.6
c m-I Figure 2. High-resolution spectra of the second component of the resonance Raman spectrum of Cu, excited with laser radiation of the Indicated frequencies.
around 16810 cm-’. Under high resolution each number of the main progression was seen to consist essentially of a doublet. Figure 2, for example, shows the second Stokes component of the main progression under high resolution as obtained with various excitation frequencies. These results point to yet another unusual feature of this spectrum which will be discussed below. The carrier of this spectrum cannot be Cu2 or Cu2X where X is an unknown impurity for the following reasons. (1)The frequency is too high for it to be a Cu-Cu bond (w”,(Cu,) = 264 cm-’). (2) Copper has two abundant isotopes, 65Cu (30.9%) and 63Cu (69.1%). The two most abundant species of a Cu2-containingmolecule would be 63Cu2Xand 63Cu65CuX.If so, the observed splitting (10.5 cm-’) between the two components of the second Stokes member of the main progression (Figure 2) is too large by about a factor of 2. If the vibrational spacing in the progression is interpreted to be the CuX vibration of the diatomic CuX or the triatomic Cu2X,then the mass of X which would produce the observed split would exceed loo0 amu! The observed split might be interpreted as being due to “Cu2X and 63Cu2X,if one assumes that the molecule 63Cu65CuXis not observed because the chosen exciting frequency coincidentally happens to be in resonance with the first two molecules but not with the third. It is for this reason that a range of exciting frequencies was used (Figure 2). That range should have covered all isotopic forms of
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The Journal of Physical Chemistty, Vol. 87, No. 3, 1983
DiLella et al.
~rY
Flgure 3. Contour map of the lowest ‘E’ surface of Cu, showing the three mlnlma and three barriers to pseudorotatlon.
TABLE I : Vibronic States of Li3a symmetry
n,= 0
na= 1
n,= 2
E’ A,
303 3 50
915 96 2
A;’ E’
416 466
E’ A,’
558 56 2
609 656 722 772 865 86 9
a n = the n u m b e r of q u a n t a of the totally symmetric vibration. Data taken from W. H. Gerber, Ph.D.Thesis.’
Cu2or Cu2X,and, while the data of Figure 2 show that one is exciting into the bound region of the excited-state potential of the carrier since one does go in and out of resonance with the two molecules producing the two components of the spectra, a central component such as should have been seen for 63Cu65CuXor 63Cu65Cuis never encountered. Consequently, we may reject the possibility that the spectrum is due to Cu2, CuX, or Cu2X. We propose that the carrier is Cu3 and that the features a t 700.0 and 710.5 cm-’ belong respectively to 63Cu3and @Cup If so, one is still hard pressed to explain the absence of a feature due to 63Cu2@Cu which should be as abundant as @Tu3. We suggest that its nonobservation as well as the irregularity in the progression are both due to the fact that Cu3 is a Jahn-Teller, fluxional molecule pseudorotating even a t matrix temperatures. To understand how this proposal accounts for the observations, we turn to the calculations of Schumacher and Gerber5 on Li3 which is predicted to exhibit the same fluxional phenomenon. In zeroth order Li3 (and Cu3) is predicted to have a 2E’ ground state of D3h symmetry. Such a molecule will be characterized by three vibrations corresponding to the three vibrational coordinates Q, of al’ symmetry and the degenerate pair (Q,, Q,) of e’ symmetry. The Jahn-Teller theorem predicts a first-order instability along the Q,, Q, surface, a surface with three equivalent minima (Figure 3). If the barriers (marked b in Figure 3) are not too high, the molecule will be fluxional, passing from minimum to minimum, never (or only infrequently) adopting the D3h configuration. The Born-Oppenheimer approximation would then not be invokable and the molecule will be characterized by a series of vibronic states of symmetries E’, A i , and Al’ (the representations spanning the direct product E’ X E’). These states will be replicated many times by including one or more quanta of the symmetric vibration. This will form a series of vibronic states of the same symmetry separated by an interval corresponding to the totally symmetric vibrational frequency, to which transitions will form something akin to a progression but a rather irregular one. So, for example, for Li3 Gerber calculates the energies shown in Table I in which one sees such a set of E’ levels (with 0, 1, and 2 quanta of the symmetric vibration) separated by a constant interval
Figure 4. Selected energy levels of C y showing the effect of isotope substitution.
(about 306 cm-l). We therefore interpret the main progression (Figure 1)as being due to a series of transitions from the electronically excited state to several vibronic states of E’ symmetry differing by a quantum of the totally symmetric vibration. The average spacing (354 cm-’) corresponds therefore to the symmetric stretching vibration of a fluxional Cu3 molecule. This still leaves the question of the absence in Figure 2 of the isotopic component due to 63Cu265Cuand 63Cu65Cu2).A molecule of the form 63Cu265Cu has C2”or C, symmetry, according to the position of the 65Cuatom. Confining ourselves to the former, for the moment, one would formally no longer have a Jahn-Teller molecule once isotope substitution takes place. By substituting a 65Cu atom in 63Cu3one causes the 2E1ground state to split into a 2B2and 2A1state in Czu. If we assume, for the moment, that the electronically excited state which is in resonance with our exciting photons is of %” symmetry in the 63Cu3 molecule (this would be one of the states formed primarily of Cu 4p orbitals), then on isotope substitution the 2E” state would divide into a 2B1and a 2A2state. The resulting splitting of the 2E’ground and 2E” excited states may be greatly in excess of what one expects for simple isotopic splitting. Hence, the 63Cu265Cu molecule may no longer be in resonance with the exciting laser photons. Moreover, if the lower states of the two sets of split states are the 2B2 for the ground and 2B1 for the excited levels, then the transition 2B1 2B2would be formally forbidden. One would have, then, the unusual situation wherein an allowed transition in 63Cu3and 65Cu3becomes formally forbidden simply through isotopic substitution. This process is illustrated in Figure 4. The secondary progression observed with this molecule we ascribe to transitions to another set of vibronic levels, perhaps also of E’ symmetry.
