Question

The combustion of gasoline produces carbon dioxide and water. Assume gasoline to be pure octane \(\left(\mathrm{C}_{8} \mathrm{H}_{18}\right)\) and calculate the mass ( in \(\mathrm{kg}\) ) of carbon dioxide that is added to the atmosphere per 1.0 \(\mathrm{kg}\) of octane burned. (Hint: Begin by writing a balanced equation for the com- bustion reaction.

Short answer

3.08044 kg of CO2 is produced per 1.0 kg of octane burned.

Step-by-step solution

Write the balanced chemical equation for the combustion of octane

The combustion of octane can be represented by the chemical equation: \[\mathrm{C}_{8}\mathrm{H}_{18} + \mathrm{O}_{2} \rightarrow \mathrm{CO}_{2} + \mathrm{H}_{2}\mathrm{O}\]To balance the equation, ensure that the number of atoms of each element is the same on both sides of the equation. The balanced equation is:\[2 \mathrm{C}_{8}\mathrm{H}_{18} + 25 \mathrm{O}_{2} \rightarrow 16 \mathrm{CO}_{2} + 18 \mathrm{H}_{2}\mathrm{O}\]

Calculate moles of CO2 produced per mole of octane

From the balanced chemical equation, 2 moles of octane (\(\mathrm{C}_{8}\mathrm{H}_{18}\)) produce 16 moles of carbon dioxide (\(\mathrm{CO}_{2}\)). Therefore, 1 mole of octane will produce 8 moles of CO2.

Convert moles of octane to mass

The molar mass of octane is \(8 \times \mathrm{C} + 18 \times \mathrm{H} = 8 \times 12.01 \, \mathrm{g/mol} + 18 \times 1.008 \, \mathrm{g/mol} = 114.23 \, \mathrm{g/mol}\).Therefore, the mass of 1.0 kg of octane is equivalent to \(\frac{1000 \, \mathrm{g}}{114.23 \, \mathrm{g/mol}} \approx 8.751 \, \mathrm{moles}\) of octane.

Calculate moles of CO2 produced from the mass of octane

1 kg of octane is 8.751 moles. Each mole produces 8 moles of CO2, so \(8.751 \, \mathrm{moles} \times 8 \, \frac{\mathrm{moles \, CO}_{2}}{\mathrm{mole \, octane}} = 70.008 \, \mathrm{moles \, CO}_{2}\).

Convert moles of CO2 to mass

The molar mass of CO2 is \(\mathrm{C} + 2 \times \mathrm{O} = 12.01 \, \mathrm{g/mol} + 2 \times 15.999 \, \mathrm{g/mol} = 44.009 \, \mathrm{g/mol}\).So, the mass of CO2 produced is \(70.008 \, \mathrm{moles} \times 44.009 \, \mathrm{g/mol} = 3080.44 \, \mathrm{g} \) or \(3.08044 \, \mathrm{kg}\).

Key concepts

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. By understanding the principles of stoichiometry, students can predict the amounts of substances consumed and produced in any given reaction.

This concept is essential when evaluating chemical reactions, such as the combustion of gasoline. The stoichiometric calculations begin with a balanced chemical equation, which serves as a recipe indicating how many moles of each reactant are needed to produce a certain number of moles of each product.

In the combustion of octane, stoichiometry allows us to calculate the mass of carbon dioxide produced from a known mass of octane. Such calculations are crucial in evaluating the environmental impact of burning fuels and finding ways to reduce greenhouse gas emissions.

Chemical Reaction Balancing

Chemical reaction balancing is a key skill in chemistry, ensuring that a chemical equation adheres to the Law of Conservation of Mass. This means that the number of atoms for each element must be the same on both sides of the equation because matter cannot be created or destroyed in an ordinary chemical reaction.

To balance a chemical equation, like the combustion of octane, coefficients are adjusted in front of the reactant and product formulas. For instance, the balanced equation for octane combustion shows a precise ratio of octane to oxygen molecules needed to produce carbon dioxide and water without leaving any atoms unaccounted for. Without a balanced equation, stoichiometric calculations cannot begin since these rely on correct proportions.

Mole Concept

The mole concept is a fundamental chemical principle that helps scientists quantify substances. A mole is a unit that represents a massive number—Avogadro's number—of particles, which is approximately 6.022 x 10²³.

Understanding the mole concept is essential when performing stoichiometric calculations. It enables a direct comparison between the number of particles in a given mass of different substances, and when combined with molar mass, it bridges the gap between the mass of a substance and the amount of substance in moles.

In the case of the octane combustion, the mole concept allows us to translate the mass of octane and carbon dioxide involved in the reaction into moles, which is crucial for determining how much product is formed from a given quantity of reactant.

Molar Mass Calculation

Molar mass is the mass of one mole of a given substance and is expressed in grams per mole (g/mol). It is calculated by summing the masses of the individual atoms within the molecule according to the periodic table.

For instance, the molar mass of octane (C₈H₁₈) involves multiplying the mass of carbon and hydrogen by their respective number of atoms, and summing them together to find the total molar mass of octane. Similarly, we calculate the molar mass of carbon dioxide (CO₂) to convert moles of carbon dioxide to grams, which then allows us to find the mass of carbon dioxide released upon the combustion of octane.

This calculation is pivotal in stoichiometric equations and environmental studies where weighing the impact of emissions is essential. Hence, mastering the calculation of molar mass is invaluable for both laboratory chemists and environmental scientists alike.