Research: Science and Education edited by
Advanced Chemistry Classroom and Laboratory
Joseph J. BelBruno Dartmouth College Hanover, NH 03755
Determination of the Rotational Barrier in Ethane by Vibrational Spectroscopy and Statistical Thermodynamics Gianfranco Ercolani Dipartimento di Scienze e Tecnologie Chimiche, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy;
[email protected] In introductory organic chemistry courses, students are taught that in ethane, because of the interactions between the hydrogen atoms on each carbon, the internal rotation is not free, but restricted or hindered by a potential energy barrier of about 12 kJ mol᎑1. Considerable activity has focused on understanding the physical origin of this barrier (1–10), and some articles have also been reported in this Journal (11– 13). In spite of the efforts, however, the question is still the object of debates (14, 15). Traditionally the barrier has been attributed to Pauli exchange repulsions, or steric hindrance, between the vicinal C⫺H bonds, but recently it has been suggested that the dominant factor is stabilization of the staggered conformation by hyperconjugation (9, 10, 15). At last the question has been settled (for the time being) by admitting that both effects significantly contribute to the barrier with a dominance, however, of steric hindrance (16). Apart from the rather complex question of the physical origin of the barrier, more elementary issues, such as the way it was discovered and measured, are practically ignored by current physical chemistry texts. In a previous article in this Journal (17), we have introduced a numerical method, namely, the finite-difference boundary-value method, for the solution of the one-dimensional Schrödinger equation and illustrated its application to the evaluation of energy levels and wave functions for hindered internal rotations. As an ideal continuation, here we wish to illustrate how the previously developed concepts and algorithms can be used to determine, in combination with vibrational spectroscopy and statistical thermodynamics, the torsional potential in ethane. We believe that the previous and the present article, taken together, beside illustrating an important, yet neglected, topic in physical chemistry, should stimulate students to become more familiar with computational methods and encourage them to freely use available databases of molecular properties. The present article is addressed to teachers and students at the upper level associated with physical chemistry, but also a less specialized audience should find the results of interest. Barrier Height from the Torsional Frequency of Ethane The ethane molecule, composed of 8 atoms has 18 (= 3N − 6) normal modes of vibrations. In fact one of these corresponds to the internal rotation of one methyl group relative to the other and thus is better described as a torsional mode. The vibrational frequencies of ethane have been extensively investigated by analyses of infrared and Raman spectra. The torsional mode, however, belonging to the A1u species under the assumption of D3d overall symmetry, is inactive www.JCE.DivCHED.org
•
both in the infrared and Raman spectrum. It is known, however, that selection rules often break down as the result of intermolecular interactions or interaction among vibrations, causing forbidden transitions to become possible. Based on this fact, Weiss and Leroi, by recording the infrared spectrum of ethane under high-pressure path-length conditions, directly observed for the first time, in 1968, the fundamental of the torsional mode at 289 cm᎑1 (18). The complete vibrational assignments for ethane (Table 1) have been reviewed by Shimanouchi and made available on the NIST Web site (19). The torsional wavenumber ν4 can be used to obtain the barrier height of internal rotation in ethane with high accuracy as follows. The potential energy, PE, for the internal rotation being a periodic function can be expanded in a Fourier series that, as a first approximation, can be truncated after the second term, yielding,
PE =
PE bar 1 − cos ( σ int ϕ) 2
(1)
where PEbar is the height of the barrier, σint is the internal symmetry number defined as the number of indistinguishable but nonidentical positions into which the molecule can be turned by internal rotation (σint = 3 for ethane), and ϕ is the torsion angle, chosen so that ϕ = 0 corresponds to a minimum (1). The curve described by eq 1 is reported in Figure 3 of our previous work (17) and in virtually all organic chemistry textbooks.
Table 1. Vibrational Frequencies of Ethane Mode No. a
ν1
ν2a ν3a ν4a ν5a ν6a ν7b ν8a ν9a ν10a ν11a ν12a
Symmetry Species
Wavenumber (cm᎑1)
A1g
2954
A1g
1388
A1g
0995
A1u
0289
A2u
2896
A2u
1379
Eg
2969
Eg
1468
Eg
1190
Eu
2985
Eu
1469
Eu
0822
a Torsional mode. bNormal modes belonging to symmetry species E are doubly degenerate.
