Question

The reusable booster rockets of the U.S. space shuttle employ a mixture of aluminum and ammonium perchlorate for fuel. A possible equation for this reaction is $$ \begin{aligned} 3 \mathrm{Al}(s)+3 \mathrm{NH}_{4} \mathrm{ClO}_{4}(s) & \longrightarrow \\ \mathrm{Al}_{2} \mathrm{O}_{3}(s)+\mathrm{AlCl}_{3}(s)+3 \mathrm{NO}(g)+6 \mathrm{H}_{2} \mathrm{O}(g) \end{aligned} $$ What mass of \(\mathrm{NH}_{4} \mathrm{ClO}_{4}\) should be used in the fuel mixture for every kilogram of \(\overline{\mathrm{Al}}\) ?

Short answer

The mass of ammonium perchlorate (NH4ClO4) required for every kilogram of aluminum (Al) in the fuel mixture is approximately 4350.72 grams, or 4.35 kg.

Step-by-step solution

Write the balanced chemical equation

The provided balanced chemical equation for the reaction between aluminum and ammonium perchlorate is: \(3 Al(s) + 3 NH_{4}ClO_{4}(s) \longrightarrow Al_{2}O_{3}(s) + AlCl_{3}(s) + 3 NO(g) + 6 H_{2}O(g)\)

Find the molar masses of Al and NH4ClO4

Next, we need to find the molar masses of aluminum (Al) and ammonium perchlorate (NH4ClO4). Using the periodic table, we have the following values: - Aluminum (Al): 26.98 g/mol - Ammonium perchlorate (NH4ClO4): 18.03 g/mol (N) + 4(1.01 g/mol) (H) + 35.45 g/mol (Cl) + 4(16.00 g/mol) (O) = 117.49 g/mol

Calculate the moles of Al

Now, we'll calculate the moles of aluminum (Al) in 1 kg of the substance: Given, mass of Al = 1 kg = 1000 g Moles = (mass) / (molar mass) Moles of Al = (1000 g) / (26.98 g/mol) = 37.067 moles

Calculate the moles of NH4ClO4 using the stoichiometric ratio

From the balanced chemical equation, we see that the stoichiometric ratio between aluminum (Al) and ammonium perchlorate (NH4ClO4) is 3:3, or simply 1:1. We have 37.067 moles of Al, so this ratio implies that we also need 37.067 moles of NH4ClO4 for the reaction.

Calculate the mass of NH4ClO4

Now that we have the moles of NH4ClO4, we can calculate its mass by using its molar mass: Mass of NH4ClO4 = (moles) × (molar mass) = 37.067 moles × 117.49 g/mol = 4350.72 g There you have it! We should use approximately 4350.72 grams, or 4.35 kg, of ammonium perchlorate (NH4ClO4) for every kilogram of aluminum (Al).

Key concepts

Chemical Reaction Equation

Understanding a chemical reaction begins with the chemical reaction equation, which conveys the substances involved and how they interact at the molecular level. The equation for the reaction of aluminum with ammonium perchlorate is:\(3 Al(s) + 3 NH_{4}ClO_{4}(s) \longrightarrow Al_{2}O_{3}(s) + AlCl_{3}(s) + 3 NO(g) + 6 H_{2}O(g)\)This notation is compact but informative—it tells us the reactants, the products, and the phases of each substance (solid, gas). Moreover, the coefficients indicate the stoichiometric ratios necessary for the reaction to occur with no leftover reactants, assuming it goes to completion.
Ensuring a balanced equation is crucial, as it reflects the law of conservation of mass where atoms are neither created nor destroyed.

Molar Mass Calculation

The molar mass of a substance is the mass of one mole of that substance, typically expressed in grams per mole (g/mol). It's a cornerstone of stoichiometry, allowing us to convert between the mass of a substance and the amount of substance in moles. To calculate it, you sum the atomic masses of each element, found on the periodic table, multiplied by the number of atoms of that element in the formula.For example:

• Aluminum (Al): 26.98 g/mol
• Ammonium perchlorate (NH4ClO4): (18.03 g/mol for N) + (4 × 1.01 g/mol for H) + (35.45 g/mol for Cl) + (4 × 16.00 g/mol for O) = 117.49 g/mol

These calculations allow us to proceed with further stoichiometric conversions by providing the 'currency exchange rate' between mass and moles.

Stoichiometric Ratio

The stoichiometric ratio is the proportion of reactants that react with each other as represented by their coefficients in the balanced chemical equation. It provides the roadmap for mole-to-mole conversions between reactants and products.
In our reaction, the stoichiometric ratio of aluminum to ammonium perchlorate is 3:3, or simplified to 1:1, meaning one mole of aluminum reacts with one mole of ammonium perchlorate.
It's essential for predicting how much of each reactant is required and what quantity of products will be produced, hence why getting the stoichiometric ratio correct is essential for practical applications such as fuel mixtures for rockets. It's a direct application of the mole concept and the balanced equation.

Mole-to-Mass Conversion

Mole-to-mass conversion is a two-step process that allows us to convert moles of a substance to mass, and vice versa, using the molar mass as the conversion factor. This is important when preparing the reactants for a chemical reaction, where precise mass measurements are typically required.
For converting moles to mass, as we did in finding the mass of NH4ClO4 needed, the formula is:\( \text{Mass of substance (g)} = \text{Moles of substance} \times \text{Molar mass of substance (g/mol)} \)Conversely, to find the number of moles when you have the mass, divide the mass by the molar mass. Knowing these conversions is vital for both laboratory work and industrial applications, as it ensures the correct proportions of reactants are used.