-
J. Phys. Chem. 1983, 87, 527-528
Several molecular orbital calculations have appeared in which the geometry of Cu3 was considered. Anderson2c finds Cu3 to be linear or close to linear. Richtsmeier et al.,2don the other hand, produce a calculation which in its essential features is in agreement with our spectroscopic results. They find Cu3 to be a Jahn-Teller, unstable molecule which opens to a molecule of C2”symmetry with an apical angle of about 66”. This form of the molecule is found to be approximately 700 cm-’ more stable than the DS form; hence, the barrier to pseudorotation may be considerably lower than this. On the other hand, the linear molecule is found to be unstable by about 1750 cm-’ with respect to the bent triatomic. (Richtsmeier et al.2ddid not consider the dynamics of Cu3but only the adiabatic surface which one obtains assuming the Born-Oppenheimer approximation; hence, a possible fluxional nature of the molecule was not addressed by them.) Those authors2d also predict the symmetric stretching frequency to be high. Although their figure, 445 cm-’, is higher than what we find, 354 cm-’, the unexpected fact that w”,,(Cu,) is greater than 3/2 times the frequency of the diatomic is predicted by those authors, that is, that the Cu-Cu force
527
constant is larger in Cu3 than in Cu2 in spite of the fact that the measured6(and calculatedM)binding energies per bond are less in the triatomic than in the diatomic. (The symmetric stretching frequency of a X3 (D3J molecule is given by (3k/m)’/2/2?r7in the simple valence force field, where k and m are the metal-metal force constant and the atomic mass. For a diatomic X2 the frequency is (2k’/ m)’l2/2?r. Hence, in the case of k k‘the ratio we-, (Cu3)/w(Cu2)would be approximately equal to (3/2)’/2. A ratio exceeding this number implies that k > k’.) Acknowledgment. We are grateful to NSERC and Imperial Oil for financial support. Discussions with Professors Ernst Schumacher and James Gole are warmly noted. Registry No. Cu3, 66771-03-7. (6) K. Hilpert and K. A. Gingerich, Ber. Bumenges. Phys. Chem., 84, 739 (1980). (7) G. Herzberg, ‘Molecular Spectra and Molecular Structure”,Vol. 11, Van Nostrand, New York, 1945. (8) G. A. Thompson and D. M. Lindsay, J. Chem. Phys., 74, 959 (1981).
COMMENTS Hydratlon Numbers by Near-Infrared Spectrophotometry. Contradictory Assumptions
Sir: Recently Hollenberg and Ifft’ reported measurement of hydration numbers by near-infrared spectrophotometry utilizing the method of Bonner and Woolsey.2 This method, because it involves contradictory assumptions, should not be regarded as valid. Problems associated with the method have been discussed el~ewhere.~This note, by spelling out assumptions that are merely suggested by Hollenberg, will indicate where unacceptable assumptions have been made. Equations provided by Hollenberg and Ifft are here identified by the same numbers that they use in ref 1. Additional equations deduced from their discussion are designated by primed numbers. The subscript “r” designates properties of pure water; “s”designates properties of any solution. The subscript “sat” will be used here to designate properties of saturated solutions that are assumed to contain no monomeric water. Four species are taken into account: monomeric water, “H20”;hydrogenbonded water, “X”, solute particles, “S”; and water of hydration, “H”. The product of molar absorptivity times path length is designated “k”. Beer’s law expressions can be written for absorbance by pure water, A,, and for absorbance by a solution, A,. (1) AI = ~ , W , O I r+ k2[XIr A , = ki[H2OIs + kz[XIs + ka[Sls + k4[HI8 (2) For solutions of interest, k3 = 0. It is assumed that k4 = 0 (2’) ~~~
(1) Hollenberg, J. L.; Ifft, J. L. J. Phys. Chem. 1982, 86,1938-41. (2) Bonner, 0.D.; Woolsey, G . B. J . Phys. Chem. 1968, 72,899-905. (3) Jayne, J. C. J. Chem. Educ. 1982,59,882-1.
Several assumptions were made in order to evaluate the various quantities in these equations. First, it was assumed that there exist very concentrated solutions, “sat”, for which [HzOIsat = 0 (2”) Next it was assumed that as the concentration of a solution changes the resulting change in concentration of species X is determined solely by the change in the volume fraction of water, f, which decreases as solute particles occupy more of the total volume of the solution. The expression used by Bonner and Woolsey2and by Hollenberg and Ifft’ is [XI, = f[XI, (2’”) (This expression does not take into account the volume occupied by water of solvation, H, and is true only if [HI = 0. This is an easily correctable error, rather than a flaw in the method.) These assumptions lead to Beer’s law expressions for very concentrated solutions: (3) Asat = k~[Xlsat= kdsat[Xlr The directly measured absorbance for a solution, Ad, is the difference between the absorbance of pure water in the reference compartment and the absorbance of a solution in a matched cell in the sample compartment. Ad = A, - As = ki([HzOlr - [H@I,) + M l - f)[XI, (4) (5) &,sat = Ar - Asat = k,[H,Olr + kZ(1 - fmJ[XIr If k1[H2O],is eliminated between eq 5 and 1 (5’) Ad@ - AI = kZ(1 - f s a t ) [ x l r - kZ[x11 The first k2 contains the molar absorptivity of species X in a very concentrated solution. The second k2 contains the molar absorptivity of species X in pure water. If the
0022-3654/83/2087-0527$01.50/00 1983 American Chemical Society
