COMMUNICATIONS TO THE EDITOR
1845
+
Wigner3 correction [l 1/24(hv/lcT)2] for small values of V , the expression derived by Yao and Zwolinski’ takes the form [I ‘ / ~ ( h v / k T ) ~corresponding ], to “tunneling factors” less than unity. The error in their treatment originates in the formulation of the Hamiltonian. They set the potential energy equal to zero at the origin and assume that the kinetic energy, as well as the potential energy, is therefore negative. They then write the Hamiltonian in the form H * = -p2/2p* - f Y 2 P (1) wheref = - f > 0. Now if H is the Hamiltonian for a regular harmonic oscillator, then Iz$ = E$ (2) ‘/2)hv. Howand the eigenvalues are given by (n ever, if we multiply (2) on both sides by - l to give
-
*
+
-fi$
=
-E,,!, = &*$
(3)
we have not thereby formulated the Schroedinger equation for the entirely different problem of a particle subject to a negative force constant. Yao and Zwolinski’s conclusion that the eigenvalues for the case of the inverted parabola are given by E* = -(n l12)hv, and their subsequent deductions from this, are therefore untenable. Actually, the correct Schroedinger equation for an inverted parabolic potential is
+
-+-
d2$ dY2
8ir2p*
h2
(E
+ 2x2p’v2q2),,!,
=
0
satisfactory state. In a recent n‘ote,’ the authors assumed that the unusual degree of freedom along the reaction coordinate is assumed to be a stationary harmonic vibration oscillation in an inverted parabolic potential. Ridley, Quickert, and Le Roy2 took exception to our treatment. For a bounded or stationary oscillation, the Virial Theorem requires that the kinetic energy and the potential energy must be of the same sign for this inverted parabolic potential. Since the potential energy for an inverted parabolic potential is negative, the kinetic energy as well as the total energy of this inverted H. 0. must be negative. The solution for the energy-reveysed H. 0. becomes straightforward and the eigenvalues are found to be E,* = - (n ‘/Z)hv, where v is real. This energy-reversed oscillator is physically realizable since it satisfies the Schroedinger equation. From the potential energy surface point of vien- and rate theory, we admit that we have erroneously introduced the negative energy oscillator into this treatment; however, OUT final expression eq 15, ref 1, although improperly derived, has certain desirable features in situations where, D # v # iu
+
*.
(1) S. 3 . Yao and B. J. Zwolinski, J . P h ~ p Chem., . 72, 373 (1968). (2) B. A . Ridley, K. A. Quickert, and D. J. Le Roy, ibid., 72, 1844 (1968).
THERMODYNAMICS RESEARCH CENTER TEXAS A&M UNIVERSITY COLLEGE STATION, TEXAS
RECEIVED RIARCH 22, 1968
(4)
where v = 1/2n (f/p*)”a. This equation can be solved if we define a new variable v* = iv whereby (4) becomes
The Primary Isotope Effect for the Replacement of
+
with complex energy eigenvalues E* = (n l/z)hvi, belonging to the Normal operator H* which is not Hermitian.
.
(3) E. P. Wigner, 2. Physik. Chem., 19B,203 (1932).
B. A. RIDLEY LASHMILLERCHEMICAL LABORATORIES UNIVERSITY OF TORONTO K. A. QUICKERT D. J. LE ROY TORONTO, ONTARIO, CANADA RECEIVED FEBRUARY 9, 1968
Reply to “ComJmenton ‘Quantum Theoretical Treatment of Equilibrium Chemical Rate Processes’ ”
Sir: From at least two points of view, namely, rate processes in condensed systems and reactions of radicals in the gas phase, the nature and magnitude of the “temperature-independent factor” is still in an un-
S. J. Yao B. J. ZWOLINSKI
N
ws.
D by Energetic Tritium Atoms
Sir: Variations in reaction yields from isotopic molecules, involving several kinds of isotope effects, have been observed for many energetic tritium atom reaction^."^ Direct measurement of the primary isotope effect for the replacement of H or D by energetic T, as in (1),6has not been reported, since all (1) H. C. Jurgeleit and R . Wolfgang, J . Amer. Chem. Soc., 8 5 , 1057 (1963). (2) E. K. C. Lee and F. S. Rowland, ibid.,85, 2907 (1963). (3) R. Wolfgang, Progr. Reaction Kinetics, 3, 97 (1965). (4) E. K. C. Lee, G. Miller, and F. S. Rowland, J . Amer. Chem. SOC., 87, 190 (1965). (5) E. K. C. Lee, J. W. Root, and F. S. Rowland, “Chemical Effects
of Nuclear Transformations,” Vol. 1, International Atomic Energy Agency, Vienna, 1965, p 5. (6) Possible primary isotope effects involving the substitution of HI, D*, and T*, which we shall designate as primary substitution isotope effects, have not yet been measured for energetic hydrogen atom species. Energetic H* or D* from nuclear recoil are some factors of 10 beyond present levels of detection. Comparisons should be feasible for photochemical sources but have not yet been made: R. M. Martin and J. E. Willard [ J . Chem. Phys., 4 0 , 3007 (1964)] have given total yields for D* f CH4 and H* CDI, representing a summation of primary substitution, primary replacement, and secondary substitution isotope effects.
