Question

In air bags used in automobiles, the gas that fills the bags is produced from the reaction of sodium azide, \(\mathrm{NaN}_{3}\) : $$ 2 \mathrm{NaN}_{3}(s) \longrightarrow 2 \mathrm{Na}(s)+3 \mathrm{~N}_{2}(g) $$ What mass of sodium azide is needed to fill a \(2.50-\mathrm{L}\) air bag with nitrogen gas at a pressure of 1140 torr and \(25^{\circ} \mathrm{C}\) ?

Short answer

The required mass of Sodium Azide \(\mathrm{NaN}_3\) to inflate the air bag is approximately 5.28 grams.

Step-by-step solution

Convert given values into standard units

First convert the volume from litres to cubic meters, the temperature from Celsius to Kelvin and the pressure from torr to atmospheres. The converted values are: \n$V = 2.5L = 0.0025 m^3$, \n$T = 25°C = 298.15 K$, and \n$P = 1140 torr = 1.5 atm$

Calculate the amount of Nitrogen gas

The Ideal Gas Law, \(PV=nRT\) can be used to calculate the number of moles (\(n\)) of the gas. Rearrange the formula to solve for \(n\), then substitute in the given values: \n\(n = PV/RT = (1.5 atm * 0.0025 m^3) / (0.0821 L atm K^−1 mol^−1 * 298.15 K) \+ 0.122 mol\)

Determine moles of Sodium Azide

From the balanced chemical equation, \(2 \mathrm{NaN}_3(s) \longrightarrow 2 \mathrm{Na}(s)+3 \mathrm{~N}_2(g)\), it's provided that 2 moles of sodium azide produces 3 moles of nitrogen. Therefore, to get the amount in moles of Sodium Azide \(\mathrm{NaN}_3\), multiply the moles of \(N_2\) by the stoichiometric ratio (2/3): \n\(moles\ of\ NaN_3 = 0.122 mol * (2/3) = 0.0813 mol\)

Calculate Mass of Sodium Azide

Finally, convert the moles of Sodium Azide into grams by multiplying by the molar mass of Sodium Azide (\(65 g/mol\)): \n\(mass\ of\ NaN_3 = 0.0813 mol * 65 g/mol = 5.28 g\)

Key concepts

Chemical Reaction

A chemical reaction is a process where substances, known as reactants, transform into different substances, called products. This transformation involves a rearrangement of atoms and molecules. In the context of the exercise, the chemical reaction uses sodium azide

• Sodium Azide (\(\text{NaN}_3\)) decomposes to form sodium metal (\(\text{Na}\)) and nitrogen gas (\(\text{N}_2\)).
• This reaction is crucial for airbag deployment as nitrogen gas rapidly inflates the airbag.
• The chemical equation for this reaction is balanced, meaning the number of each type of atom is the same on both sides of the equation.

Understanding the balanced chemical equation is key, as it shows the relationship between reactants and products. For every 2 molecules of sodium azide, 3 molecules of nitrogen are produced.

Stoichiometry

Stoichiometry is the aspect of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It allows us to use a balanced chemical equation to determine how much of a reactant is needed or how much of a product will be produced.
For instance, in the sodium azide reaction:

• The balanced equation, \(2 \text{NaN}_3 \rightarrow 2 \text{Na} + 3 \text{N}_2\), tells us that 2 moles of \(\text{NaN}_3\) produce 3 moles of \(\text{N}_2\).
• Using stoichiometry, you can find the needed quantity of sodium azide (reactant) to produce a specific amount of nitrogen gas (product).
• In the exercise, the stoichiometric ratio of sodium azide to nitrogen is 2:3, which is essential for calculating the moles of sodium azide required to generate a desired amount of nitrogen.

This is how stoichiometry helps translate theoretical chemical equations into real-world amounts.

Molar Mass

Molar mass is a measure of the mass of one mole of a substance, usually expressed in grams per mole (\(\text{g/mol}\)). It acts as a bridge between the macroscopic scale that we can measure, such as grams, and the microscopic scale of atoms and molecules. For sodium azide, the molar mass is essential to convert between moles and grams.
Here's how it works in the exercise:

• The molar mass of sodium azide is 65 g/mol.
• To find out the mass in grams of sodium azide, multiply the moles calculated from stoichiometry by its molar mass.
• For example, if you have calculated 0.0813 moles of sodium azide, multiplying by 65 g/mol gives you a mass of 5.28 grams.

This conversion is crucial when preparing a specific quantity of a compound required for chemical reactions such as the one used in airbags.