344
6 Molecular Systems
experimentally, and the contributions from the translational and rotational motions,
together with those contributions from the 17 vibrational modes of ethane for which
spectroscopic data are available are subtracted from it, the remainder must be the
contribution to C V (T ) associated with the hindered rotational motion.
From an examination of the Newman projection of the two CH 3 groups, it is clear
that there is maximal repulsion between them when they are eclipsed, and minimal
repulsion between them when they are staggered, so that the internal interaction
potential has cos 3φ symmetry for 0 ≤ φ ≤ 2π . This case has been examined
in detail by Ercolani [44]. As the eclipsed and staggered forms of ethane have
different interaction energies, rotation about the C–C bond will be nonuniform,
and the rotational motion is thereby hindered. On the one hand, for energies small
enough that the barrier, V 3 , to free internal rotation is very large, the result will be
(torsional) vibrations of the CH 3 groups, while on the other hand, for very high
energies, V 3 will be comparatively small, and essentially free rotational motion will
take place. For energies lying between these extremes, the motion will be much
more complicated: it is this more complicated motion that is referred to as hindered
rotational motion.
We shall briefly examine the two limiting cases discussed earlier, for which the
relative energy of rotation is either much smaller or much greater than the barrier
height, V 3 , associated with Eq. (6.6.5). In the former case, the angle φ will be
confined near the minimum of one of the three equivalent potential wells of V (φ):
without loss of generality, we may choose the centre of this well as the origin for φ.
With this choice of origin φ will necessarily be small, and we may expand cos 3φ,
retaining only the first two terms, to obtain
V (φ) =
1
2 k φ φ
2 ,
which is a SHO potential energy with force constant k φ =
9
2 (
1
2 V 3 /I r )
1
2 . The
accepted barrier height for C 2 H 6 is V 3 = 12.35 kJ mol
−1 or V 3 = 2.0505 × 10 −20 J
in SI units. For C 2 H 6 , I r is given simply by
1
2 I CH 3 , with I CH 3 the moment-of-inertia
of a rigid methyl group undergoing rotation about the ethane C–C bond.
For rotational energies much greater than V 3 , i.e., lying well above the top of the
barrier, we may replace V (φ) by its angular average
1
2 V 3 , to approximate Eq. (6.6.5)
by the Schrödinger equation for a free rotor, but with a shift of the zero of the energy
scale upwards by
1
2 V 3 . The energy levels will thus be
E k =
¯
h 2 k 2
2π
+
1
2 V 3 ,
k = 0, ±1, ±2, . . . .
We should hence expect the energy eigensolutions of Eq. (6.6.5) to be similar to
those of a triply degenerate SHO for energies near the bottom of the wells and to be
similar to those of a shifted free rotor for energies well above the top of the potential
energy barriers. Such a comparison has been given by Pitzer [43] and by Ercolani
[44], and is also shown in Fig. 6.21.
6 Molecular Systems
experimentally, and the contributions from the translational and rotational motions,
together with those contributions from the 17 vibrational modes of ethane for which
spectroscopic data are available are subtracted from it, the remainder must be the
contribution to C V (T ) associated with the hindered rotational motion.
From an examination of the Newman projection of the two CH 3 groups, it is clear
that there is maximal repulsion between them when they are eclipsed, and minimal
repulsion between them when they are staggered, so that the internal interaction
potential has cos 3φ symmetry for 0 ≤ φ ≤ 2π . This case has been examined
in detail by Ercolani [44]. As the eclipsed and staggered forms of ethane have
different interaction energies, rotation about the C–C bond will be nonuniform,
and the rotational motion is thereby hindered. On the one hand, for energies small
enough that the barrier, V 3 , to free internal rotation is very large, the result will be
(torsional) vibrations of the CH 3 groups, while on the other hand, for very high
energies, V 3 will be comparatively small, and essentially free rotational motion will
take place. For energies lying between these extremes, the motion will be much
more complicated: it is this more complicated motion that is referred to as hindered
rotational motion.
We shall briefly examine the two limiting cases discussed earlier, for which the
relative energy of rotation is either much smaller or much greater than the barrier
height, V 3 , associated with Eq. (6.6.5). In the former case, the angle φ will be
confined near the minimum of one of the three equivalent potential wells of V (φ):
without loss of generality, we may choose the centre of this well as the origin for φ.
With this choice of origin φ will necessarily be small, and we may expand cos 3φ,
retaining only the first two terms, to obtain
V (φ) =
1
2 k φ φ
2 ,
which is a SHO potential energy with force constant k φ =
9
2 (
1
2 V 3 /I r )
1
2 . The
accepted barrier height for C 2 H 6 is V 3 = 12.35 kJ mol
−1 or V 3 = 2.0505 × 10 −20 J
in SI units. For C 2 H 6 , I r is given simply by
1
2 I CH 3 , with I CH 3 the moment-of-inertia
of a rigid methyl group undergoing rotation about the ethane C–C bond.
For rotational energies much greater than V 3 , i.e., lying well above the top of the
barrier, we may replace V (φ) by its angular average
1
2 V 3 , to approximate Eq. (6.6.5)
by the Schrödinger equation for a free rotor, but with a shift of the zero of the energy
scale upwards by
1
2 V 3 . The energy levels will thus be
E k =
¯
h 2 k 2
2π
+
1
2 V 3 ,
k = 0, ±1, ±2, . . . .
We should hence expect the energy eigensolutions of Eq. (6.6.5) to be similar to
those of a triply degenerate SHO for energies near the bottom of the wells and to be
similar to those of a shifted free rotor for energies well above the top of the potential
energy barriers. Such a comparison has been given by Pitzer [43] and by Ercolani
[44], and is also shown in Fig. 6.21.