Vol. 82 No. 11 November 2005
•
Journal of Chemical Education
1703
Research: Science and Education
Equation 1 can be further expanded in Maclaurin series truncated after the second-order term,
PE ≈
PE bar σ int2 ϕ 2 4
(2)
so as to approximate the region of the minimum to a parabola. With this potential, the Schrödinger wave function is that of a simple harmonic oscillator with a force constant of PEbarσint2兾2. Accordingly the energy levels are given by Ev = (v +
2
) hc ν
ν =
σ int 2π c
1
v = 0, 1, 2, …
(3)
PEbar 2 Ir
(4)
with
where ∼ ν is the wavenumber of the oscillator in cm᎑1, PEbaris the barrier height in J per molecule, Ir is the reduced moment of inertia in kg m2, and c is the speed of light in cm s᎑1. The reduced moment of inertia of ethane,1 calculated by simple geometrical considerations (17), is 2.6129 × 10᎑47 kg m2. Since the selection rule for the excitation of the harmonic oscillator is ∆v = ±1, the absorption wavenumber of the torsional mode, ν4, coincides with the wavenumber of the oscillator, thus from eq 4 the value of PEbar can be estimated as 1.721× 10᎑20 J (or 10.4 kJ mol᎑1). To obtain a more accurate value of the barrier height, the energy levels for the hindered internal rotation must be evaluated by solving the Schrödinger equation shown in eq 5, for a number of PEbarvalues near the value estimated above: −
h 2 d2 ψ PE bar + 1 − cos ( σ int ϕ) ψ = E ψ (5) 2 Ir dϕ2 2
Accordingly, eq 5 has been numerically integrated by the method outlined in our previous work (17), subdividing the
Figure 1. Plot of the calculated fundamental torsional wavenumber, ν4 (0 → 1), as a function of the barrier height, PEbar. The calculated points (filled circles) are joined by a spline curve. Interpolation of the curve at the experimental value, ν4 = 289 cm᎑1, is shown by the empty circle and the dashed lines.
1704
Journal of Chemical Education
•
interval ᎑π ≤ ϕ ≤ π in 1440 mesh points, for PEbarvalues in the range 9 ≤ PEbar ≤ 14 kJ mol᎑1. Both the energy levels corresponding to v = 0 and v = 1 are practically threefold degenerate (17)2; the energy difference between these two levels yields the fundamental torsional wavenumber ν4 (0 → 1). A plot of the calculated ν4 (0 → 1) as a function of the barrier height, PEbar is shown in Figure 1. By interpolating the curve at the experimental value ν4 = 289 cm᎑1, a value of PEbar = 12.35 kJ mol᎑1 is obtained. Solving eq 5 for PEbar= 12.35 kJ mol᎑1 allows the calculation of the transitions ν4 (1 → 2) = 255 cm᎑1 for A and ν4(1 → 2) = 259 cm᎑1for E.3 These are in very good agreement with the experimental values of 255 cm᎑1 and 258 cm᎑1, respectively (18 ), thus proving that the simple cosine function (eq 1) provides a good description of the potential energy curve for internal rotation in ethane. Barrier Height from Heat Capacity of Ethane Thermodynamic functions may be derived for an ideal gas by using statistical mechanical models with spectroscopic data. These functions are usually more accurate than the corresponding values obtained by direct evaluation using thermodynamic experiments, at least for simple molecules such as ethane. However, since a reliable experimental torsional frequency for ethane has not been available until 1968, such calculations were not feasible before that year. Here we wish to illustrate the statistical thermodynamic approach that led in the 1930s to the discovery and evaluation of the rotational barrier of ethane. The notion of a hindered rotation in ethane may have been proposed for the first time by Ebert in 1929 (22) and then Wagner in 1931 (23). Nielsen, in 1932 (24), and successively Teller and Weigert, in 1933 (25), presented solutions for the quantum mechanical problem of internal rotation. Eucken and Parts, in 1933, found that their heat capacity data, owing to uncertainties about the vibrational frequencies, may be fitted either by the assumption of free rotation or with a torsional vibration of 298 cm᎑1 (26). Using the calculations of Teller and Weigert, Eucken and Weigert, in 1933, found that the best fit is obtained with a barrier of 1.3 kJ mol᎑1 (27). Although the discovery of hindered rotation in ethane is often credited to Kemp and Pitzer (1936), actually their principal merit is to have provided the first realistic value of the barrier height (PEbar = 13.18 kJ mol᎑1) from