Question

Sodium azide, the explosive compound in automobile air bags, decomposes according to the following equation: $$ 2 \mathrm{NaN}_{3}(\mathrm{s}) \rightarrow 2 \mathrm{Na}(\mathrm{s})+3 \mathrm{N}_{2}(\mathrm{g}) $$ What mass of sodium azide is required to provide the nitrogen needed to inflate a 75.0 -L bag to a pressure of 1.3 atm at \(25^{\circ} \mathrm{C} ?\)

Short answer

172.3 g of sodium azide is required.

Step-by-step solution

Convert Pressure to Standard Units

To utilize the Ideal Gas Law, convert the given pressure of 1.3 atm to pascals (Pa) because SI units are used in gas calculations. The conversion is 1 atm = 101325 Pa: \(1.3 \text{ atm} \times 101325 \text{ Pa/atm} = 131722.5 \text{ Pa}\).

Convert Temperature to Kelvin

The Ideal Gas Law requires temperature in Kelvin. Convert the given temperature of \(25^{\circ} \text{C}\) to Kelvin: \(25 + 273.15 = 298.15 \text{ K}\).

Use Ideal Gas Law to find Moles of Nitrogen

Apply the Ideal Gas Law \(PV = nRT\) to find the moles of nitrogen gas (\(n\)). Use \(R = 8.314 \text{ J/mol}\cdot\text{K}\): \(n = \frac{PV}{RT} = \frac{131722.5 \text{ Pa} \times 75.0 \text{ L}}{8.314 \text{ J/mol}\cdot\text{K} \times 298.15 \text{ K}}\). First, convert \(75.0 \text{ L}\) to cubic meters: \(75.0 \text{ L} = 0.075 \text{ m}^{3}\). Hence, \(n = \frac{131722.5 \times 0.075}{8.314 \times 298.15} \approx 3.97 \text{ moles of } \text{N}_2\).

Calculate NaN3 Moles using Stoichiometry

From the balanced equation, \(2 \text{ NaN}_3\) decomposes to form \(3 \text{ N}_2\). Thus, \(\text{Moles of } \mathrm{NaN}_3 = \frac{2}{3} \times 3.97 \text{ moles of } \mathrm{N}_2 \approx 2.65 \text{ moles of } \mathrm{NaN}_3\).

Convert Moles of NaN3 to Mass

The molar mass of \(\text{NaN}_3\) is \(22.99 + 3 \times 14.01 = 65.02 \text{ g/mol}\). So, the mass of sodium azide required is \(2.65 \times 65.02 \approx 172.3 \text{ g}\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation used in chemistry to relate the pressure, volume, temperature, and number of moles of a gas.The law is expressed as \( PV = nRT \), where:

• \(P\) is pressure
• \(V\) is volume
• \(n\) is the number of moles
• \(R\) is the ideal gas constant
• \(T\) is temperature in Kelvin

To solve problems using the Ideal Gas Law, ensure that units are consistent. Pressure is often given in atmospheres (atm) and should be converted to pascals (Pa) if using the SI unit system. Volume in liters (L) can be converted into cubic meters (m³). Temperature should always be in Kelvin, which can be obtained by adding 273.15 to the Celsius temperature. For example, \(25^{\circ} \text{C} = 298.15 \text{ K}\). By substituting the given values into the Ideal Gas Law, you can solve for the number of moles of gas, \(n\).

Chemical Equations

Chemical equations are symbolic representations of chemical reactions.They display the reactants and products along with their quantities, often using stoichiometric coefficients.A balanced chemical equation has equal numbers of each type of atom on both sides, maintaining the Law of Conservation of Mass. In the decomposition of sodium azide \(2 \mathrm{NaN}_3(s) \rightarrow 2 \mathrm{Na}(s)+3 \mathrm{N}_2(g)\), two moles of sodium azide produce three moles of nitrogen gas.This balanced equation indicates that for every two molecules of sodium azide that decompose, three molecules of nitrogen gas are formed. The coefficients in the equation provide the necessary ratios for stoichiometric calculations, which are used to predict the amounts of products formed from given reactants.

Molar Mass

Molar mass is the mass of one mole of a substance, generally expressed in grams per mole (g/mol).It is crucial for converting moles to grams and vice versa. To calculate the molar mass of a compound, sum up the atomic masses of each element in the formula. For sodium azide \(\text{NaN}_3\), the molar mass is calculated as:

• Sodium (Na): 22.99 g/mol
• Nitrogen (N): 14.01 g/mol, and there are three nitrogen atoms, hence \(3 \times 14.01\)

So, the molar mass of sodium azide is \(22.99 + 42.03 = 65.02 \text{ g/mol}\).Molar mass serves as a conversion factor when calculating the mass required for certain moles of a substance, helping to understand the weight of chemical quantities used in reactions.

Stoichiometric Calculations

Stoichiometric calculations are vital in chemical reactions for determining the proportions of reactants and products. These calculations rely on the balanced chemical equation.Start with the known information, such as moles or mass of a reactant or product. Convert this to moles (if it isn’t already) using molar mass. Use the stoichiometric coefficients from the balanced equation to calculate moles of another reactant or product. In our sodium azide example, we determine the moles of \(\mathrm{NaN}_3\) needed to produce a specific amount of \(\mathrm{N}_2\). Since the equation gives the ratio of \(2 \mathrm{NaN}_3 : 3 \mathrm{N}_2\), the moles of sodium azide required can be calculated using:\[\text{Moles of } \mathrm{NaN}_3 = \frac{2}{3} \times \text{Moles of } \mathrm{N}_2\].Finally, convert the moles back into grams using the molar mass if mass is required, completing the stoichiometric conversion.