Balance the chemical equation below using the smallest possible whole number stoichiometric coefficients. \( \mathrm{CH}_{3}\left(\mathrm{CH}_{2}\right)_{6} \mathrm{CH}_{3}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \)

To balance the chemical equation for the combustion of octane (\( \mathrm{C}_{8}\mathrm{H}_{18} \)), we start with the unbalanced equation: \[ \mathrm{C}_{8}\mathrm{H}_{18}(\mathrm{g}) + \mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{g}) + \mathrm{H}_{2}\mathrm{O}(\mathrm{g}) \] 1. **Count the number of carbon (C) and hydrogen (H) atoms in the reactants:** - There are 8 carbon atoms and 18 hydrogen atoms in octane. 2. **Balance the carbon atoms:** - Since there are 8 carbon atoms in octane, we need 8 \( \mathrm{CO}_{2} \) molecules: \[ \mathrm{C}_{8}\mathrm{H}_{18} + \mathrm{O}_{2} \rightarrow 8 \mathrm{CO}_{2} + \mathrm{H}_{2}\mathrm{O} \] 3. **Balance the hydrogen atoms:** - There are 18 hydrogen atoms in octane, which means we need 9 \( \mathrm{H}_{2}\mathrm{O} \) molecules (since each water molecule has 2 hydrogen atoms): \[ \mathrm{C}_{8}\mathrm{H}_{18} + \mathrm{O}_{2} \rightarrow 8 \mathrm{CO}_{2} + 9 \mathrm{H}_{2}\mathrm{O} \] 4. **Count the total number of oxygen atoms needed on the product side:** - From \( 8 \mathrm{CO}_{2} \), we get \( 8 \times 2 = 16 \) oxygen atoms. - From \( 9 \mathrm{H}_{2}\mathrm{O} \), we get \( 9 \times 1 = 9 \) oxygen atoms. - Total oxygen atoms needed = \( 16 + 9 = 25 \) oxygen atoms. 5. **Determine the number of \( \mathrm{O}_{2} \) molecules required:** - Since each \( \mathrm{O}_{2} \) molecule contains 2 oxygen atoms, we need \( \frac{25}{2} = 12.5 \) \( \mathrm{O}_{2} \) molecules. 6. **Convert to whole numbers:** - To eliminate the fraction, we can multiply the entire equation by 2: \[ 2 \mathrm{C}_{8}\mathrm{H}_{18} + 25 \mathrm{O}_{2} \rightarrow 16 \mathrm{CO}_{2} + 18 \mathrm{H}_{2}\mathrm{O} \] The balanced equation is: \[ \boxed{2 \mathrm{C}_{8}\mathrm{H}_{18} + 25 \mathrm{O}_{2} \rightarrow 16 \mathrm{CO}_{2} + 18 \mathrm{H}_{2}\mathrm{O}} \]