entropy and heat capacity data, thus definitely ruling out the commonly held belief that the internal rotation is free (or nearly free) (28). Successively Pitzer reported an even more accurate value (PEbar = 12.03 ± 0.52 kJ mol᎑1) (1). Molar heat capacities at constant pressure of gaseous ethane at various temperatures have been determined by many investigators. Selected experimental values corrected to the state of ideal gas, reported for convenience in the second column of Table 2, are from the NIST Web site (29).4 In the rigid rotator-harmonic oscillator-internal rotator approximation the heat capacity Cp0is given by a sum of contributions detailed in eq 6 (30): Cp 0 = CV ( trans ) + CV ( rot ) +
Vol. 82 No. 11 November 2005
∑ CV ( vib )i i
•
+ CV ( tor ) + R (6)
www.JCE.DivCHED.org
Research: Science and Education
The first four terms are the translational, rotational, vibrational, and torsional contributions, respectively, to heat capacity, while the last term is the molar gas constant that accounts for the change in external energy, pV, with temperature. For a nonlinear polyatomic molecule both the translational and rotational contributions are, to a good approximation, equal to (3兾2)R, in accordance with the principle of equipartition of energy, which states that for each squared term in the expression for the energy of the molecule, there is a contribution RT兾2 to the molar internal energy and R兾2 to the molar heat capacity. The vibrational contributions, in contrast, do not follow this principle at ordinary temperatures, and the contribution of each normal mode must be calculated by Einstein heat capacity equation (30):
priate approximation is to replace the sum by an integral: +∞
Q fr
1 = σ int =
h c νi kT
2
(kT )
exp hc νi
exp hc νi (k T ) − 1
2
(7)
By substitution in eq 6 and rearrangement, the experimental contribution of torsional mode to heat capacity is obtained as, CV ( tor ) = C p 0 − 4R − R i
hc νi kT
exp hc νi
2
(kT )
exp hc νi (k T ) − 1
2
(8)
where the summation is extended to all the normal modes with the exclusion of the torsional mode ν4 that is treated separately (of course the doubly degenerate modes in Table 1 must be counted twice in the summation). The resulting experimental torsional contributions to heat capacity in ethane are reported in the third column of Table 2 and plotted in Figure 2 in units of R. Before considering the model of hindered internal rotation, it is useful to consider the two limit models of free rotation and torsional harmonic oscillation because both are amenable to analytic solution. If there is no energy barrier to torsional motion, we have a free internal rotation. The Schrödinger equation for free rotation can be easily integrated to yield the following energy levels (17), Ei =
h2 j 2 2 Ir
j = 0, ±1, ± 2, ...
−∞
(2 π Ir k T )
1
h2 j2 dj 2 Ir k T (11)
2
σ int h
Table 2. Experimental Molar Heat Capacity at Constant Pressure of Ethane Corrected to the State of Ideal Gas, Cp0, and Contribution of Torsional Mode, CV(tor) Cp0/(J K᎑1 mol᎑1)
T/K
CV (vib) i = R
exp −
CV(tor)/(J K᎑1 mol᎑1)
189.20
41.66
6.63
209.30
43.25
7.13
229.65
45.08
7.56
249.90
47.27
8.08
250.15
47.17
7.96
272.00
49.68
8.38
272.07
49.51
8.20
279.00
50.66
8.64
292.00
52.14
8.71
302.70
53.27
8.63
335.82
57.40
8.77
347.65
58.91
8.79
359.75
60.38
8.71
364.78
61.04
8.72
373.60
62.10
8.63
387.55
63.89
8.60
451.95
72.43
8.70
520.55
80.08
7.71
(9)
where the i label assumes the values: 1 when j = 0; 2j when j > 0; 1 − 2j when j < 0. If one or both of the rotating groups has symmetrically placed atoms, then only certain values of j are allowed. The number of equivalent orientation is given by the internal symmetry number, σint, that plays an entirely equivalent role to that of the symmetry number in the partition function for overall rotation. The partition function, Q, for a free rotator (fr) is then (30):
Q fr
1 = σ int
+∞
exp − j = −∞
h2 j 2 2 Ir kT
(10)
Since, excluding very low temperatures, the spacing between occupied energy levels is small with respect to kT, an appro-
www.JCE.DivCHED.org
•
Figure 2. Contribution of torsional mode to heat capacity of ethane. Filled circles: experimental points. Dotted line: free rotator. Dashed ∼ = 118 cm᎑1. Solid curve: curve: simple harmonic oscillator with ν restricted rotator with PEbar= 11.74 kJ mol᎑1.
