Index of Hydrogen Deficiency

When organic chemists look at a chemical formula or a chemical structure, they instinctively determine the index of hydrogen deficiency of the compound, the formal charge of each atom, and the oxidation level of each atom in the molecule. It's part of the way they think about molecules. In order to figure out these three pieces of information, they apply a set of simple rules. The discussion that follows explains these rules.

Degrees of Unsaturation and the Index of Hydrogen Deficiency

The valence of an atom specifies the number of bonds which that atom forms to other atoms. Generally an atom's valence is dictated by the filled shell rules; the difference between the number of electrons in a filled valence shell and the number of valence electrons assigned to an atom equals the valence of that atom. For example, in hydrogen, an element in Period 1, a filled shell contains 2 electrons. Since a hydrogen atom contains 1 electron in its valence shell, it needs only one more electron to fill its valence shell. Hydrogen is univalent. It takes eight electrons to fill the valence shell of elements in Period 2 of the periodic table. Thus nitrogen, which has 5 electrons in its valence shell needs three more electrons to fill the shell. It acquires those electrons by forming three bonds to other atoms; nitrogen is trivalent.

Carbon is tetravalent. This means that it forms four bonds to other atoms in order to fill its valence shell. There are several ways it can do this:

• Carbon can form a single bond to each of four different atoms. In methane, CH₄, the carbon atom is bonded to four hydrogen atoms. In methanol, CH₃OH, the carbon is bonded to three hydrogen atoms and an oxygen atom.
• Carbon can form a double bond to one atom and a single bond to each of two other atoms. This is the case in ethene, H₂C=CH₂, where each carbon is doubly bonded to the other and singly bonded to each of two hydrogen atoms.
• Carbon can form a triple bond to one atom and a single bond to another. Ethyne (acetylene) is an example. It contains two carbon atoms that are connected by a triple bond; each carbon is also bonded to a hydrogen atom. (Draw a Lewis structure for ethyne.)
• Carbon can form a double bond to each of two different atoms. In carbon dioxide, CO₂, the carbon is doubly bonded to both oxygen atoms.

When a carbon atom uses its four valence electrons to form bonds to four other atoms, the carbon is said to be saturated. The carbon in methane is one example. The carbons in ethane, C₂H₆, are another. Methane and ethane both conform to the formula C_nH_(2n+2). They are called saturated hydrocarbons; they contain only hydrogen and carbon, and each carbon atom is bonded to four other atoms.

It shouldn't be surprising that an alkene like ethene in which each carbon atom is bonded to only three atoms is classified as an unsaturated hydrocarbon. The general formula for alkenes that contain one double bond is C_nH_2n. Notice that such an alkene contains two fewer hyrdogen atoms than an alkane with the same number of carbon atoms. This is equivalent to 1 molecule of dihydrogen, and the degree of unsaturation of ethene is said to equal 1. Figure 1 indicates why this is so.

Figure 1

Saturated and Unsaturated Hydrocarbons

In ethane each carbon uses its four valence electrons to form bonds to four other atoms. In ethene they use them to form bonds to three other atoms; rather than sharing one of their valence electrons with another hydrogen, C₁ and C₂ share them with each other as indicated by the blue color coding in the Figure.

Extension of this idea reveals that the degree of unsaturation of ethyne, C₂H₂, is 2 since C₂H₆- C₂H₂= H₄, which is equivalent to 2 molecules of dihydrogen. The triple bond accounts for 2 degrees of unsaturation in ethyne

The concept of saturation may be applied to any atom or molecule. Consider methanol, H₃C-O-H, again, but this time focus on the oxygen atom rather than the carbon. Oxygen is divalent. In methanol it is bonded to two other atoms and is therefore described as a saturated atom. In fact, the whole molecule is considered to be saturated, which means that the degree of unsaturation of the molecule is zero. To understand this, compare the molecular formulas of methane and methanol; CH₄ vs CH₄O. The number of hydrogen atoms is the same in these two 1-carbon molecules.

Contrast the case of methanol with that of formaldehyde, CH₂O, another 1-carbon molecule. The degree of unsaturation in this molecule equals 1 ( CH₄- CH₂O = H₂). Formaldehyde contains a carbon-oxygen double bond.

The molecular formula of cyclopropane is C₃H₆. Its degree of unsaturation is 1. But cyclopropane is a saturated hydrocarbon! Every atom is saturated. Cyclopropane doesn't contain any multiple bonds. Still, it has less than the maximum number of hydrogen atoms that a 3-carbon compound can contain. It's not that the method of calculating the degree of unsaturation does not work, but rather that cyclopropane is not an acyclic saturated hydrocarbon. To appreciate how a compound can have a less than the maximum number of hydrogens and still be considered saturated, examine Figure 2. Then compare the process shown in Figure 2 with the alternative outlined in Figure 1.

Figure 2

Alkanes and Cycloalkanes

To differentiate between hydrogen deficiency that arises from multiple bonds in a molecule and hydrogen deficiency due to the presence of rings in a structure, chemists defined the term index of hydrogen deficiency. It is a more general term than degree of unsaturation, indicating the total number of multiple bonds and/or rings in a molecule. While the degree of unsaturation of cyclopropane is zero, its index of hydrogen deficiency is 1. To extend this distinction, the index of hydrogen deficiency of cyclobutene is 2 while its degree of unsaturation is 1; it contains 1 double bond and one ring.

Rule 1: Calculating the Index of Hydrogen Deficiency

To calculate the index of hydrogen deficiency of a compound, subtract the number of hydrogens in its empirical formula from the number of hydrogen atoms in a saturated acyclic hydrocarbon containing the same number of carbon atoms, i.e. C_nH_(2n+2). Divide the difference by 2.

Each multiple bond and each ring contributes 1 degree of hydrogen deficiency to a compound:

Structural Unit	Contribution to Index of Hydrogen Deficiency
double bond	1
triple bond	2
ring	1

Taken by itself the index of hydrogen deficiency is not especially useful. In combination with other data however, it can provide important insights into chemical structure. Consider the following two examples.

Hexene, an alkene, and cyclohexane, an alkane, have the same molecular formula, C₆H₁₂, and the same index of hydrogen deficiency, 1.

A simple way to differentiate an alkene from an alkane involves treatment of a sample with a dilute solution of dibromine, Br₂, dissolved in carbon tetrachloride, CCl₄. This is called the Beilstein test. Dibromine is an orange-red liquid. It reacts rapidly with alkenes to produce compounds that are colorless. When a solution of dibromine in carbon tetrachloride is added to an alkene, the orange-red color disappears instantly. Alkanes and cycloalkanes do not react under comparable conditions, and the orange-red color persists. So a Beilstein test will immediately determine whether a hydrocarbon that has an index of hydrogen deficiency of 1 is an alkene or a cycloalkane.

Now imagine that you're a natural products chemist who has isolated a terpene from the bark of a tree. Elemental analysis indicates its molecular formula to be C₁₀H₁₆. Two isomers that fit this data are shown in Figure 3.

Figure 3

Two Isomeric Terpenes

In order to differentiate between these alternatives, you perform an experiment; you measure the number of moles of dihydrogen that are required to completely react with one mole of your sample. Like dibromine, diydrogen reacts with the double bond(s) in alkenes. Although the index of hydrogen deficiency of both compounds is 3, the degree of unsaturation of structure 1 is 1, while that of structure 2 is 2. If your compound has structure 1, it will react with one mole of dihydrogen. Structure 2 will consume two moles. Measuring the volume of dihydrogen consumed is simple. So is calculating the number of moles of dihydrogen that is equivalent to that volume. It entails manipulation of the ideal gas law.