Degree of unsaturation

The formula for degree of unsaturation is:

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

where n_i is the number of atoms with valence v_i.[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.

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.

For hydrocarbons, the DBE (or IHD) tells us the number of rings and/or extra bonds in a non-saturated structure, which equals to the number of hydrogen pairs that are required to make the structure saturated, simply because joining two elements to form a ring or adding one extra bond in a structure reduces the need for two H's. (For non-hydrocarbons, the elements in a pair can include any elements in the lithium family and the fluorine family in the periodic table.) A popular 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 carbons, each of the carbon chained to the structure with a single bond requires a pair of hydrogen atoms 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 carbons needs one extra hydrogen – that is why 1 is added to the formula. (A branch’s terminal doesn’t need an extra hydrogen added in the calculation because an hydrogen at where the branch attached to must have been replaced, which is true for a branch terminated with any element.)

3) Except the terminal nitrogens, each nitrogen in the chain only requires one H attached to it, which is half a pair of hydrogens– that is why +N/2 is in the formula, which gives a value of 1 for every two nitrogens. (This is also true if an nitrogen is added into the structure, whether it is inserted to a backbone chain, attached to a terminal to replace an hydrogen, or branched out from a carbon to replace an hydrogen.)

4) The H/2 represents the number of hydrogen pairs because it gives a value of 1 for every two hydrogen atoms. It is subtracted in the formula to count how many pairs of hydrogen atoms are missing in the unsaturated structure, which tells us the degree of hydrogen deficiency. (No hydrogen pair is missing if IHD = 0, which corresponds to no H-deficiency.)

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 – (G₁ + G₇ – G₅ )/2

Young, Paul R. (2000). Practical Spectroscopy. ISBN 0-534-37230-9.

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