Vol. 82 No. 11 November 2005
•
Journal of Chemical Education
1705
Research: Science and Education
The molar heat capacity at constant volume depends on partition function according to eq 12 (30):
C V = 2RT
∂ ln Q ∂ T
+ RT 2 V
∂ 2 lnQ ∂ T 2
V
(12)
Introducing eq 11 into eq 12, it is found that for a free internal rotation CV(tor) = R兾2. This result is in accordance with the principle of equipartition of energy in that there is only one squared term in the expression for the energy of a free internal rotator, namely, the kinetic energy term (17). When plotted in Figure 2, it appears evident that this result does not fit the experimental points. The conclusion is that the internal rotation in ethane is far from being free in the examined temperature range. In order to test the model of the harmonic oscillator, we have fitted the contribution of the torsional mode, reported in Table 2, to eq 7 by a nonlinear least-squares method. The best fit is obtained with a torsional wavenumber of 118 ± 30 cm᎑1 and the corresponding calculated curve is shown in Figure 2. Apart from the fact that this value is largely different from the experimental torsional wavenumber (ν4 = 289 cm᎑1), it is evident that the fit is not good, thus we have to conclude that the torsional motion in ethane is not a simple harmonic oscillation. Having established that the torsional motion in ethane is neither a free rotation nor a torsional oscillation, the alternative is that it is a hindered rotation. Before discussing the way of obtaining the partition function for the case of hindered rotation, it is useful to recall the behavior of torsional energy levels in the ethane molecule (17). Near the potential minima, the wave function can be approximated to that of a simple harmonic oscillator, thus the restricted rotator energy levels will approach the harmonic oscillator levels, given by eqs 3 and 4, at energies well below the top of the barrier. At high energies, well above the top of the barrier, one can replace the potential shown in eq 1 by
its average value PEbar兾2. The resulting Schrödinger equation is that of a free rotator in which the zero of the energy scale has been displaced by PEbar兾2. The corresponding energy levels, given by eq 13, will be approached by the restricted rotator at energies well above the top of the barrier: Ei =
h2 j 2 PE bar + 2 Ir 2
Q hr =
1 σ int
r
exp − i =1
Journal of Chemical Education
•
∞
Ei kT
exp −
+ hr
i = r +1
Ei kT
sfr
(14)
where the two summations take into account the numerically evaluated energy levels of the hindered rotator from 1 to r and the energy levels of the shifted free rotator from r + 1 onwards, respectively. The second summation is conveniently evaluated as the difference between the complete partition function of the shifted potential free rotator and the contribution of the corresponding first r levels; thus eq 14 becomes Q hr =
1 σ int
r
exp − i =1
h
1706
(13)
Figure 3 illustrates the relation between harmonic oscillator energy levels, shifted free rotator energy levels, and the energy levels evaluated by solving eq 5 numerically for the case of ethane with PEbar = 12.35 kJ mol᎑1 (or 1032 cm᎑1). To obtain highly accurate values of the partition function of the hindered rotator we have exploited the fact that, as evidenced in Figure 3, the hindered rotator (hr) energy levels merge into the shifted free rotator (sfr) energy levels on increasing the quantum level. By assuming that merging occurs at a definite quantum level, say r, we can evaluate the partition function, Qhr, by the following equation,
Ei kT
( 2 π Ir kT )
Figure 3. Potential functions, on the left, for a harmonic oscillator (A), ethane restricted rotation (B), and a shifted potential free rotator (C) in the range ᎑70° ≤ ϕ ≤ 70°. Consider that in the range ᎑180° ≤ ϕ ≤ 180° three minima are present (not shown for space economy) approximated by three harmonic oscillators. The corresponding energy levels (A, B, C) are shown on the right.
j = 0, ±1, ± 2, ...
1
+ exp − hr
PE bar × 2k T
( r −1)// 2
2
exp −
− j = −( r −1)/ 2
h2 j 2 2 Ir k T
(15)
It is interesting to note that if PEbar >> kT, only the first few levels beneath the top of the barrier are occupied, then eq 15 tends to the partition function of a harmonic oscillator, and the torsional motion may be treated like any other vibration. On the contrary if PEbar
