Degree of unsaturation

In the analysis of the molecular formula of organic molecules, the degree of unsaturation (also known as the index of hydrogen deficiency (IHD), double bond equivalents, or unsaturation index[1]) is a calculation that determines the total number of rings and π bonds. A formula is used in organic chemistry to help draw chemical structures. It does not give any information about those components individually—the specific number of rings, or of double bonds (one π bond each), or of triple bonds (two π bonds each). The final structure is verified with use of NMR, mass spectrometry and IR spectroscopy, as well as qualitative inspection. It is based on comparing the actual molecular formula to what would be a possible formula if the structure were saturated—having no rings and containing only σ bonds—with all atoms having their standard valence.

General formula

The formula for degree of unsaturation is:

${\displaystyle \mathrm {DU} =1+{\tfrac {1}{2}}\sum n_{i}(v_{i}-2)}$

where ni is the number of atoms with valence vi.[2]

That is, an atom that has a valence of x contributes a total of x − 2 to the degree of unsaturation. The result is then halved and increased by 1.

Simplified formulae

For certain classes of molecules, the general formula can be simplified or rewritten more clearly. For example:

${\displaystyle {\text{Double bond equivalent}}=(a+1)-{\frac {b-c+f}{2}}}$

where

a = number of carbon atoms in the compound
b = number of hydrogen atoms in the compound
c = number of nitrogen atoms in the compound
f = number of halogen atoms in the compound

or

${\displaystyle \mathrm {rings+\pi \ bonds} =C-{\frac {H}{2}}-{\frac {X}{2}}+{\frac {N}{2}}+1\,}$

where C = number of carbons, H = number of hydrogens, X = number of halogens and N = number of nitrogens,[3] gives an equivalent result.

In either case, oxygen and other divalent atoms do not contribute to the degree of unsaturation, as 2 − 2 = 0.

Explanation

IHD tells us how many hydrogen pairs are missing from its saturated structure, which is related to the number of multi bonds and/or rings in a non-saturated organic chemical structure, simply because adding one bond or joining two elements to form a ring in the structure reduces the need for two H’s. A simple form of the formula is as follows:

IHD = C + 1 + N/2 – H/2 – X/2

where C, N, H and X represent the number of carbon, nitrogen, hydrogen and halogen atoms, respectively. Each of the terms on the RHS can be explained, respectively, as follows:

1) Except the terminal C’s, each of the C’s chained to the structure requires a pair of H’s attached to it -- that is why the number C is in the formula, which actually represents the number of hydrogen pairs requires for that number of carbons in a saturated structure. (This is also true if a carbon is added into the structure, whether it is inserted to a backbone chain, attached to a terminal to replace an H, or branched out from a carbon to replace an H.)
2) Each of the two terminal C’s needs one extra H – that is why 1 is added in the formula. (A branch’s terminal doesn’t need an extra H added in the calculation because an H at where the branch attached to must have been replaced, which is true for any branch terminals, including C, N and O’s.)
3) Except the terminal N’s, each N in the chain only requires one H attached to it, which is half a pair of H’s – that is why +N/2 is in the formula, which gives a value of 1 for every two N’s. (This is also true if an N is added into the structure, whether it is inserted to a backbone chain, attached to a terminal to replace an H, or branched out from a carbon to replace an H.)
4) The H/2 represents the number of hydrogen pairs because it gives a value of 1 for every two H’s. It is subtracted in the formula to count how many pairs of H’s are missing from the saturated structure, which tells us the degree of hydrogen deficiency. (No hydrogen pair is missing if IHD = 0, which corresponds to no H-deficiency. Keep in mind that if one N is added to the structure, 1/2 should be added to the N/2 and subtracted from H/2 in the formula, which means that the IHD remains the same.)
5) The presence of X/2 is for a reason similar to H/2.

Adding an oxygen atom in the structure requires no hydrogen added, which is why the number of oxygen atoms does not appears in the formula. Furthermore, the formula can be generalised to include all elements of Group I (the hydrogen and lithium family), Group IV (the carbon family), Group V (the nitrogen family) and Group VII (the fluorine family) of CAS A group in the periodic table as follows:

IHD = G4 + 1 + G5 /2 – G1 /2 – G7 /2

Or simply,

IHD = G4 + 1 + (G5 – G1 – G7 )/2

References

1. ^ Sparkman, David O. Mass Spectrometry Desk Reference. Pittsburgh: Global View Pub. p. 54. ISBN 0-9660813-9-0.
2. ^ Badertscher, M.; Bischofberger, K.; Munk, M.E.; Pretsch, E. (2001). "A Novel Formalism To Characterize the Degree of Unsaturation of Organic Molecules". Journal of Chemical Information and Modeling. 41 (4): 889. doi:10.1021/ci000135o.
3. ^ Organic structural spectroscopy, chapter 1.

Young, Paul R. Practical Spectroscopy. ISBN 0-534-37230-9.