Editing Structural isomer (section)

==Structural equivalence and symmetry==
===Structural equivalence===
Two molecules (including polyatomic ions) A and B '''have the same structure''' if each atom of A can be paired with an atom of B of the same element, in a one-to-one way, so that for every bond in A there is a bond in B, of the same type, between corresponding atoms; and vice versa.<ref name=mumb2018/> This requirement applies also to complex bonds that involve three or more atoms, such as the [[delocalized bond]]ing in the benzene molecule and other aromatic compounds.

Depending on the context, one may require that each atom be paired with an atom of the same isotope, not just of the same element.

Two molecules then can be said to be structural isomers (or, if isotopes matter, structural isotopomers) if they have the same molecular formula but do not have the same structure.

===Structural symmetry and equivalent atoms===
Structural symmetry of a molecule can be defined mathematically as a [[permutation]] of the atoms that exchanges at least two atoms but does not change the molecule's structure. Two atoms then can be said to be structurally [[equivalence relation|equivalent]] if there is a structural symmetry that takes one to the other.<ref name=faul2010/>

Thus, for example, all four hydrogen atoms of [[methane]] are structurally equivalent, because any permutation of them will preserve all the bonds of the molecule.

Likewise, all six hydrogens of [[ethane]] ({{chem|C|2|H|6}}) are structurally equivalent to each other, as are the two carbons; because any hydrogen can be switched with any other, either by a permutation that swaps just those two atoms, or by a permutation that swaps the two carbons and each hydrogen in one methyl group with a different hydrogen on the other methyl. Either operation preserves the structure of the molecule. That is the case also for the hydrogen atoms in [[cyclopentane]], [[allene]], [[2-butyne]], [[hexamethylenetetramine]], [[prismane]], [[cubane]], [[dodecahedrane]], etc.

On the other hand, the hydrogen atoms of [[propane]] are not all structurally equivalent. The six hydrogens attached to the first and third carbons are equivalent, as in ethane, and the two attached to the middle carbon are equivalent to each other; but there is no equivalence between these two [[equivalence class]]es.

===Symmetry and positional isomerism===
Structural equivalences between atoms of a parent molecule reduce the number of positional isomers that can be obtained by replacing those atoms for a different element or group. Thus, for example, the structural equivalence between the six hydrogens of [[ethane]] {{chem|C|2|H|6}} means that there is just one structural isomer of [[ethanol]] {{chem|C|2|H|5|OH}}, not 6. The eight hydrogens of [[propane]] {{chem|C|3|H|8}} are partitioned into two structural equivalence classes (the six on the methyl groups, and the two on the central carbon); therefore there are only two positional isomers of propanol ([[1-propanol]] and [[2-propanol]]). Likewise there are only two positional isomers of [[butanol]], and three of [[pentanol]] or [[hexanol]].

===Symmetry breaking by substitutions===
Once a substitution is made on a parent molecule, its structural symmetry is usually reduced, meaning that atoms that were formerly equivalent may no longer be so. Thus substitution of two or more equivalent atoms by the same element may generate more than one positional isomer.

The classical example is the derivatives of [[benzene]]. Its six hydrogens are all structurally equivalent, and so are the six carbons; because the structure is not changed if the atoms are permuted in ways that correspond to flipping the molecule over or rotating it by multiples of 60 degrees. Therefore, replacing any hydrogen by chlorine yields only one [[chlorobenzene]]. However, with that replacement, the atom permutations that moved that hydrogen are no longer valid. Only one permutation remains, that corresponds to flipping the molecule over while keeping the chlorine fixed. The five remaining hydrogens then fall into three different equivalence classes: the one opposite to the chlorine is a class by itself (called the ''para'' position), the two closest to the chlorine form another class (''ortho''), and the remaining two are the third class (''meta''). Thus a second substitution of hydrogen by chlorine can yield three positional isomers: [[1,2-Dichlorobenzene|1,2- or ''ortho''-]], [[1,3-Dichlorobenzene|1,3- or ''meta''-]], and [[1,4-Dichlorobenzene|1,4- or ''para''-dichlorobenzene]].

