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 $$ 3 \mathrm{Al}(s)+3 \mathrm{NH}_{4} \mathrm{ClO}_{4}(s) \longrightarrow $$ $$ \mathrm{Al}_{2} \mathrm{O}_{3}(s)+\mathrm{AlCl}_{3}(s)+3 \mathrm{NO}(g)+6 \mathrm{H}_{2} \mathrm{O}(g) $$ What mass of \(\mathrm{NH}_{4} \mathrm{ClO}_{4}\) should be used in the fuel mixture for every kilogram of Al?

Short answer

To determine the mass of ammonium perchlorate (NH₄ClO₄) needed for every kilogram of aluminum (Al), we calculate the mass ratio between Al and NH₄ClO₄ using their molar masses and the balanced chemical equation. We find that for 1 kg of Al, approximately 4.36 kg of NH₄ClO₄ is needed in the fuel mixture.

Step-by-step solution

Identify the balanced chemical equation

The balanced chemical equation for the reaction between aluminum (Al) and ammonium perchlorate (NH₄ClO₄) is given by: \[ 3 \mathrm{Al}(s)+3 \mathrm{NH}_{4} \mathrm{ClO}_{4}(s) \longrightarrow \mathrm{Al}_{2} \mathrm{O}_{3}(s)+\mathrm{AlCl}_{3}(s)+3 \mathrm{NO}(g)+6 \mathrm{H}_{2} \mathrm{O}(g) \]

Calculate the mole ratio between Al and NH₄ClO₄

From the balanced chemical equation, we can see that the mole ratio between Al and NH₄ClO₄ is 3:3, which can be simplified to 1:1. This means that for every 1 mole of aluminum, 1 mole of ammonium perchlorate is required.

Find the molar masses of Al and NH₄ClO₄

We need to determine the molar masses of Al and NH₄ClO₄ to convert the mole ratio into a mass ratio. - The molar mass of Al (Aluminum) is 26.98 g/mol. - The molar mass of NH₄ClO₄ (Ammonium perchlorate) can be calculated as follows: - N (Nitrogen) has a molar mass of 14.01 g/mol. - H (Hydrogen) has a molar mass of 1.008 g/mol. - Cl (Chlorine) has a molar mass of 35.45 g/mol. - O (Oxygen) has a molar mass of 16.00 g/mol. NH₄ClO₄ = (1 × 14.01) + (4 × 1.008) + (1 × 35.45) + (4 × 16.00) = 117.53 g/mol

Calculate the mass ratio between Al and NH₄ClO₄

We know that the mole ratio between Al and NH₄ClO₄ is 1:1. Now, we can calculate the mass ratio using the molar masses of Al and NH₄ClO₄: For every 1 mole of Al (26.98 g), we need 1 mole of NH₄ClO₄ (117.53 g). So, the mass ratio between aluminum and ammonium perchlorate is 26.98:117.53.

Determine the mass of NH₄ClO₄ needed for 1 kg of Al

We need to calculate the mass of NH₄ClO₄ required for every kilogram (1000 grams) of Al. We can use the mass ratio of Al and NH₄ClO₄ to determine this: For 26.98 g of Al, we need 117.53 g of NH₄ClO₄. For 1 g of Al, we need \(\frac{117.53}{26.98} \) g of NH₄ClO₄. For 1000 g (1 kg) of Al, we need \(\frac{117.53}{26.98} \times 1000 \)g of NH₄ClO₄. Thus, the required mass of NH₄ClO₄ for 1 kg of Al is approximately 4355.41 g or 4.36 kg.

Key concepts

Molar Mass Calculation

The concept of molar mass is central to understanding stoichiometry in chemical reactions. Molar mass is the mass of a given substance (chemical element or chemical compound) divided by the amount of substance in moles. It is expressed in units of grams per mole (g/mol). Calculating the molar mass involves adding up the atomic masses of each element present in a compound.

