Problem 180 - Chemistry The Science in Context

Question

Here is the overall reaction in an automobile air bag: $$ \begin{aligned} 20 \mathrm{NaN}_{3}(s)+6 \mathrm{SiO}_{2}(s)+4 \mathrm{KNO}_{3}(s) & \rightarrow \\ 32 \mathrm{N}_{2}(g)+5 \mathrm{Na}_{4} \mathrm{SiO}_{4}(s)+\mathrm{K}_{4} \mathrm{SiO}_{4}(s) \end{aligned} $$ Calculate how many grams of sodium azide (NaN \(_{3}\) ) are needed to inflate a \(40 \times 40 \times 20 \mathrm{cm}\) bag to a pressure of 1.25 atm at a temperature of \(20^{\circ} \mathrm{C} .\) How much more sodium azide is needed if the air bag must produce the same pressure at \(10^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

Answer: 2.67 grams more sodium azide is needed.

Step 1: Calculate the volume of the airbag in liters

The dimensions of the airbag are given in centimeters. Convert them to meters and then calculate the volume in liters: $$ V = (40 \times 40 \times 20) cm^3 = 32000 \, cm^3 = 32 \, L $$

Step 2: Find the moles of N₂ gas needed at 20°C

First, convert the temperature to Kelvin: $$ T = 20^{\circ}C + 273.15 = 293.15 \, K $$ Using the Ideal Gas Law equation, solve for n: $$ n_{N_2} = \frac{PV}{RT} = \frac{(1.25 \, atm) \times (32\, L)}{(0.0821\, L \, atm \, K^{-1} \, mol^{-1}) \times (293.15 \, K)} $$ Calculate the value of n: $$ n_{N_2} \approx 1.656 \, mol $$

Step 3: Calculate the moles of NaN3 needed

Now using the stoichiometry of the reaction, we can find out how many moles of NaN3 are needed: $$ \frac{20 \, moles\, NaN_{3}}{32 \, moles \, N_{2}} = \frac{x \, moles \, NaN_{3}}{1.656 \, moles\, N_{2}} $$ Solve for x: $$ x \approx 1.034 \, moles\, NaN_{3} $$

Step 4: Convert moles of NaN3 to grams

Finally, we will convert the moles of NaN3 to grams. The molar mass of NaN3 is approximately 65 g/mol: $$ 1.034 \, moles \, NaN_{3} \times \frac{65\, g}{1 \, mole} \approx 67.21 \, g $$ So, 67.21 grams of NaN3 are required to inflate the airbag at 20°C.

Step 5: Find the moles of N₂ gas needed at 10°C

Now we will do the same calculations for 10°C. Convert the temperature to Kelvin: $$ T = 10^{\circ}C + 273.15 = 283.15 \, K $$ Using the Ideal Gas Law equation, solve for n: $$ n_{N_2} = \frac{PV}{RT} = \frac{(1.25 \, atm) \times (32\, L)}{(0.0821\, L \, atm \, K^{-1} \, mol^{-1}) \times (283.15 \, K)} $$ Calculate the value of n: $$ n_{N_2} \approx 1.723 \, mol $$

Step 6: Calculate the moles of NaN3 needed

Now using the stoichiometry of the reaction, we can find out how many moles of NaN3 are needed: $$ \frac{20 \, moles\, NaN_{3}}{32 \, moles \, N_{2}} = \frac{y \, moles \, NaN_{3}}{1.723 \, moles\, N_{2}} $$ Solve for y: $$ y \approx 1.075 \, moles\, NaN_{3} $$

Step 7: Convert moles of NaN3 to grams

Finally, we will convert the moles of NaN3 to grams: $$ 1.075 \, moles \, NaN_{3} \times \frac{65\, g}{1 \, mole} \approx 69.88 \, g $$ So, 69.88 grams of NaN3 are required to inflate the airbag at 10°C.

Step 8: Calculate the additional amount of sodium azide needed

To find the additional amount of NaN3 needed, subtract the amount needed at 20°C from the amount needed at 10°C: $$ 69.88 \, g - 67.21 \, g = 2.67 \, g $$ Therefore, 2.67 grams more sodium azide is needed if the airbag must produce the same pressure at 10°C compared to 20°C.

Key Concepts

Stoichiometry

Stoichiometry is the study of the quantitative relationships between the amounts of reactants and products in a chemical reaction. In a chemical equation, coefficients are used to show these relationships.
For example, the airbag reaction involves the conversion of sodium azide (\(\text{NaN}_3\)) into nitrogen gas (\(\text{N}_2\)) and other solid compounds. The balanced chemical equation gives you the ratio in which every reactant and product participates.

• 20 moles of NaNdef\({3}\) corresponds to 32 moles of Ndef\({2}\).
• Using this ratio, we can deduce how much NaNdef\({3}\) we need to produce a certain amount of Ndef\({2}\).

In stoichiometry, it's essential to balance the equation first to understand these relationships fully. From the given stoichiometric coefficients, we can calculate the required amounts of reactants when a specific amount of product is needed, aiding in practical applications such as inflating airbags.

Sodium Azide

Sodium azide (\(\text{NaN}_3\)) is an ionic compound used as a propellant in airbags. It rapidly decomposes to produce a large volume of nitrogen gas, which is critical in the fast deployment of airbags during a collision. Here are some key features:

• Sodium azide is solid at room temperature.
• Under thermal decomposition, one mole of NaNdef\({3}\) produces one and a half moles of nitrogen gas (Ndef\({2}\)).
• It must be handled with care due to its toxic and potentially explosive nature when contaminated.

In an airbag application, precise measurements of NaNdef\({3}\) ensure that adequate nitrogen is produced to inflate the bag quickly and safely within milliseconds of impact.

Chemical Reactions

Chemical reactions involve the transformation of substances through the breaking and forming of chemical bonds. Each reaction has reactants and products determined by a balanced chemical equation. In the case of airbags, the key reaction is the decomposition of sodium azide:

• This is a gas-evolving reaction, producing nitrogen gas to inflate the airbag.
• The reaction is endothermic; energy is absorbed, aiding the stability of the generated gas mixture and materials.
• Catalysts are often used to initiate such reactions rapidly to ensure a timely response during a car crash.

Understanding the details of reactions helps in calculating requirements and predicting outcomes, such as how much reactant is needed under different conditions, like varying temperatures.