Problem 67 - Chemistry An Atoms First Approach

Question

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 \(\mathrm{STP} ?\)

Step-by-Step Solution

Answer

To find the mass of sodium azide (NaN3) needed to inflate an airbag to 70.0 L at STP, we follow these steps: 1. Use the ideal gas law (\(PV = nRT\)) to find the moles of N2 gas needed. 2. Convert moles of N2 to moles of NaN3 using the stoichiometry of the balanced chemical equation: \(\displaystyle 2\mathrm{NaN}_{3}(s) \longrightarrow 2 \mathrm{Na}(s) + 3 \mathrm{N}_{2}(g)\) 3. Calculate the mass of NaN3 required by multiplying the moles of NaN3 by the molar mass of NaN3 (65.02 g/mol). Upon completing these steps, we will find the mass of sodium azide needed to inflate the airbag to 70.0 L at STP.

Step 1: Calculate the moles of N2 gas needed to inflate the airbag

Firstly, we need to calculate the moles of nitrogen gas (N2) needed to inflate the airbag to 70.0 L at STP. At STP, temperature (T) = 273.15 K, pressure (P) = 1 atm, and the ideal gas constant (R) = 0.0821 L·atm/mol·K. Using the ideal gas law equation, PV = nRT, we can find the moles of N2 gas: n = PV/RT

Step 2: Convert moles of N2 to moles of NaN3

To find the moles of NaN3 needed, we must look at the stoichiometry of the balanced chemical equation for the decomposition of sodium azide: $$2\mathrm{NaN}_{3}(s) \longrightarrow 2 \mathrm{Na}(s) + 3 \mathrm{N}_{2}(g)$$ From the equation, we observe that 2 moles of NaN3 produce 3 moles of N2. Therefore, we can find the moles of NaN3 required by setting up a proportion. moles of NaN3 = (moles of N2) × (2 moles of NaN3 / 3 moles of N2)

Step 3: Convert moles of NaN3 to mass

To calculate the mass of NaN3 needed, we will use the molar mass of sodium azide. The molar mass of NaN3 can be calculated as follows: Molar mass of NaN3 = 22.99 g/mol (Na) + 3 × 14.01 g/mol (N) = 65.02 g/mol Finally, we can calculate the mass of NaN3 required: mass of NaN3 = moles of NaN3 × molar mass of NaN3 After following these steps and carrying out the calculations, we will find the mass of sodium azide needed to inflate the airbag to 70.0 L at STP.

Key Concepts

Ideal Gas Law

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.

Molar Mass Calculation

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.

Chemical Reaction Stoichiometry

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.