Question

When ammonium nitrate explodes, nitrogen, steam, and oxygen gas are produced. If the explosion is carried out by heating one kilogram of ammonium nitrate sealed in a rigid bomb with a volume of one liter, what is the total pressure produced by the gases before the bomb ruptures? Assume that the reaction goes to completion and that the final temperature is \(500^{\circ} \mathrm{C}\) (3 significant figures).

Short answer

Answer: The total pressure produced by the gases is approximately 2.79 x 10^7 Pa.

Step-by-step solution

Write the balanced equation for the ammonium nitrate reaction

The reaction of ammonium nitrate decomposing is given by: NH\(_4\)NO\(_3\) \(\rightarrow\) N\(_2\) + 2H\(_2\)O + \(\frac{1}{2}\)O\(_2\)

Determine the number of moles of ammonium nitrate

We are given 1 kilogram (1000 grams) of ammonium nitrate. Using the molar mass of NH\(_4\)NO\(_3\) (80.04 g/mol), we can find the number of moles: Moles (NH\(_4\)NO\(_3\)) = \(\frac{1000 \text{ g}}{80.04 \text{ g/mol}} \approx 12.49 \text{ mol}\)

Determine the total number of moles of gases produced

From the balanced equation, each mole of NH\(_4\)NO\(_3\) produces 1 mole of N\(_2\), 2 moles of H\(_2\)O, and 0.5 moles of O\(_2\). Thus, the total moles of gases produced are: Moles of gases = moles of N\(_2\) + moles of H\(_2\)O + moles of O\(_2\) Moles of gases = \(1\cdot12.49 + 2\cdot12.49 + 0.5\cdot12.49 \approx 43.73 \text{ mol}\)

Convert the final temperature from Celsius to Kelvin

We need to convert the final temperature from Celsius to Kelvin in order to use the Ideal Gas Law equation. Temperature (K) = \(500^{\circ}\mathrm{C}\) + 273.15 = 773.15 K

Calculate the total pressure produced by the gases

Now we have all the information needed to calculate the pressure using the Ideal Gas Law equation: \(PV=nRT\) Where \(P\) is the pressure, \(V\) is the volume (1 L = 0.001 m³), \(n\) is the number of moles (43.73 mol), \(R\) is the gas constant (\(8.314 \text{ J/(mol·K)}\)), and \(T\) is the temperature (773.15 K). We'll solve for pressure: \(P = \frac{nRT}{V} = \frac{43.73 \text{ mol} \cdot 8.314 \text{ J/(mol·K)} \cdot 773.15 \text{ K}}{0.001 \text{ m³}}\) \(P \approx 2.79 \times 10^7 \text{ Pa}\) So the total pressure produced by the gases before the bomb ruptures is approximately \(2.79 \times 10^7 \text{ Pa}\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a critical concept in chemistry and physics, providing a relationship between the pressure (P), volume (V), temperature (T), and the number of moles (n) of an ideal gas. Represented by the equation
\( PV = nRT \),
where R is the ideal gas constant, the law allows us to predict the behavior of a gas under different conditions. For instance, when examining the aftermath of an ammonium nitrate explosion, we can use the Ideal Gas Law to estimate the pressure exerted by the gases just before the containment, or in this case 'the bomb', fails. Understanding how to manipulate this equation is vital for accurately predicting the final state of the system, given initial conditions or, as in our exercise, the final conditions just before a dramatic change, like a rupture.

Chemical Reaction Stoichiometry

Stoichiometry relates to the quantitative aspects of chemical reactions, serving as a method to calculate relative quantities of reactants and products. For a reaction to be balanced, the number of atoms for each element must be conserved. In our ammonium nitrate explosion example, the stoichiometric coefficients of nitrogen, water, and oxygen gas generated from the decomposition of ammonium nitrate are dictated by the balanced chemical equation.
By knowing these coefficients, we can determine how many moles of each gas are produced from a given amount of reactant. This precise calculation is vital for predicting the total volume of gases released during the explosion, which subsequently helps in the calculation of pressure using the Ideal Gas Law.

Molar Mass Calculation

The molar mass is the mass of one mole of a substance and is typically expressed in grams per mole (g/mol). It plays a fundamental role in converting between the mass of a substance and the number of moles. Calculating the molar mass of ammonium nitrate, NH₄NO₃, requires adding the atomic masses of nitrogen, hydrogen, and oxygen as found in the periodic table.
As showcased in the ammonium nitrate explosion problem, molar mass enables us to convert the mass of ammonium nitrate used in the reaction (1000 grams) to moles, which is an essential step in predicting the outcome of the reaction using stoichiometry and the Ideal Gas Law.

Gas Pressure Calculation

The calculation of gas pressure within a sealed container, like the bomb in our example, involves understanding how the physical properties of gases interact. The Ideal Gas Law incorporates these properties to calculate the pressure exerted by gases within a volume at a certain temperature. When gases are produced in a reaction within a confined space, their pressure increases as they collide with the container's walls. As the temperature rises, the kinetic energy of the gas molecules increases, leading to more frequent and forceful collisions and thus higher pressures.
By applying the Ideal Gas Law to the resultant moles of gases at a specific temperature, we can compute the pressure just before the 'bomb' ruptures. This type of calculation is essential in fields like engineering and safety to prevent catastrophic failures.