Air bags are activated when a severe impact causes a steel ball to compress a spring and electrically ignite a detonator cap. This causes sodium azide (NaN, ) to decompose explosively according to the following reaction:$$2 \mathrm{NaN}_{3}(s) \longrightarrow 2 \mathrm{Na}(s)+3 \mathrm{N}_{2}(g)$$ What mass of \(\mathrm{NaN}_{3}(s)\) must be reacted to inflate an air bag to \(70.0 \mathrm{L}\) at \(\mathrm{STP} ?\)

Understanding the ideal gas law is crucial when working with gases, such as determining the amount of a gas needed to inflate an airbag. The ideal gas law equation, represented by PV = nRT, relates the pressure (P), volume (V), moles of gas (n), temperature (T), and the ideal gas constant (R). At standard temperature and pressure (STP), which are 273.15 K and 1 atm respectively, one mole of any ideal gas will occupy 22.4 L.

To apply the ideal gas law to our problem, we first identify the known values: a volume of 70.0 L, a temperature of 273.15 K, a pressure of 1 atm, and the ideal gas constant of 0.0821 L·atm/mol·K. We use the equation to solve for the unknown quantity, which is the moles of nitrogen gas (N₂) needed to inflate the airbag. With this information, we can further explore the chemical process that generates the gas.

The molar mass of a compound, which is the mass of one mole of that compound, is essential for converting between the mass of a substance and the moles of the substance. In our example, we need to calculate the molar mass of sodium azide (NaN₃) to determine the mass that must decompose to produce a set volume of nitrogen gas.

To calculate the molar mass, we sum the atomic masses of all the atoms in a molecule. For NaN₃, sodium (Na) has an atomic mass of approximately 22.99 g/mol, and nitrogen (N) has an atomic mass of 14.01 g/mol. With three nitrogen atoms in sodium azide, the calculation is as follows:Molar mass of NaN₃ = (1 × 22.99 g/mol) + (3 × 14.01 g/mol) = 65.02 g/mol.

Knowing the molar mass allows us to convert moles of NaN₃ into grams, which is a more practical unit for measuring and handling substances in a laboratory or an engineering setting, such as filling an airbag.

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It allows for the calculation of quantities of reactants required to produce a set amount of product, based on the balanced chemical equation. The decomposition of sodium azide (NaN₃) in airbags illustrates this:

2NaN₃(s) → 2Na(s) + 3N₂(g)

Here, the stoichiometry of the reaction tells us that 2 moles of NaN₃ produce 3 moles of nitrogen gas (N₂). By understanding the stoichiometry, we set up a ratio that connects the moles of N₂ needed with the moles of NaN₃ required. For our airbag, using the ideal gas law gives us the moles of N₂. Once known, we multiply by the appropriate stoichiometric factor, in this case, (2 moles of NaN₃ / 3 moles of N₂), to find the moles of NaN₃ that must decompose. This step is the cornerstone of calculating reactive material in chemistry and engineering applications, highlighting the importance of understanding reaction stoichiometry for practical applications.