NOTES
energy electron-beam technique. Their curve for CaH4shows a hint of a discontinuity near 6 ev that could indicate an unresolved structure. It might be expected that absorption of the 2062-A line could result in both excitation and dissociation. In the photolysis of acetylene both excited-molecule and free-radical chain mechanisms have been suggested to account for the observed products, and similar reaction schemes could be proposed for photolysis of methylacetylene. In addition to the gas phase reactions, wall reactions certainly occur and may play the major role in the formation of the observed products. The process of polymer formation obviously does not occur uniquely in the gas phase since it does not result in a solid precipitating to the bottom of the vessel. Instead, the polymer is observed on the inner wall and may continue to polymerize after condensation. In view of these considerations any more detailed speculation into the mechanism is probably unwarranted at this time. Acknowledgment. A. G. wishes to thank NATO for a fellowship and Rensselaer Polytechnic Institute for hospitality. We also wish to thank Dr. B. A. Thompson for her assistance in analyzing some of the results and in preparation of the manuscript. This work was carried out in part under a research grant from the National Aeronautics and Space Administration.
2721
and semiclassical methods and with the methods of Vestal, et ~ 1 . ~Lin, 3 et u Z . , ~ Haarhoffls and Thiele,6 as applied to the molecules of acetylene, cyclopropane, and t-butyl chloride. The computer program for exact counting was so designed that the number of states, Wexsot(E), up to a maximum energy was counted and simultaneously the numbers of states falling within energy intervals AE, starting at 0 ev and ending at the maximum energy, were obtained. An empirical formula, which had been previously propo~ed,~ was used for calculating the coefficients, up, in the formulation of Vestal, et uLla for the cases of cyclopropane and t-butyl chloride. In the case of acetylene, all rP values were calculated exactly. The quantities ( Y ) in the expression of Lin, et al., were chosen to be 200, 150, and 112 cm-l for acetylene, cyclopropane, and t-butyl chloride, respectively, which will make Y ~ , / ( Y ) values close to integers. The determination of the root for 8 in the expression was successfully accomplished by using
5'T?..
A Numerical Comparison of Methods for Computing Statistical Complexions
of Harmonic Oscillators -6
by James C. Tou The Dow Chemical Company, Chemical Physics Research Laboratory, Midland, Michigan (Received December 9, 1966)
To find a. correct expression for calculating the number of states, W ( E ) , for a collection of harmonic oscillators is one of the most important prerequisites to the computation of the rate constant112 of an isolated system. A number of approaches to finding a continuous function for discrete energy level has been proposed in the past few Thiele's formulation has been showns to correlate the first few terms with those given by Schlag and Sandsmark' and by Whitten and Rabinovitch.8 In this study, a numerical comparison is made of exact counting with the classical
-
(ev)
-
Figure 1. W(E)/WeXact(E)E plot for cyclopropane ( E . = 2.21 ev). Frequencies 749 (2), 878 (3), 1118 (7), 1478 (3), 8221 (6) cm-l, grouped by Schlag and Sandsmark:' dots, calculated points.
(1) R. A. Marcus and 0. K. Rice, J . Phys. Colloid Chem., 55, 894 (1951). (2) H. M. Rosenstock, M. B. Wallenstein, A. L. Wahrhaftig, and H. Eyring, Proc. Nail. Acad. Sci. U.S.,38, 664 (1952). (3) M. Vestal, A. L. Wahrhaftig, and W. H. Johnston, J . Chem. Phys., 37, 1276 (1962). (4) 9. H. Lin and H. Eyring, ibid., 43, 2153 (1965). (6) (a) P. C. Haarhoff, Mol. Phys., 6,337 (1963); (b) P.C.Haarhoff, ibid., 7, 101 (1963). (6) E. Thiele, J . Chem. Phys., 39, 3258 (1963). (7) E. W. Schlag and R. A. Sandsmark, ibid., 37, 168 (1962). (8) G . Z. Whitten and B. 9. Rabinovitch, ibid., 38, 2466 (1963). (9) J. C.Tou, L. P. Hills, and A. L. Wahrhaftig, ibid.. 45, 2129 (1966).
Volume 71, Number 8 July 1987
NOTES
2722
Table I: Comparison of the Calculated Values and the Exact Values of W ( E )for Acetylene (Ez = 0.702 ev)" Classical
Semiclassical
Vestal
Lin
0 1.006 x lo-' 1.287 x 10-3 2.199 x 10-2 1.648 x 10-1 2.815 2.109 x 10 1.006 X 10' 3.603 X 102 2.699 X lo3 1.287 X l o 4
8.457 2.148 X 10 4.889 X 10 1.021 x 104 1.986 X 104 6.382 x 102 1.735 X lo3 4.162 X 10' 9.058 X 103 3.446 X lo4 1.058 X lo6
1 6.667 2.502 X 10 6.862 X 10 1.594 X lo* 6.324 X lo2 1.928 X 10' 4.944 x 10* 1.12 x 104 4.422 X lo4 1.372 X 10'
4.373 1.673 X 10 3.654 X 10 1.085 X lo2 4.317 X lo2 1.322 X lo3 3.415 X lo3 7.808 X 10' 3.152 X lo4 9.992 X lo6
Energy,
Exact
ev
0.0 0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2 1.6 2.0
1 5 15 36 94 390 1.198 X 3.163 X 7.333 x 2.969 X 9.521 X
lo3 lo3 103 lo4 lo4
*..
