How many liters of \(\mathrm{O}_{2},\) density \(1.43 \mathrm{g} / \mathrm{L},\) are needed for the complete combustion of \(1.00 \mathrm{L} \mathrm{C}_{8} \mathrm{H}_{18},\) density 0.700 \(\mathrm{g} / \mathrm{mL} ?\)

Understanding the fundamentals of chemical equation balancing is essential for solving problems in chemistry, including combustion reactions. A balanced chemical equation represents a reaction with the same number of atoms for each element on both the reactant and product sides. This balance adheres to the law of conservation of mass, which states that matter cannot be created or destroyed in a chemical reaction.

In our example with the combustion of octane (\(C_8H_{18}\)), the balanced equation is critical for predicting the amount of each reagent and product involved. The equation tells us that 2 moleculs of octane react with 25 molecules of diatomic oxygen to produce 16 molecules of carbon dioxide and 18 molecules of water. This stoichiometric relationship allows us to compute quantities needed or produced from the combustion, which is precisely what's needed to solve the given problem.

The mole concept is a cornerstone of quantitative understanding in chemistry. One mole is a unit that represents Avogadro's number (\(6.022 \times 10^{23}\)) of particles, which could be atoms, molecules, ions, or electrons. When we speak about the mole, we are dealing with a standardized number of particles that can be used to translate between atomic scale events and measurable, real-world quantities.

In our combustion problem, we use the molar mass of octane to convert the mass of a known volume into moles. The molar mass is the weight of one mole of a substance and is expressed in grams per mole. Once we determine the number of moles of octane, the balanced chemical equation provides the mole ratio needed to calculate the required moles of oxygen. This step is vital as it represents the bridge between qualitative chemical formulas and the quantitative aspect required to perform calculations.

When dealing with gases, the gas laws and density are key concepts for converting between the moles of a gas and its volume. Density is defined as mass per unit volume of a substance. For gases, the density can vary greatly with temperature and pressure, but under controlled conditions, it can be used as a conversion factor.

In the given problem, oxygen gas has a density of 1.43 g/L, which helps us move from mass to volume. We first calculated the amount of oxygen required for the reaction in moles using stoichiometry and then translated it into a mass. Subsequently, utilizing the known density of the gas, we converted the mass into a volume. This follows from the ideal gas law equation (PV = nRT) where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature. Although pressure and temperature are not explicitly mentioned, their effects are encapsulated in the density value provided for the oxygen gas.