{| align="center"
|-
| style="text-align:center;" | [[File:O-Dichlorobenzene-3D-balls.png|100px]]
| style="text-align:center;" | [[File:M-Dichlorobenzene-3D-balls.png|100px]]
| style="text-align:center;" | [[File:P-Dichlorobenzene-3D-balls.png|90px]]
|-
| style="padding-left:20px;padding-right:20px;text-align:center;" | ''ortho''-Dichlorobenzene
| style="padding-left:20px;padding-right:20px;text-align:center;" | ''meta''-Dichlorobenzene
| style="padding-left:20px;padding-right:20px;text-align:center;" | ''para''-Dichlorobenzene
|-
| style="padding-left:20px;padding-right:20px;text-align:center;" | 1,2-Dichlorobenzene
| style="padding-left:20px;padding-right:20px;text-align:center;" | 1,3-Dichlorobenzene
| style="padding-left:20px;padding-right:20px;text-align:center;" | 1,4-Dichlorobenzene
|}

For the same reason, there is only one [[phenol]] (hydroxybenzene), but three [[benzenediol]]s; and one [[toluene]] (methylbenzene), but three [[toluol]]s, and three [[xylene]]s.

On the other hand, the second replacement (by the same substituent) may preserve or even increase the symmetry of the molecule, and thus may preserve or reduce the number of equivalence classes for the next replacement. Thus, the four remaining hydrogens in ''meta''-dichlorobenzene still fall into three classes, while those of ''ortho''- fall into two, and those of ''para''- are all equivalent again. Still, some of these 3 + 2 + 1 = 6 substitutions end up yielding the same structure, so there are only three structurally distinct [[trichlorobenzene]]s: [[1,2,3-Trichlorobenzene|1,2,3-]], [[1,2,4-Trichlorobenzene|1,2,4-]], and [[1,3,5-Trichlorobenzene|1,3,5-]].

{| align="center" class="skin-invert-image"
|-
| style="text-align:center;" | [[File:1,2,3-trichlorobenzene.svg|90px]]
| style="text-align:center;" | [[File:1,2,4-trichlorobenzene.svg|90px]]
| style="text-align:center;" | [[File:1,3,5-Trichlorobenzene.svg|120px]]
|-
| style="padding-left:20px;padding-right:20px;text-align:center;" | 1,2,3-Trichlorobenzene
| style="padding-left:20px;padding-right:20px;text-align:center;" | 1,2,4-Trichlorobenzene
| style="padding-left:20px;padding-right:20px;text-align:center;" | 1,3,5-Trichlorobenzene
|}

If the substituents at each step are different, there will usually be more structural isomers. [[Xylenol]], which is benzene with one hydroxyl substituent and two methyl substituents, has a total of 6 isomers:

{| align="center" class="skin-invert-image"
|-
| style="text-align:center;" | [[File:2,3-Xylenol.svg|100px]]
| style="text-align:center;" | [[File:2,4-Xylenol.svg|100px]]
| style="text-align:center;" | [[File:2,5-dimethylphenol.png|150px]]
|-
| style="padding-left:20px;padding-right:20px;text-align:center;" | 2,3-Xylenol
| style="padding-left:20px;padding-right:20px;text-align:center;" | 2,4-Xylenol
| style="padding-left:20px;padding-right:20px;text-align:center;" | 2,5-Xylenol
|-
| style="text-align:center;" | [[File:2,6-dimethylphenol.png|160px]]
| style="text-align:center;" | [[File:3,4-Xylenol.svg|100px]]
| style="text-align:center;" | [[File:3,5-Xylenol.svg|160px]]
|-
| style="padding-left:20px;padding-right:20px;text-align:center;" | 2,6-Xylenol
| style="padding-left:20px;padding-right:20px;text-align:center;" | 3,4-Xylenol
| style="padding-left:20px;padding-right:20px;text-align:center;" | 3,5-Xylenol
|}