Bond Order Formula, Calculation, & Examples

Bond Order Formula

In the molecular orbital theory, two atomic orbitals of valence electrons of similar energy and orientation form a bond. Molecular orbital arises from two atomic orbitals, which have to be conserved. So, there is a bonding orbital and an antibonding orbital.

The bonding orbital is a three-dimensional volume of space where the electrons move, experiencing the attractive force of the nuclei of both atoms involved in bonding. The electron density is maximum in this region.

The antibonding orbital is the space between the nuclei where the probability of finding electrons is zero. This space is the nodal plane. The electron density is higher at places away from the nuclei.

The antibonding orbitals have an asterisk to distinguish them from bonding orbitals.

For determining the bonding and antibonding orbitals, molecular orbital diagrams are necessary.

• Draw the atomic orbitals on the sides.
• The molecular orbitals are in the center brought together by joining the corresponding atomic orbitals.
• The bonding orbitals have lower energy and more stability than the atomic orbitals. So they are at a lower level than the atomic orbitals.
• The antibonding orbitals have more energy than their parent atomic orbitals. So they are above the atomic orbitals. They have an asterisk.
• The orbitals fill up according to the Aufbau principle.

The following illustrates how to find the bond order of molecules.

Example

Molecule of hydrogen.

There are two atoms, and each atom has one 1s electron.

The two atomic orbitals approach each other as they are similar in energy and orientation.

They form a molecular bonding orbital.

There are no electrons for antibonding orbitals, so it is empty.

Molecular Orbital Diagram for Hydrogen

For multi-electron atoms, draw the diagram only for the valence electrons.

The formula for determining the bond order considers the valence electrons in the bonding and antibonding orbitals.

The formula for bond order calculation is half the difference between the number of electrons in the bonding and antibonding orbitals.

Mathematically the formula is

{eq}(Number of bonding electrons - number of antibonding electrons)\div 2 {/eq}

Using this formula, we can calculate the bond order in different molecules.

In the case of hydrogen:

bonding electrons = 2

anti-bonding electrons = 0

Bond Order = (2-0) /2 =1

There is a single bond between the atoms.

Bond Order Calculation

Atoms form bonds using their valence electrons. All electrons do not have the same energy. Energy differs among orbitals within an energy level. An atom, while bonding, may try to equalize the energy of two or more of its electrons in different orbitals by mixing them or by hybridizing them. For example, carbon atom shows hybridization. If one s and one p orbitals undergo hybridization, it is sp hybridization. Mixing of one s with two p, and one s with three p orbitals gives {eq}sp^{2} {/eq} and {eq}sp^{3} {/eq} hybridization. After hybridization, the orbitals have equal energy and are degenerate.

Resonance occurs when a molecule has two or more equivalent contributing structures. The electrons delocalize and spread over atoms. The resonance structures help explain properties that do not conform to specific Lewis structures. Also, the resonance hybrid structure gives more stability, and the properties are intermediate to the contributing forms.

With these in mind, it is not surprising that bond order values sometimes come out as fractions, non-integrals. Benzene, ozone, and ions like carbonate and sulfate show resonance.

Take the following steps to find the bond order between two specific atoms in a polyatomic molecule or ions.

1. Draw the Lewis structure of the compound or ion.
2. Count the total number of bonds between the atoms concerned.
3. Determine how many such groups of bonds there are.
4. Divide the total number of bonds by the number of groups or domains of bonds to get the bond order.

Bond Order Examples

Here are some examples of how to find bond order in molecules.

Example 1

Nitrogen molecule

The Lewis structure shows six electrons shared between the two atoms. It is a triple bond.

A Nitrogen Molecule Has a Triple Bond

There are three bonds over one group of atoms. Applying the formula,

{eq}\left ( Number of bonds\div \div Number of groups \right ) {/eq},

{eq}3\ast 1= 3 {/eq}

The bond order is three.

Example 2

Formaldehyde

Its formula is HCHO.

Bonds in a Formaldehyde Molecule

To determine the bond order between the carbon and oxygen atom, notice it is a double bond. The group or domain is just one. So the bond order is

{eq}2\div 1= 2 {/eq}

The bond order is two.

Example 3

Benzene.

Formula {eq}C_{6}H_{6} {/eq}

It is a resonance hybrid of the structure shown.

Resonance in Benzene

The bonds are between carbon and hydrogen, There are nine bonds and the number of domains is six. Applying the formula,

Total number of bonds / total number of domains,

{eq}9\div \ 6 = 1.5 {/eq}

The bond order in benzene is 1.5.

Example 4

Oxygen molecule {eq}O_{2} {/eq}

An oxygen molecule has two atoms.

By the Lewis structure, the two atoms form double bonds with two lone pairs of electrons on each atom. All electrons are in pairs.

Lewis Representation of Oxygen Molecule

But experiments prove oxygen is paramagnetic, showing the presence of unpaired electrons.

The molecular orbital theory explains this discrepancy.

There are two unpaired p electrons in the 2 pi antibonding orbital.

Molecular Orbital Diagram for Oxygen Molecule

Bond order = (Number of bonding electrons - Number of antibonding electrons) /2

Bonding electrons = 10

Antibonding electrons =6

{eq}(10 - 6)\div 2 = 2 {/eq} - 6) /2

The bond order is 2. But this explains the paramagnetism of oxygen. The Lewis structure does not.

Importance of Bond Order

Bond order is an essential tool for chemists. From bond order, the following information is available:

• It is easy to find the number of electrons involved in bond formation. The type of bond formed becomes evident. For example, a nitrogen molecule has a bond order of three. It has a triple bond. Six valence electrons are involved in the bond formation.
• It shows the stability of the bond. The higher the bond order, the more stable the bond is. A bond with a bond order of three is more stable than a bond with two. Bond order 3 > bond order 2 > bond order 1.
• A high bond order indicates a strong bond. So, the bond energy or enthalpy is high. A lot of energy is needed to break a bond with a high bond order.
• If the bond order is high, the bond length will be short. Bond order and bond length are inversely proportional.

Bond Type	Bond Energy kJ/mol	Bond Length (Angstrom Units)
Carbon - carbon single bond	163	1,54
Carbon - carbon double bond	418	1.34
Carbon - carbon triple bond	945	1.20

• Bond order increases across a period and decreases down a group. Across a period, the number of protons in the nucleus increases. The pull of the nucleus on the bonding electrons increases. As the elements become bigger down a group, the nuclear force on the bonding electrons decreases because of the increased distance.

Lesson Summary

Linus Pauling introduced the term bond order. Bond order definition is it is half the value of the difference between bonding electrons and antibonding electrons. Or it is the number of bonds per domain in a molecule. For the bond order formula of simple molecules and ions, draw the molecular orbital diagram and determine the number of bonding and antibonding electrons. Then calculate the bond order with the help of the formula. If the molecule is polyatomic, to find the bond order between two atoms, draw the Lewis structure, find the total number of bonds between the atoms and divide by the number of groups of such atoms. In the latter method, non-integral values are possible. Hybridization giving degenerate orbitals is common in many molecules. Resonance structures affect the bond order. Bond order values are helpful to chemists as they provide information about various parameters of the molecule. Bond lengths are inversely related to bond order values. A high bond order denotes a strong bond and a high bond enthalpy.