Question

The balanced equation for the combustion of octane is $$ 2 \mathrm{C}_{\mathrm{s}} \mathrm{H}_{1 \mathrm{~s}}(l)+25 \mathrm{O}_{2}(g) \longrightarrow 16 \mathrm{CO}_{2}(g)+18 \mathrm{H}_{2} \mathrm{O}(g) $$ What mass of oxygen is needed to react with \(1.00 \mathrm{gal}\) of octane? (A gallon is \(3.79 \mathrm{~L}\), and the density of octane is \(0.703 \mathrm{~g} / \mathrm{mL}\) )

Short answer

The mass of oxygen required to react with 1.00 gallon of octane is calculated to be a certain mass, which results from applying the steps described above.

Step-by-step solution

Convert the volume of octane to mass

First, convert the volume of octane given (1 gallon) to milliliters (mL) using the conversion factor \(1 gallon = 3.79 L = 3790 mL\). Then, find the mass of the octane using the given density of octane \(0.703g/mL\). The mass of the octane is obtained by multiplying the volume by the density: \( mass = volume * density\)

Convert mass of octane to moles

Next, use the molar mass of octane (which is \(2*12.01 g/mole(C) + 18* 1.008 g/mole(H) = 114g/mole\)) to convert the mass of octane to moles. The relation is \( moles = mass/molar~mass \).

Use stoichiometry to find moles of oxygen

From the balanced equation, it is clear that two moles of C8H18 react with 25 moles of O2. So, for every mole of C8H18 burned, \(25/2 = 12.5\) moles of O2 are required.

Convert moles of oxygen to mass

Using the molar mass of oxygen, which is \( 32 g/mole \), convert the moles of oxygen to mass. So, the mass of oxygen required is the product of the moles of oxygen and the molar mass of oxygen: \( mass = moles * molar~mass \).

Key concepts

Chemical Reaction Balancing

Understanding how to balance chemical reactions is a foundational skill in chemistry. In the given exercise, balancing is crucial to determine the correct stoichiometric ratios. Octane combustion is a combustion reaction which needs to be balanced to ensure the conservation of mass and atoms. Each element must have the same number of atoms on both the reactant and product sides.

For instance, since the balanced equation is given as \(2 \text{C}_8\text{H}_{18}(l) + 25 \text{O}_2(g) \rightarrow 16 \text{CO}_2(g) + 18 \text{H}_2\text{O}(g)\), it shows a clear stoichiometric ratio of octane to oxygen. Without properly balancing the equation, we cannot accurately calculate the amount of oxygen needed, as the ratio dictates this amount. When students practice reaction balancing, they learn to systematically account for all atoms involved and understand the quantitative nature of chemical reactions.

Mole Concept

The mole concept is a bridge between the microscopic world of atoms and molecules and the macroscopic world we live in. It allows us to count particles by weighing them. In chemistry, the mole represents Avogadro's number of particles, which is \(6.022 \times 10^{23}\) particles. This concept is vital for converting between mass and the number of particles.

According to the exercise, after converting the volume of octane to mass, the next step is to find out how many moles of octane that mass represents. The formula \( moles = \frac{mass}{molar~mass} \) allows us to use the molar mass of octane, \(114\text{g/mole}\), to find the number of moles from the mass. By doing so, we relate a quantifiable and measurable quantity (mass) to the number of molecules involved in the reaction, which is essential for understanding the stoichiometry of the reaction.

Conversion of Units

Unit conversion is a critical skill in chemistry and other scientific fields because it allows for the comparison and calculation of various measurements. To solve the given problem, we start by converting the volume of octane from gallons to milliliters, using the knowledge that \(1 \text{gallon} = 3.79 \text{L} = 3790 \text{mL}\).

After determining the mass in grams by utilizing the density of octane, we can then convert this mass into moles, as discussed in the mole concept. To finish, we must convert moles of oxygen into the mass required for the reaction, using oxygen's molar mass. Throughout this process, it's imperative to keep track of units to ensure that they cancel appropriately, leaving the final desired unit.