+
Volume 72, Number 6 M a g 1968
1846
COMMUNICATIONS TO THE EDITOR
+ RH (D) +RT* + H (D)
T*
(1)
previous experiments have also involved simultaneous isotopic variation in R. Thus, the substitution of T-for-H us. T-for-D in CHd-CD4 mixtures involves possible secondary isotope effects from bond formation with CH3 us. CD3 in addition to the primary replacement of H V S . D;6 the existence of an appreciably higher yield of CHaT from CH3F than of CD3T from CD3F (1.21 f 0.04) illustrates that secondary isotope eflects upon energetic substitution reactions can be quite ~ u b s t a n t i a l . ~We have now performed a direct measurement of the primary replacement isotope effect for H us. D through measurements with CHX3 and CDX3 as in (2), in which the X groups are idenT*
+ X3CH (U)
--j
+ H (D)
X&T*
(2)
tical, and the only isotopic variant is the H us. D in the tertiary position. The chief experimental problem, that of normalization of the tritium flux for comparison of two molecules, is readily solved through the measurement of T* reactions with X as an intramolecular monitor for the energetic tritium atom flux. Our experiments have involved recoil tritium irradiation of various isotopic isobutanes, isolation of the tritium-labeled isobutane from other radioactive products, and then measurement of the fraction of the tritium in the tertiary position in each molecule. Compasisons of the relative yields in tertiary us. primary positions for (CH3),CH us. (CH3)3CDand for (CDJ&H vs. (CDa)sCD furnish two separate estimates of the primary replacement isotope effect. The measurements have been carried out for irradiations in both gas and liquid phases, with the results shown in Table I. While the data show a marked phase effect in all samples, all measurements show a consistent primary replacement isotope effect of -1 -25. The original specific activity of isobutane-t was calculated from the consecutive measurement of the radioactivity and thermal conductivity response of Table I : Primary Replacement Isotope Effect for Recoil Tritium Reactions with Isotopic Isobutanes
_--Gas phasea---RC
Compound
---Liquid
phaseb--
Isotope
Isotope effed
R
eff eot
(CH3)aCH (CH8)sCD
0.87 0 69
1 . 2 6 3= 0.04
o,81 0'75
1 1 . 2 3 f 0.06
(CD3)sCH
1.20 0.97,O. 95
1 . 2 5 i 0.04
o.78 0"9
1 1 . 2 7 =t0.06
(CDs)aCD
I
a Sample contents: 2 cm of aHe, 6 ern of 02,70 cm of isobutane. Sample contents: LiF, 10 mole % butadiene, isobutane. R = (tritium activity per tertiary C-H or C-D bond)/(tritium activity per primary C-H or C-D bond); estimated error: &2% (gas), -13% (liquid). Isotope effect = Rtert B/Rtert. D for molecules with the same isotopic content in t h e primary position.
T h e Journal of Physical Chemistry
the isobutane peak from a radio gas ~hromatograph.~ The tertiary-T was similarly measured after complete exchange of primary-?' by washing the isobutane with sulfuric acid.*v9 Separate tests established that tertiary-" exchange was negligible under conditions permitting complete exchange (>99.9%) of the primary-?' in isobutane.10-12 The smaller inass of the H atom furnishes less inertial resistance to the oncoming energetic T atom than does the D atom, and the H atom can be set into more rapid motion in the initial stages of collision. (A free H atom struck by an energetic tritium atom would recoil with 25% greater velocity and 22y0 less kinetic energy than for the corresponding collision of free D with an energetic T atom.) While the final energy and velocity of an atom freed in a substitution reaction will depend upon the detailed interactioiis during the entire reaction, the primary replacement isotope effect probably reflects chiefly the initial eaSe of displacement of the H and D atoms. Our measurements are most valid for correlation with hydrogen replacement isotope effects for other systems with three heavy substituents on the central carbon atom, and may not apply directly to molecules such as methane for which rapid relaxation of additional substituents may be intimately involved in the detailed reaction. The overall substitution isotope effect in CHJ us. CDI (1.33 =k 0.04)6 presumably includes an additional contribution from the relatively greater rate of relaxation of CH3 than for CD3.5 The comparison of the relative activities of the tertiary and primary positions of isobutane has been used earlier as a test of the possible steric effect of methyl substituents upon the substitution rea c t i ~ n . ~ - ~The l , l ~phase differences observed in our experiments make it difficult to estimate the magnitude of such substituent effects without a precise explanation of the origin of the phase differences themselves. Alternate postulates, without clear preference, can be suggested as: (a) the gas-phase results represent a (7) J. K. Lee, E. K. C. Lee, B. Musgrave, Y.-N. Tang, J. TV. Root, and F. S. Rowland, A n a l . Chem., 34, 741 (1962). (8) J. W. Otvos, D. P. Stevenson, C. D. Wagner, and 0. Beeck, J . Arner. Chem. Soc., 73,5741 (1951). (9) A. Odel, A. Rosenberg, R. Fink, and R. Wolfgang, J . Chern. Phys., 40, 3730 (1964). (10) Only small variations were found in the exchange rates for different isotopic isobutanes, and these caused no difficulties in the exchange of primary-T without appreciable loss of tertiary-T. (11) A. Rosenberg, Ph.D. Thesis, Yale University, 1964. (12) Attempts t o use these techniques for study of the intramolecular distribution of tritium in isobutane-t from the reactions of 2.8 eV T* + (CH8)sCH have been unsuccessful so far because of a rapid bromine-photocatalyzed exchange of TBr with the tertiary position of isobutane. (13) The energy deposition following T-for-H in c-CaHs is essentially the same as that following T-for-D in c-CaDs. (Unpublished experiments by A . Hosaka and F. S. Rowland.) (14) The value for tertiary/primary in i-C4H10 of 0.79 0.04 (gas phase) reported in ref 9 is in semiquantitative agreement with the corresponding value of 0.87 f 0.02 in Table I, although outside the limits of error of each.
