Chemistry Reference
In-Depth Information
Problem 2.3:
Find and confirm (by forming
X
−
1
X
) the inverse operations for (i)
S
4
3
,
(ii)
C
5
2
and (iii)
S
6
3
.
2.4.3 The Order of the Product; Operations that Commute
Equipped with the full set of symmetry operations it is now possible to check that any
group of molecular operations is closed, and so we return to the problem of ethane in its
staggered conformation. Ethane contains a set of equivalent
C
2
axes and a set of equivalent
dihedral mirror planes
σ
d
. For the purposes of this exercise these must be distinguished so
that the exact equivalence of the products of operations may be discerned. The notation
to be used is given in Figure 2.10. From the diagram of the molecule in the space-filling
representation (Figure 2.10a) the
C
2
axes can seen to be equivalent: they all go through
the centre of the C C bond and are perpendicular to it. Looking down the
C
3
axes, it can
also be seen that each
C
2
axis lies in-between C H bonds that are at opposite ends of the
molecule. With the atoms labelled as shown in Figure 2.10b we can distinguish the result
of each
C
2
operation. For example
C
2
A
will cause H
3
to be swapped with H
6
,but
C
2
B
will
cause H
3
to swap with H
4
.
2
C
C
σ
d
A
H
4
H
1
σ
d
C
H
3
H
5
H
6
H
2
2
B
C
2
A
C
σ
B
Figure 2.10
Labelling used for the sets of equivalent C
2
axes and dihedral mirror planes for
the purposes of constructing the multiplication table for ethane. Note that the
σ
d
planes have
been labeled so that A initially contains H
1
, B initially contains H
2
and C initially contains H
3
,
the C
2
axes have also been labeled for the
σ
d
plane to which they are perpendicular.
As before, when generating products of operations for the multiplication table, the sym-
metry elements are thought of as fixed in space, set by the global axis system. For example,
C
2
A
will alter the positions of the hydrogen atoms but not the location of
C
2
B
.
Applying each pair of operations in turn gives the full multiplication table shown in
Table 2.4. This table is considerably more complex than that for the examples of water and
ammonia used earlier because the number of individual operations is greater. However, the
rules set out for the products of the principal axis with vertically orientated mirror planes
can also be seen to apply to the dihedral planes in this case.
