Chapter 9, Problem 9.93QA

Interpretation Introduction

To estimate:

The difference in ${∆H}_{rxn}^o$ values between the incomplete combustion of one mole of ethane to carbon monoxide and water vapor and the complete combustion of ethane to carbon dioxide and water vapor using average bond energies.

Answer to Problem 9.93QA

Solution:

The difference in ${∆H}_{rxn}^o$ values between the incomplete and complete combustion of one mole of ethane is $554 kJ$.

Explanation of Solution

1) Concept:

Bond energy is the amount of energy required when one mole of bonds in the gas phase is broken.

This can be used to calculate the heat of reaction (enthalpy change) by combining the $∆H$ values (bond energies) for bonds that are broken and bonds that are formed.

$∆H$ values are positive for bonds broken.

$∆H$ values are negative for bonds formed.

${∆H}_{rxn}^o=\sum {∆H}_{f bond breaking}^o-\sum {∆H}_{f bond forming}^o$

The balanced equation for the incomplete combustion of ethane to carbon monoxide and water vapor is

$2C_2{H}_6\left(g\right)+5O_2\left(g\right)\to 4CO\left(g\right)+{6H}_{2}O\left(g\right)$

The balanced equation for the complete combustion of ethane to carbon dioxide and water vapor is

$2C_2{H}_6\left(g\right)+7O_2\left(g\right)\to 4C{O}_2\left(g\right)+{6H}_{2}O\left(g\right)$

2) Formula:

${∆H}_{rxn}^o=\sum {∆H}_{f bond breaking}^o-\sum {∆H}_{f bond forming}^o$

3) Given:

Bond energies from Appendix 4: Table A4.1

Bonds	Bond energy ($kJ/mol$)
$C=O$	$799$
$C-H$	$413$
$O-H$	$463$
$C\equiv O$	$1072$
$C-C$	$348$
$O=O$	$498$

4) Calculations:

a) ${∆H}_{rxn}^o$ for incomplete combustion:

$2C_2{H}_6\left(g\right)+5O_2\left(g\right)\to 4CO\left(g\right)+{6H}_{2}O\left(g\right)$

	Bonds	Number of bonds (mol)	Bond energy (kJ/mol)	$\Delta H$
Bonds broken	$O=O$	$5$	$498 kJ/mol$	$5 mol \times 498 kJ/mol$
	$C-H$	$12$	$413 kJ/mol$	$12 mol \times 413 kJ/mol$
	$C-C$	$2$	$348 kJ/mol$	$2 mol \times 348 kJ/mol$
Bonds formed	$O-H$	$12$	$463 kJ/mol$	$-\left(12 mol \times 463 kJ/mol\right)$
	$C\equiv O$	$4$	$1072 kJ/mol$	$-\left(4 mol \times 1072 kJ/mol\right)$

${∆H}_{rxn}^o=\left[\left(5 mol \times 498 kJ/mol\right)+\left(12 mol \times 413 kJ/mol\right)+\left(2 mol \times 348 kJ/mol\right)\right]-\left[\left(12 mol \times 463 kJ/mol\right)+\left(4 mol \times 1072 kJ/mol\right)\right]$

${∆H}_{rxn}^o=8142 kJ-9844 kJ$

${∆H}_{rxn}^o=-1702 kJ$

${∆H}_{rxn}^o$ for 1 mol of ethane is $-851 kJ$.

b) ${∆H}_{rxn}^o$ for complete combustion:

$2C_2{H}_6\left(g\right)+7O_2\left(g\right)\to 4C{O}_2\left(g\right)+{6H}_{2}O\left(g\right)$

	Bonds	Number of bonds (mol)	Bond energy (kJ/mol)	$\Delta H$
Bonds broken	$O=O$	$7$	$498 kJ/mol$	$7 mol \times 498 kJ/mol$
	$C-H$	$12$	$413 kJ/mol$	$12 mol \times 413 kJ/mol$
	$C-C$	$2$	$348 kJ/mol$	$2 mol \times 348 kJ/mol$
Bonds formed	$O-H$	$12$	$463 kJ/mol$	$-\left(12 mol \times 463 kJ/mol\right)$
	$C=O$	$8$	$799 kJ/mol$	$-\left(8 mol \times 799 kJ/mol\right)$

${∆H}_{rxn}^o=\left[\left(7 mol \times 498 kJ/mol\right)+\left(12 mol \times 413 kJ/mol\right)+\left(2 mol \times 348 kJ/mol\right)\right]-\left[\left(12 mol \times 463 kJ/mol\right)+\left(8 mol \times 799 kJ/mol\right)\right]$

${∆H}_{rxn}^o=9138 kJ-11948 kJ$

${∆H}_{rxn}^o=-2810 kJ$

${∆H}_{rxn}^o$ for 1 mol of ethane is $-1405 kJ$.

The difference between ${∆H}_{rxn}^o$ of incomplete and complete combustion of 1 mole of ethane is

${∆H}_{rxn}^o=1405 kJ-851 kJ=554 kJ$

Conclusion:

We calculated ${∆H}_{rxn}^o$ for 1 mole of ethane for both incomplete and complete combustion and then took their difference.