Question

Sometimes it is more convenient to measure the amount of a liquid reactant in volume units instead of mass units. Given that \(1.00 \mathrm{gal}\) is \(3.79 \mathrm{~L}\) and the density of methanol is \(0.793 \mathrm{~g} / \mathrm{mL}\), what mass of oxygen is needed to react with \(1.00\) gal of methanol? The balanced equation is $$ 2 \mathrm{CH}_{3} \mathrm{OH}(l)+3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g) $$

Short answer

The mass of oxygen needed to react with 1.00 gal of methanol can be found using unit conversion and reaction stoichiometry.

Step-by-step solution

Conversion of Volume of Methanol to Moles

Before calculations can be done according to the reaction stoichiometry, the amount of methanol needs to be converted to moles. First, the volume of methanol in gallons should be converted to liters, using the given conversion factor 1.00 gal = 3.79 L. The volume in liters then can be converted to mL. After that, the amount of methanol in mL can be converted to grams using the density of methanol (0.793 g/ml). Finally, the mass in grams can be converted to moles using the molar mass of methanol.

Use Reaction Stoichiometry to Calculate Required Mass of Oxygen

The balanced chemical equation indicates the mole ratio between methanol and oxygen. It shows that every 2 moles of methanol react with 3 moles of oxygen. Hence, to find the moles of oxygen needed, multiply the moles of methanol by the ratio 3/2. Consequently, the moles of oxygen can be converted to grams using the molar mass of molecular oxygen.

Conclude

Finally, when all steps are accurately followed, what can be found is the required mass of oxygen in grams that are needed to react with the given amount of methanol. This hence answers the posed question.

Key concepts

Density Calculation

Density is a measurement of how much mass is contained in a given volume. In chemistry, the density of a substance often acts as a bridge to convert between mass and volume units. To calculate density, use the formula: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]For example, in our exercise, the density of methanol is given as 0.793 g/mL. This means that each milliliter of methanol weighs 0.793 grams. To find the mass of methanol when its volume is known, multiply its volume by the density value:

• Convert gallons to liters: 1 gal = 3.79 L
• Convert liters to milliliters: 3.79 L = 3790 mL
• Calculate mass: 3790 mL \(\times\) 0.793 g/mL = mass in grams

This process allows you to move from the measurement of a liquid's volume to understanding its mass, crucial for stoichiometric calculations in reactions.

Molar Mass

Molar mass is the mass of one mole of a given substance, usually expressed in grams per mole (g/mol). It serves as a conversion factor between the mass of a substance and the moles of a substance. To find the molar mass, sum up the atomic masses of all atoms in a given molecular formula. For methanol (\(\text{CH}_3\text{OH}\)),

• Carbon (C): 12.01 g/mol
• Hydrogen (H): 1.01 g/mol (for each H atom, so 3 H from \(\text{CH}_3\) and 1 H from \(\text{OH}\))
• Oxygen (O): 16.00 g/mol

This results in a total molar mass of:\[ 12.01 + 4 \times 1.01 + 16.00 = 32.05 \text{ g/mol} \]Knowing the molar mass, you can convert the mass of a substance to the number of moles, which is necessary for stoichiometry. For example, once we find the mass of methanol, divide it by 32.05 g/mol to obtain the number of moles, crucial in linking the mass of methanol to the amount of oxygen needed in a reaction.

Chemical Reaction

Chemical reactions involve the breaking and forming of bonds between atoms to create new substances. They follow balanced chemical equations, which illustrate the conservation of mass and the stoichiometry of the reaction. In the equation given:\[ 2 \text{CH}_3\text{OH}(l) + 3 \text{O}_2(g) \rightarrow 2 \text{CO}_2(g) + 4 \text{H}_2\text{O}(g) \]The numbers in front of each chemical formula, known as coefficients, represent the mole ratio of the reactants and products. This equation tells us that 2 moles of methanol react with 3 moles of oxygen. Using this ratio, you can determine how much of each reactant is required, based on the balanced equation.

• Moles of methanol are converted using its molar mass.
• Use the mole ratio to find moles of oxygen needed: for every 2 moles of \(\text{CH}_3\text{OH}\), 3 moles of \(\text{O}_2\) are required.
• Convert moles of oxygen to grams using its molar mass (32.00 g/mol for \(\text{O}_2\)).

By following these stoichiometric principles, you can calculate how much oxygen is needed to completely react with the methanol used in the exercise.