Question

Ammonium nitrate can be used as an effective explosive because it decomposes into a large number of gaseous products. At a sufficiently high temperature, ammonium nitrate decomposes into nitrogen, oxygen, and steam. (a) Write a balanced equation for the decomposition of ammonium nitrate. (b) If \(1.00 \mathrm{~kg}\) of ammonium nitrate is sealed into a \(50.0\) - \(\mathrm{L}\) steel drum and heated to \(787^{\circ} \mathrm{C}\), what is the pressure in the drum, assuming \(100 \%\) decomposition?

Short answer

Answer: The pressure in the drum after 100% decomposition of ammonium nitrate at 787℃ is 73.16 atm.

Step-by-step solution

Write the balanced equation for the decomposition of ammonium nitrate

The decomposition of ammonium nitrate produces nitrogen, oxygen, and water vapor (steam). The chemical equation for this reaction is: NH4NO3(s) -> N2(g) + 2H2O(g) + 1/2O2(g) This equation is already balanced.

Convert mass of ammonium nitrate to moles

To calculate the pressure in the drum, we first need the number of moles of each gas. Since we know the mass of ammonium nitrate (1.00 kg), we can convert this to moles using its molar mass. Molar mass of NH4NO3 = 14.01 (N) + 1.01 (H) * 4 + 16.00 (O) * 3 = 80.05 g/mol 1.00 kg of ammonium nitrate = 1000 g Number of moles = (1000 g) / (80.05 g/mol) = 12.49 mol

Calculate the moles of each gas produced

Using the balanced chemical equation, we can determine the number of moles of each gas produced from the decomposition of ammonium nitrate: Moles of N2 = 12.49 mol Moles of H2O = 12.49 mol * 2 = 24.98 mol Moles of O2 = 12.49 mol * 1/2 = 6.245 mol Total moles of gases = 12.49 + 24.98 + 6.245 = 43.71 mol

Calculate the pressure using the Ideal Gas Law

To calculate the pressure, we will use the Ideal Gas Law formula: PV = nRT Where P is the pressure, V is the volume (50.0 L), n is the number of moles (43.71 mol), R is the universal gas constant (0.0821 L atm/mol K), and T is the temperature (787℃ + 273.15 = 1060.15 K). Solving for P: P = nRT/V = (43.71 mol) * (0.0821 L atm/mol K) * (1060.15 K) / (50.0 L) P = 73.16 atm The pressure in the drum after 100% decomposition of ammonium nitrate at 787℃ is 73.16 atm.

Key concepts

Chemical Reaction Balancing

Understanding the balance of chemical equations is essential to the study of chemistry. In chemical reactions, the law of conservation of mass dictates that atoms are neither created nor destroyed. This means that the same number and type of atoms must be present before and after the reaction.

For the decomposition of ammonium nitrate, the chemical reaction is given as:
NH4NO3(s) -> N2(g) + 2H2O(g) + 1/2O2(g)
This reaction is balanced when the number of atoms for each element is equal on both sides of the equation. For nitrogen, we have one nitrogen in NH4NO3, which yields one N2 molecule on the product side. In the case of oxygen, we have three oxygen atoms in NH4NO3 and a total of three oxygen atoms on the right side when combining the oxygen in H2O and O2. Hydrogen atoms are also balanced with four hydrogen atoms in NH4NO3 turning into four hydrogen atoms in 2H2O molecules.

When balancing equations, it's important to adjust coefficients (the numbers before the chemical formulas), and sometimes add fractional coefficients—as we see with the 1/2O2. Balancing is a critical step because it ensures the stochiometry of the reaction is correct, which is necessary for subsequent calculations, such as mole-to-mole conversions.

Moles and Molar Mass Conversion

Moles and molar mass are foundational concepts in chemistry that deal with quantifying substances. A mole is a unit that represents 6.022 x 10^23 particles (Avogadro's number), and the molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol).

For the given problem, the molar mass of ammonium nitrate (NH4NO3) is calculated by summing the atomic masses of its constituent atoms:
Molar mass of NH4NO3 = 14.01 (N) + 4 * 1.01 (H) + 3 * 16.00 (O) = 80.05 g/mol
To convert the mass of ammonium nitrate into moles, the formula used is:
Number of moles = Mass (g) / Molar mass (g/mol)
In this case, we convert the given 1.00 kg (or 1000 g) of ammonium nitrate to moles, resulting in 12.49 moles. This conversion is pivotal as it allows us to determine how much of each gas is produced during the decomposition, based on the stoichiometry provided by the balanced chemical equation. Understanding moles and molar mass conversion is critical when predicting the outcome of a chemical reaction and is integral to experiments and industrial applications.

Ideal Gas Law Calculation

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV=nRT, where P represents pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. This law is a crucial part of chemical thermodynamics and is often used to predict the behavior of gases under various conditions.

To calculate the pressure of the gases produced from the decomposition of ammonium nitrate in the problem, we first need to convert the temperature to Kelvin by adding 273.15 to the Celsius temperature, resulting in 1060.15 K. The volume of the steel drum is given as 50.0 L.
Using the Ideal Gas Law:P = \(\frac{nRT}{V}\) = \(\frac{43.71 mol \times 0.0821 \frac{L \cdot atm}{mol \cdot K} \times 1060.15 K}{50.0 L}\)
The calculation yields a pressure of 73.16 atm for the gases inside the drum, assuming 100% decomposition and ideal gas behavior. It's important to note that real gases may not adhere strictly to the Ideal Gas Law, but it often serves as a good approximation for many gases under a range of temperatures and pressures.