*
COMAIUSICATIOXS TO THE EDITOR
1847
perturbation of the more-correct liquid-phase results by isotope effects in rates of secondary decomposition reactions; or (b) the liquid phase results indicate a condensed phase limitation upon the motions necessary for completion of some substitution reactions in the tertiary position. I n any event, the primary replacement isotope effect is independent of phase, and thereby of thc choice between these alternate mechanisms.
of a bidimensional condensation in the nth layer, me write
and
(15) This research has been supported by AEC Contract No. AT(11-1)-34, Agreement N o . 126.
With an appropriate thermodynamic definition of the adsorbed layer, utn) and constitute approximate THOMAS SM.41L DEP LRTMENT O F CHEhllSTRY15 values of the molar energy and entropy of the outer F. S. ROWLAND UKIVERSITY OF CALIFORNIA condensed layer; u(")and s(") represent the same I R V ~ NCALIFORNIA E, 92664 quantities relative to the outer layer of the solid adRECEIVED FEBRUARY 20, 1968 sorbat e. Formula 1 differs from that of Singleton and Halsey2r3by the addition of the entropic term -1/R(s(") - de)), I n this preliminary report on our experimental Bidimensional Condensation n-ork, we wish to sholy the importance of this term. in Adsorbed Layers We have studied the adsorption of lirypton and xenon on a certain number of halides having a layerSi?: I n a paper to appear,l we have derived formulas like structure, namely SiClz, CoC12, FeC12, CdC12, expressing the logarithm of the ratio of the vapor presCdBr,, CdI,, and PbI,. The results reported here sure of the adsorbate P ( - ) to the transition pressure in concern only the condensation of the first layer of an adsorbed film P ( n )and its first derivative as a funckrypton on the basal face of these substrates: typical tion of the inverse absolute temperature. I n the case isotherms for nickel chloride are plotted in Figure 1. The coefficients of the regression lines (eq 1) are detwmined by least-squares analysis. Thus the ratio P(m)/P(ljcan be split into its energetic and entropic CI 71. contributions. The results thus obtained are given in Table I, which also contains the crystallographic parameter a of the hexagonal or pseudo-hexagonal lattices of the adsorbents and, for comparison, the diameter of the krypton atom (distance between two krypton atoms in the (111) plane of the fcc solid krypton). From these data two conclusions can be drawn. First, the entropic term of formula 1 may be as im(1) Y. Larher, J . Chim. Phys., in press. (2) J. H. Singleton and G. D. Halsey, Can. J . Chem., 33, 184 (1955). (3) L. J. Slutsky and G. D. Halsey in "Physical Chemistry, An
Pressurr ( t o r r )
Figure 1.
Adsorption isotherms of krypton on nickel chloride.
Advanced Treatise," E. H. Eyring, D. Anderson, and 117, Jost, Ed,, Academic Press, New York, N. Y., 1966.
Table I : Respective Contributions of Energy and Entropy to the Ratio P ( m ) / P ( of l ) the Vapor Pressure of the Adsorbate to the Condensation Pressure of the First Layer of Krypton on Different Adsorbents
Adsorbent
KiClz COClp
FeClz CdClz CdBrz CdIz PbII
pW
Temp, OK
p 0
75.69 75.34 75.13 82.52 88.44 88.35 88.52
26.9 22.4 19.4 21.3 35.7 53.1 38.2
Energetic contribution to P@)/P(l)
5.13 4.78 4.85 9.20 26.9 41..8 12.45
Entropic contribution to P ) / N
a of t h e adsorbent,
Diameter of the krypton atom, A
5.25 4.68 4.00 2.32 1,325 1.27 3.07
3.543 3.544 3,579 3,854 3.95 4.24 4.555
4.07
A
volume 7% .\'umber
6
May 1968
