Q. how to calculate average bond enthalpy

Definition. The average bond enthalpy of a given bond is the average energy required to break one mole of that bond in the gas phase, usually reported in kilojoules per mole. To estimate the enthalpy change of a reaction using average bond enthalpies, use the rule that the reaction enthalpy is approximately the energy needed to break all bonds in the reactants minus the energy released when new bonds are formed in the products.

General formula. If you list every bond that is broken in the reactants and every bond that is formed in the products, the estimate is

\[
\Delta H_{\text{rxn}} \approx \sum \text{(bond enthalpies of bonds broken)} \;-\; \sum \text{(bond enthalpies of bonds formed)}.
\]

Step 1. Write and balance the chemical equation. For a specific example, consider the combustion of methane. I will write the balanced equation in words so that no arrow symbol is used. Methane plus oxygen yields carbon dioxide plus water. The balanced stoichiometry is one mole of CH4 plus two moles of O2 yields one mole of CO2 plus two moles of H2O.

Step 2. Identify which bonds are broken and which are formed, and count how many of each. For CH4 the bonds broken are four C–H bonds. For 2 O2 the bonds broken are two O=O bonds. For the products, CO2 contains two C=O double bonds. For 2 H2O there are four O–H bonds (each H2O has two O–H bonds, and there are two H2O molecules).

Step 3. Use a table of average bond enthalpies. Typical approximate values (in kJ mol^{-1}) are:

\(D_{\text{C–H}} = 413 \ \text{kJ mol}^{-1}\).

\(D_{\text{O}= \text{O}} = 498 \ \text{kJ mol}^{-1}\).

\(D_{\text{C}=\text{O}} = 799 \ \text{kJ mol}^{-1}\) (in CO2, per C=O double bond).

\(D_{\text{O–H}} = 463 \ \text{kJ mol}^{-1}\).

Step 4. Compute the total energy required to break all bonds in the reactants. Sum each bond enthalpy multiplied by its count. For the reactants:

\[
\text{Energy to break reactant bonds} = 4\cdot D_{\text{C–H}} \;+\; 2\cdot D_{\text{O}=\text{O}}.
\]

Substitute the numerical values:

\[
\text{Energy to break reactant bonds} = 4\cdot 413 \;+\; 2\cdot 498 = 1652 \;+\; 996 = 2648 \ \text{kJ}.
\]

Step 5. Compute the total energy released when product bonds form. For the products:

\[
\text{Energy released by forming product bonds} = 2\cdot D_{\text{C}=\text{O}} \;+\; 4\cdot D_{\text{O–H}}.
\]

Substitute the numerical values:

\[
\text{Energy released by forming product bonds} = 2\cdot 799 \;+\; 4\cdot 463 = 1598 \;+\; 1852 = 3450 \ \text{kJ}.
\]

Step 6. Apply the general formula to estimate the reaction enthalpy. The estimated enthalpy change is the energy required to break bonds minus the energy released on formation of new bonds.

\[
\Delta H_{\text{rxn}} \approx 2648 \;-\; 3450 = -802 \ \text{kJ}.
\]

Step 7. Interpret the sign and accuracy. The negative sign means the reaction is exothermic, releasing about 802 kJ per mole of CH4 burned under this estimation. Note that this is an approximation because average bond enthalpies are average values taken from many different molecules and do not perfectly represent the exact electronic environment in a particular molecule. For methane combustion the tabulated standard enthalpy of combustion is about -890 kJ mol^{-1}, so the bond enthalpy estimate is reasonably close but not exact.

How to find an unknown average bond enthalpy. If a reaction enthalpy is known experimentally and every bond enthalpy except one is known, you can rearrange the general formula to solve for the unknown bond enthalpy. Suppose the unknown bond type has total count n in the bonds broken or formed. Rearrangement gives

\[
D_{\text{unknown}} \approx \frac{\sum \text{(known bond enthalpies of bonds broken)} \;-\; \sum \text{(known bond enthalpies of bonds formed)} \;-\; \Delta H_{\text{rxn}}}{n},
\]

where you treat bonds formed by adding their energies on the product side and bonds broken on the reactant side as shown earlier. Be careful with the sign of ΔH_{\text{rxn}} when substituting numbers.

Summary checklist for calculating an average bond enthalpy estimate of a reaction enthalpy.

1. Balance the chemical equation. 2. List all bonds broken in reactants and count them. 3. List all bonds formed in products and count them. 4. Obtain average bond enthalpies from a table. 5. Compute total energy to break reactant bonds. 6. Compute total energy released when product bonds form. 7. Use \(\Delta H_{\text{rxn}} \approx \sum D_{\text{broken}} – \sum D_{\text{formed}}\). 8. If needed, solve algebraically for an unknown bond enthalpy.