18.2 Index of Hydrogen Deficiency or Unsaturation

The Index of Hydrogen Deficiency (IHD), also known as the degree of unsaturation, is a calculation used in organic chemistry to determine the total number of rings and pi ($\pi$) bonds within a molecule from its molecular formula. It works by comparing the number of hydrogen atoms in the given compound to the number of hydrogen atoms in its corresponding saturated, acyclic counterpart.

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Concept of Unsaturation

In a fully saturated, acyclic hydrocarbon (an alkane with no rings), the number of hydrogen atoms is related to the number of carbon atoms by the formula:

$\text{Number of hydrogens}=(2\times \text{number of carbons})+2$

This is the maximum number of hydrogens a given carbon skeleton can hold.

Each multiple bond (a double or triple bond) or ring structure present in a molecule reduces the hydrogen count by two. Each of these structural features is referred to as a degree of unsaturation.

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Calculating the Index of Hydrogen Deficiency (IHD)

The IHD for a molecule can be calculated directly from its molecular formula using the following general equation:

$\text{IHD}=\frac{(2C+2+N)-(H+X)}{2}$

Where:

• $C$ = number of carbon atoms
• $N$ = number of nitrogen atoms
• $H$ = number of hydrogen atoms
• $X$ = number of halogen atoms (F, Cl, Br, I)

Oxygen and sulfur atoms do not affect the calculation and can be ignored.

This calculation provides the sum of multiple bonds and rings. It does not distinguish between them. For example, an IHD of 1 could mean one double bond or one ring. An IHD of 2 could mean two double bonds, one triple bond, two rings, or one ring and one double bond. This simple calculation is a crucial first step when determining the structure of an unknown molecule.

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Worked Examples

Example 18.1: Ethene (${\mathrm{C}}_2{\mathrm{H}}_4$)

Calculate the degree of unsaturation in ethene (${\mathrm{C}}_2{\mathrm{H}}_4$).

1. Write the given values:

   • Number of Carbons ($C$) = 2
   • Number of Hydrogens ($H$) = 4
   • Number of Nitrogens ($N$) = 0
   • Number of Halogens ($X$) = 0

2. Apply the formula: $\text{IHD}=\frac{(2C+2+N)-(H+X)}{2}$

3. Show the calculation: $\text{IHD}=\frac{(2\times 2+2+0)-(4+0)}{2}$ $\text{IHD}=\frac{(4+2)-4}{2}=\frac{6-4}{2}=\frac{2}{2}$ $\text{IHD}=1$

The degree of unsaturation for ethene is 1. This corresponds to the one double bond ($C=C$) in its structure.

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Possible Questions/Answers

This section is based on CONCEPT ASSESSMENT EXERCISE 18.1.

• Q: Calculate the degree of unsaturation in benzene (${\mathrm{C}}_6{\mathrm{H}}_6$).

  A:

  • Given: $C=6$, $H=6$
  • Calculation: $\text{IHD}=\frac{(2\times 6+2)-6}{2}=\frac{14-6}{2}=\frac{8}{2}=4$
  • The IHD of benzene is 4. This accounts for its structure, which contains one ring and three double bonds.

• Q: Calculate the degree of unsaturation in ethyne (${\mathrm{C}}_2{\mathrm{H}}_2$).

  A:

  • Given: $C=2$, $H=2$
  • Calculation: $\text{IHD}=\frac{(2\times 2+2)-2}{2}=\frac{6-2}{2}=\frac{4}{2}=2$
  • The IHD of ethyne is 2. This corresponds to the one triple bond ($C\equiv C$) in its structure.

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Summary

• The Index of Hydrogen Deficiency (IHD) or degree of unsaturation quantifies the number of $\pi$ bonds and rings in a molecule.
• A saturated, acyclic alkane has an IHD of 0 and follows the formula $C_n{H}_{2n+2}$.
• Each ring or double bond contributes 1 to the IHD.
• Each triple bond contributes 2 to the IHD.
• The general formula for calculation is: $\text{IHD}=\frac{(2C+2+N)-(H+X)}{2}$
• Significance: IHD is a fundamental tool in analytical chemistry, particularly in combination with spectroscopic methods like IR Spectroscopy→ and Mass Spectrometry, to quickly narrow down possible structures for an unknown compound.

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References

(Derived from FBISE textbook)