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The symmetry relationship
between similar groups in a molecule
determines whether those groups will have the same reactivity
or whether those groups will have different reactivity.
In addition to the symmetry relationship between groups,
we must also pay attention
to whether these groups are being reacted upon
by chiral or achiral reagents.
Let's begin by considering a pair
of similar functional groups
that are being reacted upon by an achiral reagent.
If these groups can be interchanged
by any symmetry operation,
their reactivity toward an achiral reagent
will be equivalent.
A few examples will help to clarify this idea.
Let's begin with this molecule of acetone.
The two hydrogen atoms are related by this mirror plane
and to an achiral base, they're equivalent.
In other words, each of these protons would be removed
with equal likelihood if the base is achiral.
The situation is similar for the two hydroxyl groups
in this molecule,
which are related by the reflection operation.
To an achiral base,
these two hydroxyl groups are equivalent,
so removal of the proton on either of these hydroxyl groups
by that achiral base is going to be equally probable.
In this molecule, the two electrophilic sites,
the alkyl bromides, are related by a C2 axis.
To an achiral nucleophile,
each of those groups are equivalent
and substitution at either carbon by an SN2 process
would take place with equal probability.
In the case of this enolate anion,
all of the atoms lie in a common plane.
That plane defines the space above
and the space below the enolate anion
as being equivalent since those two spaces
are related by a reflection through this plane.
And so, for an incoming electrophile,
if that electrophile happens to be an achiral electrophile,
attack on the enolate from above
is equivalent to attack from below.
When similar groups are related by some kinds of symmetry,
in particular reflection symmetry
or inversion symmetry,
the reaction can either be the same or it can be different.
It'll be the same if those similar groups
are reacting with achiral reagents.
Those similar groups will behave differently.
They'll react differently
if they're reacting with chiral reagent.
Let's take a simple example of the deprotonation
of these two hydroxyl groups.
Since those similar groups are related by a reflection plane,
they're going to be equivalent
to an achiral base like this diethyl amid.
In that case, deprotonation of either one or two
will be equally likely.
However, because these similar groups are related
by a reflection symmetry,
their behavior will be different
if they're being deprotonated by a chiral base,
like the one that's shown here.
In the case of the chiral base,
removal of proton on hydroxyl group 1
will be different than removal of the proton
on hydroxyl group 2.
The differential reactivity of these two hydroxyl groups
is the result of a pair of diastereomeric transition states
that result from removal of proton 1 or proton 2
by this chiral base.
In contrast, if an achiral base is used,
the two transition states from removal of proton 1
or removal of proton 2 are enantiomeric.
Diastereomeric transition states have different energies,
whereas enantiomeric transition states are equal in energy.
If similar groups are related by a rotation,
there is no difference in their reactivity
no matter whether the reagent
that is reacting with those groups
is chiral or achiral.
For example, in this molecule of acetone,
the protons on the methyl group labeled A
are related by a two-fold rotation
about the carbonyl axis to the protons
on the methyl group labeled B.
Because these similar groups
are related by a rotation operation,
they cannot be distinguished whether the base is achiral
or even this chiral base that's shown here.
Deprotonation of both sets of protons is equally likely
regardless of the reagent that's used.