Haarhoff
Thiele
3.774 x 10-1 3.739 1.437 X 10 4.041 X 10 9.547 x 10 3.882 X lo2 1.208 X loa 3.157 X loJ 7.281 X lo3 2.976 X lo4 9.518 X lo4
3.774 x 10-1 3.739 1.437 X 10 4.041 X 10 9.547 x 10 3.882 X lo* 1.208 X 10' 3.157 X 10' 7.281 X 10' 2.976 X lo4 9.518 X lo4
Frequencies 612 (2), 729 (2), 1974,3287, 3374 grouped by P. C. Haarhoff, Mol. Phgs., 8,49 (1964).
Table 11: Comparison of the Calculated Values and the Exact Values of " ( E ) for &Butyl Chloride ( E , = 3.24 evY Semiclassical
Energy,
ev
0.0 0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2 1.6 2.0
Exact
Classical
0
1
38 725 8.944 X 7.506 X 2.767 X 5.975 x 8.726 X 9.673 x 6.616 X 2.562 X
-0
9.227 x 2.015 X 6.341 X 1.385 X 4.357 x 1.343 X 9.517 x 2.994 X 9.227 x
lo3
lo4 10' 107 lo8 loQ 10" 1013
10-3' 10-ao 10-19 10-15
10-11 10-0 10-4
10-1
3.395 x 1.012 x 2.924 X 8.191 X 2.23 X 1.526 X 9.469 X 5.281 X 2.823 X 6.28 X 1.092 X
107 108 lo8 lo8 100 10'0 1010 1011 1012 1018 101'
Vestal
1 9.617 X 1.494 X 1.611 X 1.327 X 5.146 X 1.185 X 1.889 X 2.279 X 1.787 X 7.652 X
Lin
Haarhoff
...
10 lo3
lo4 lo6 10' lo8
loo 1Olo lo1* lOl3
4.959 x 9.752 X 1.898 X 8.935 X 3.191 x 6.639 X 9.584 X 1.057 X 7.148 X 2.740 x
10 10' lo4
lo4
10' 10' los 1O1O 10" 1013
2.186 X 4.967 X 1.100 X 2.381 X 5.036 X 2.117 X 8.251 X 3.003 X 1.028 X 1.027 X 8.652 X
los loo 1Olo 1Olo 1O1O 10" 10" 10l2 lOl3
1014 10"
Thiele
1.666 2.081 1.472 1.134 8.246 2.846 5.964 8.694 9.671 6.624 2.563
X X X X X X X X X X X
10 lo2 lo3 lo4
lo4
10' lo7 lo8 lo* 10" lOI3
Frequencies 339 (7), 523 (2), 818 (l), 1066 (€9,1431 (9), 2962 (9) cm-* are grouped near the arithematic mean of those given by
J. C. Evans and G. Y - S . Lo, J . Am. Chem. Soc., 88, 2118 (1966).
Golden's sectional search method.I0 The more exact form of Haarhoff's f ~ r m u l a t i o n ,a~ ~five-term power series in E,/(E Ez)2,was used in the calculation of W ( E ) . In order to simplify the computer evaluation of W ( E )in Thiele's equation,6 we write in the same notation as Thiele's
+
~
~
(
=8
y) ~ ~ v 1 ( 3 )
and
.II 8 [-$1/2h~t)8.D,,] 1
DVl@) = zs;=v
2=1
A t , sYz
= Zs;=u
$1.
i=l
The quantities DU'@)were calculated by arranging the Af,8ip,as an s X [(s/2) 11 matrix when s was even. If s is odd, the matrix is s X 4 2 , with the last column Af,(3-l,z).Then, each term in the first row was combined with each term in the second row, the products corresponding to each value to (a s2)/2 summed and a new matrix formed by deleting the first two rows
+
+
The Journal of Phyaical Chemistrql
and substituting as a new first row these sums ordered by the value of (SI s2)/2. This procedure was continued until all the elements had been used. The final matrix has the dimension 1 X [(s/2) 11 with each element corresponding to a Dy'(*). ALGOL B-5500computer programs were written for all the calculations. The calculated results of the number of states for acetylene and t-butyl chloride with energy 0-2 ev are tabulated in Table I and Table 11, respectively, and those for cyclopropane with energy 0-2 ev are depicted as the ratios of W ( E )to Wexaot(E) in Figure l. In order to show the general feature of each method without excessive use of computer time, the ratios of W ( E ) for the continuous approximations to W,,(E) of cyclopropane over the energy range 0-10 ev are
+
+
(10) D. J. Wilde, "Optimum Seeking Method," Prentice-Hall, Inc., Englewood Cliff8, N. J., 1964, Chapter 2.