For example, ammonium perchlorate (\( \text{NH}_4\text{ClO}_4 \)) consists of nitrogen (N), hydrogen (H), chlorine (Cl), and oxygen (O). The atomic masses are as follows:

• Nitrogen (N) = 14.01 g/mol
• Hydrogen (H) = 1.008 g/mol
• Chlorine (Cl) = 35.45 g/mol
• Oxygen (O) = 16.00 g/mol

To calculate the molar mass of ammonium perchlorate:

• 1 nitrogen: \(1 \times 14.01 = 14.01\)
• 4 hydrogens: \(4 \times 1.008 = 4.032\)
• 1 chlorine: \(1 \times 35.45 = 35.45\)
• 4 oxygens: \(4 \times 16.00 = 64.00\)

Adding these values gives a total molar mass of:\[ \text{NH}_4\text{ClO}_4 = 14.01 + 4.032 + 35.45 + 64.00 = 117.53 \text{ g/mol} \] This value is essential for converting the chemical equation into practical applications, such as the formulation of rocket fuels.

Balanced Chemical Equation

A balanced chemical equation is one in which the number of atoms for each element is the same on both sides of the equation. This reflects the law of conservation of mass, which states that mass cannot be created or destroyed in a chemical reaction. Balancing equations is a fundamental skill in chemistry that ensures stoichiometric calculations are accurate.

Consider the reaction between aluminum (\( \text{Al} \)) and ammonium perchlorate (\( \text{NH}_4\text{ClO}_4 \)):\[ 3 \text{Al}(s) + 3 \text{NH}_4\text{ClO}_4(s) \rightarrow \text{Al}_2\text{O}_3(s) + \text{AlCl}_3(s) + 3 \text{NO}(g) + 6 \text{H}_2\text{O}(g) \] In this equation, the coefficients (the numbers in front of chemical formulas) are adjusted to ensure that the same number of each type of atom appears on both sides.

For example:

• Aluminum (Al) appears 3 times on both sides.
• Ammonium perchlorate's nitrogen (\( \text{N} \)), hydrogen (\( \text{H} \)), chlorine (\( \text{Cl} \)), and oxygen (\( \text{O} \)) atoms also match in number from reactants to products.

Balancing the equation is crucial to accurately calculate moles and masses in stoichiometric problems and ensure the reaction is feasible for practical applications.

Mass Ratio Calculation

Mass ratio calculations are important to connect mole ratios from balanced equations to the actual masses of substances used in reactions. This involves using the molar masses of the reactants and products. In our scenario, the equation indicates a 1:1 mole ratio between aluminum (\( \text{Al} \)) and ammonium perchlorate (\( \text{NH}_4\text{ClO}_4 \)).

To convert this mole ratio to a mass ratio, we use their respective molar masses:

• Molar mass of Aluminum (\( \text{Al} \)) = 26.98 g/mol
• Molar mass of Ammonium perchlorate (\( \text{NH}_4\text{ClO}_4 \)) = 117.53 g/mol

Thus, for every mole of aluminum that reacts, you need one mole of ammonium perchlorate, and their mass ratio becomes:\[\frac{26.98 \text{ g Al}}{117.53 \text{ g NH}_4\text{ClO}_4} = \frac{1 \text{ Al}}{4.36 \text{ NH}_4\text{ClO}_4}\] This means for every 26.98 grams of aluminum, 117.53 grams of ammonium perchlorate is needed. Knowing the mass ratio allows for scaling the reaction up or down based on desired quantities, which is particularly useful in industrial applications like fuel synthesis.

Rocket Fuel Composition

The reaction studied here is directly related to the composition of solid rocket fuels, particularly those employed in the space shuttle's reusable booster rockets. Solid rocket propellants, like the one formed by aluminum and ammonium perchlorate, are designed to produce thrust by rapidly converting stored chemical energy into heat and gas expansion.

The primary roles of the components in this fuel system are:

• Aluminum (\( \text{Al} \)) serves as a fuel. It burns at a high temperature, releasing a significant amount of energy, which contributes to thrust.
• Ammonium perchlorate (\( \text{NH}_4\text{ClO}_4 \)) acts as an oxidizer. It provides the necessary oxygen to allow the aluminum to combust in a low-oxygen environment.

This kind of blend is what creates the high-energy release needed for propulsion. The balanced chemical reaction, mass ratios, and composition knowledge are vital in formulating efficient and effective rocket propellants, given the demanding requirements of space exploration. The design must be precisely tailored to ensure optimal combustion and maximizing thrust, illustrating the practical significance of stoichiometry in real-world applications.