CO~WKJSICATIONS TO THE EDITOR
1844 Table I: The Estimation of 0°K compressibility ( P O )from Eq I
Liquid
Benzene” n-Octaneb Carbon tetrach 1ori de‘ Acetoned
Temp, QC
(d In Pi/dOt,
(daldT) V I
PO, cm2/dyne
PO, cm2/dynea
OK-1
OK-2
(this paper)
(ref I )
41 45 38.4
-0 52 x 10-3 -0.93 X - 0 . 4 6 x 10-3
4.4 5.9 7.5
10-8 10-8 10-6
9
32.5
-0.36 X
1 . 6 x 10-6
15
x x x
x x 3.2 x 8
10-12 10-l2 1O-l2
X
4.6
x
pa, cm%/dyne (adiabatic) (ref 4 )
11.9
x
10-12
14.9 X 8.3
x
12.7
x
10-l2
a Data from R. E . Gibson and J. F. Kincaid, J . Amer. Chem. Soc., 60, 513 (1938). The Tait equation with their constants waa used for interpolation where necessary. Data of H. ,If. Eduljee, D . M. Newitt, and K. E. Weale, J . Chem. Soc., 3086 (1951). Tait equation used as above. Data from R. E. Gibson and 0. H. Loeffler, J . Amer. Chem. Soc., 63, 898 (1941) (as above). Data from International Critical Tables, T’ol. 111, interpolated by computed cubic equation. Difficulty was found in fitting these results either to a Tait or Huddleston equation.
temperature. The results obtained are given in Table I. The results obtained here may be conveniently compared with those given in ref 1 and by Kudryavtsev and Samgina14whose modulus is, however, adiabatic and not isothermal. I n view of the several approximations and of the rather crude model used, the agreement between the results seems reasonable. Differences from the results of Kudryavtsev cannot be due to their use of an adiabatic measurement since they generally obtained higher values of Po. In the case of acetone and octane the lower value given by the van der Waals equation1 is probably due to use of high pressure (5000-7000 atm) at the point where the equation is applied to determine Po. (This is calculated from the intercept where V = Vo.) Bridgman’s measurements at very high pressures (10,000-40,000 atm)5 certainly indicate that PO must be nonlinear with pressure over a wide range. Recently, studies of the effect of pressure on viscosity have also led to proposals that the molar volume at 0°K has a finite ~ompressibility.~~7 Hogenboom, et al.,’ have proposed that V ofor ann-C(15) alkane should be assigned a compressibility equivalent to that of the solid near the melting point, Le., PO 25 X 10l2. This value is higher than any proposed here, but the coinpressibility of the solid may well be above that of the smaller molar volume at 0°K since it does contain some free space. Nevertheless, the higher figure agrees with the requirements of the viscosity theories. However, the fact that a value of ,Boof the right order can be obtained by this method does suggest that the approach used may have some validity. Physically speaking, it appears that U is not a specific function of V . As the temperature rises a t constant volume, so does the total pressure on the molecules. I n this way the occupied volume is reduced and some eIastic energy is stored.
Universita, Genova, Italy, for his assist’ancein carrying out this work.
Acknowledgment. The authors wish to thank Dr. A. Turturro of the Institute di Chimica Industriale,
(2) D. J. Le Roy, B. A. Ridley, and K. A. Quickert, Faraday Society Discussion on the Molecular Dynamics of the Chemical Reactions of Gases, Toronto, 1967.
T h e Journal of Physical Chemistry
(4) B. B. Kudryavtsev and G. A. Samgina, Russ. J . P h y s . Chem., 39, 478 (1985). ( 5 ) P. W. Bridgman, Pioc. Am. Acad. Arts Sci., 76, 72 (1948). (6) 8 . J. Matheson, J . Chem. Phys., 44, 895 (1966). (7) D. L. Hogenboom, (1987).
W.Webb, and A. J. Dixon, ibid.,
46, 2586
DEPARTMENT OF POLYMER A N D FIBRE SCIEXCE R. N. H.\WARD THEUNIVERSITY OF NANCHESTER INSTITUTEB. 11.PARKER OF SCIENCE A N D TECHNOLOGY MANCHESTER I, ENGLASD
RECEIVED FEBRUARY 7 , 1968
Comment on “Quantum Theoretical Treatment
of Equilibrium Chemical Rate Processes”
Sir: I n a recent Kote under the above title, Yao and Zwolinski‘ proposed a modified formulation of transition state theory in which the usual kT/h term in the expression for the rate constant is replaced by ~ e - ~ ~ / ~ ’ ‘ ~ / (1 - e-hv’kT), equivalent to multiplying it by the factor (hn/2kT)/sinh(hv/2kT), in which i; is the frequency of passage across the barrier and v is the absolute magnitude of the imaginary frequency i v of a harmonic oscillator corresponding to an inverted parabolic potential along the reaction path. Their approach is similar in some respects to that used by us2 in which the kT/h term is multiplied by the factor (hv/2kT)/sin (hv/2kT). I n both cases the correction factor derives from the extraction from the statistical mechanical expression for the equilibrium constant of a partition function Qso* corresponding to motion in the reaction coordinate. However, whereas our expression2 gives the (1) S. J. Yao and B. J. Zwolinski, J . P h y s . Chem., 72, 373 (1968).
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