NOTES
2723
T
tion of W ( E ) and the need to include the high-order Ez) if E is not very large as comterms of E,/(E pared with E,.
+
1.
Acknowledgment. The author wishes to express his sincere gratitude to Drs. A. L. Wahrhaftig and S. H. Lin for many suggestions and to Dr. C. W. Lee and Mr. D. W. Liou for discussions on computer programming. (11) E. W. Schlag, R. A. Sandsmark, and W. G. Valance, J. Phys. Chem., 69, 1431 (1965). 0.0 1.0 a.o 3.0 4.0 5 . 0 8.0 7.0 8.0 8.0 10.0
-
-'E
(ev)-+
Figure 2. W ( E ) /W,,(E) E plot for cyclopropane (E, = 2.21 ev). Frequencies 749 (2), 878 (3), 1118 (7), 1478 (3), 3221 (6) cm-1, grouped by Schlag and Sandsmark.' A, WeX&3)/W8@) calculated by Schlag and Sandsmark'.
plotted in Figure 2 and the values of the ratios of Wexact(E) as obtained by Schlag and Sandsmark? are marked with solid triangular points. From the data shown in this paper, it is apparent that the calculated values of W ( E ) by the Vestal, equation are always larger than the exact values except a t the lower energy bound ( E = 0 ev). This may be due to the approximation used in the derivation and the curve-fitted up value. The equation of Lin, et al., gave the results which are close to the exact values. It is obvious that Thiele's equation gave almost the same results as the exact values even in the low-energy region. The accuracy of the equation was also reported by Schlag, Sandsmark, and Valance." The equations of both Haarhoff and Thiele gave exactly the same results in the case of acetylene. Here, there are only four terms which need to be considered in both methods. It was found that the first four terms = 0, 2, 4, 6) in the two expressions are identical with each other. If a11 the rest of the terms are also equal, which they should be since the two methods involve the same basic approximation, then the two expressions are exactly identical. In the methods of Haarh ~ f f of , ~ Thiele,E and of Schlag and Sandsmark? the ratio E , / ( E EZ)is important. The higher the excitation energy E the less important the high-power Ez). In Figure 2, the coincidence terms of E z / ( E of the curves of' Haarhoff and Thiele above -2 ev shows the unimportance of the high-power terms (v > 8) of E z / ( E E,) when this quantity is less than one-half. It is shown in Table I1 that the values calculated by Haarhoff's equation give much larger values of W(E)/W,,,,,(E) for t-butyl chloride (E, = 3.24 ev) than for cyclopropane (E, = 2.21 ev), but that Thiele's equation gives excellent results in both cases. This indicates the importance of E, on the calcula(IJ
+
+
+
Infrared Spectroscopic Studies on Hydrogen Bonding between Alcohols and Ethers
by Izumi Motoyama and Charles H. Jarboel Department of Pharmacology, University of Louisville, Louisville, Kentucky (Received December 19, 1966)
Hydrogen bonding between various proton donors and ethers is well known and several ether-alcohol reactions have been studied in some detai1.2s3 Despite the volume of infrared spectroscopic work on equilibrium constants and shifts of hydroxyl group frequencies, there have been few reports dealing with the thermodynamic properties of such systems. In particular, systematic study on the thermodynamic consequences of progressive structural changes in the ether and alcohol components is lacking. We are interested in this aspect of hydrogen bonding because of its importance to drug action4 and we have examined a number of ether-alcohol systems with the objective of determining how structure affects the over-all reaction. The alcohols used were methyl, ethyl, n-propyl, isopropyl, n-butyl, isobutyl, t-butyl, and neopentyl. Ethyl and isopropyl ethers were used as the hydrogenbonding bases. This selection of substances permitted studying the synthesis of shielding and inductive effects as result from methyl substitution in both donor and acceptor. This work was carried out at concentrations prohibiting alcohol self-association and over the temperature range 21.7-48.1'. (1) To whom inquiries should be directed.
(2) (a) E. D.Becker, Spectrochim. Acta, 17, 436 (1961); (b) L. J. Bellamy, G . Eglinton, and J. F. Morman, J. Chem. SOC.,4762 (1961); ( c ) B.B. Bhowink and S. Basu, Trans. Faraday Soc., 59,813 (1963); (d) J. H.Walkup, J. Lyford, G . Marquardt, and G . W. Robinson, Trans. Kentucky Acad. Sci., 24, 101 (19G3). (3) (a) D.L. Powell and R. West, Spectrochim. Acta, 20, 983 (1964); (b) R.West, et al., J. Am. Chem. SOC.,8 6 , 3227 (1964). (4) T.Kitao and C. H. Jarboe, J. Ore. Chem., in press.
Volume 71 Number 8 July 1967 I
