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The sodium azide required for automobile air bags is made by the reaction of sodium metal with dinitrogen monoxide in liquid ammonia: $$\begin{aligned} 3 \mathrm{N}_{2} \mathrm{O}(\mathrm{g})+4 \mathrm{Na}(\mathrm{s})+\mathrm{NH}_{3}(\ell) & \rightarrow \\ & \mathrm{NaN}_{3}(\mathrm{s})+3 \mathrm{NaOH}(\mathrm{s})+2 \mathrm{N}_{2}(\mathrm{g}) \end{aligned}$$ (a) You have \(65.0 \mathrm{g}\) of sodium, a \(35.0-\mathrm{L}\). flask containing \(\mathrm{N}_{2} \mathrm{O}\) gas with a pressure of 2.12 atm at \(23^{\circ} \mathrm{C}\) and excess ammonia. What is the theoretical yield (in grams) of NaNg? (b) Draw a Lewis structure for the azide ion. Include all possible resonance structures. Which resonance structure is most likely? (c) What is the shape of the azide ion?

Step by step solution

01

Identify Molar Masses and Balance the Equation

Identify the molar masses of each compound involved in the reaction:- Sodium (\( \mathrm{Na} \)): 22.99 g/mol- Dinitrogen monoxide (\( \mathrm{N}_2\mathrm{O} \)): 44.01 g/mol- Sodium azide (\( \mathrm{NaN}_3 \)): 65.01 g/molCheck that the chemical equation is balanced as follows:\[3 \mathrm{N}_{2}\mathrm{O}(\mathrm{g}) + 4 \mathrm{Na}(\mathrm{s}) + \mathrm{NH}_{3}(\ell) \rightarrow \mathrm{NaN}_{3}(\mathrm{s}) + 3 \mathrm{NaOH}(\mathrm{s}) + 2 \mathrm{N}_{2}(\mathrm{g}) \]

02

Calculate the Moles of Sodium

Calculate the moles of sodium (\( \mathrm{Na} \)) available using its mass.For sodium:\[\text{Moles of } \mathrm{Na} = \frac{65.0 \text{ g}}{22.99 \text{ g/mol}} = 2.83 \text{ moles} \]

03

Calculate the Moles of Dinitrogen Monoxide

Use the ideal gas law to find the moles of \( \mathrm{N}_2\mathrm{O} \):The ideal gas law: \[PV = nRT\]Where:\(P = 2.12 \text{ atm}\)\(V = 35.0 \text{ L}\)\(R = 0.0821 \text{ L atm/mol K}\)\(T = 23^{\circ} \text{C} + 273.15 = 296.15 \text{ K}\)Solving for \(n\):\[n = \frac{PV}{RT}\]\[n = \frac{2.12 \times 35.0}{0.0821 \times 296.15} \approx 3.06 \text{ moles} \]

04

Determine the Limiting Reactant

Compare the mole ratios from the balanced equation:- 3 moles of \( \mathrm{N}_{2}\mathrm{O} \) react with 4 moles of \( \mathrm{Na} \)Calculate the needed \( \mathrm{N}_{2}\mathrm{O} \) and \( \mathrm{Na} \):\[\frac{3}{4} \times 2.83 = 2.12 \text{ moles of } \mathrm{N}_{2}\mathrm{O} \text{ needed}\]Since 3.06 moles of \( \mathrm{N}_{2}\mathrm{O} \) are available and only 2.12 moles are needed, sodium is the limiting reactant.

05

Calculate Theoretical Yield of Sodium Azide

Using stoichiometry, calculate the amount of \( \mathrm{NaN}_{3} \) produced:From the equation: 4 moles of \( \mathrm{Na} \) produce 1 mole of \( \mathrm{NaN}_{3} \).\[\text{Moles of } \mathrm{NaN}_{3} = \frac{2.83}{4} = 0.7075 \text{ moles}\]Convert moles of \( \mathrm{NaN}_{3} \) to grams:\[\text{Mass of } \mathrm{NaN}_{3} = 0.7075 \times 65.01 = 45.99 \text{ grams}\]

06

Draw Lewis Structures for Azide Ion

The azide ion (\( \mathrm{N}_{3}^{-} \)) can have multiple resonance structures involving N-N triple, double, and single bonds. Draw three possible structures:1. \( [N \equiv N - N]^{-} \)2. \( [N = N = N]^{-} \)3. \( [N - N \equiv N]^{-} \)The most likely structure is the one with a N-N triple bond (\( [N \equiv N - N]^{-} \)) and more even charge distribution.

07

Determine the Shape of Azide Ion

The shape of the azide ion is linear due to its three-atom structure and resonance, in which the central nitrogen atom forms bonds with the two nitrogen atoms at the ends.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stoichiometry

Stoichiometry is the study of quantitative relationships in chemical reactions. It allows us to predict how much product will be formed in a reaction based on the amounts of reactants used. For example, in the reaction used to produce sodium azide for airbags, stoichiometry helps us determine how much sodium azide will form from given amounts of sodium and dinitrogen monoxide.
\[3\ \mathrm{N}_{2}\mathrm{O}(\mathrm{g}) + 4\ \mathrm{Na}(\mathrm{s}) + \mathrm{NH}_{3}(\ell) \rightarrow \mathrm{NaN}_{3}(\mathrm{s}) + 3\ \mathrm{NaOH}(\mathrm{s}) + 2\ \mathrm{N}_{2}(\mathrm{g})\]
To use stoichiometry, follow these steps:

• First, ensure the chemical equation is balanced, showing equal numbers of each type of atom on both sides.
• Convert reactant masses to moles using their molar masses.
• Use mole ratios from the balanced equation to calculate moles of desired product.
• Convert moles of product to grams if needed.

This method ensures the precise amount of a chemical substance is used for the desired reaction.

Limiting Reactant

In chemical reactions, the limiting reactant is the substance that is entirely consumed first, stopping the reaction from continuing. This reactant determines the maximum amount of product formed. Identifying the limiting reactant is crucial to accurately calculate the theoretical yield of a reaction.
For our sodium azide example, sodium was found to be the limiting reactant. By comparing the mole ratio from the balanced equation:\[3\ \mathrm{N}_{2}\mathrm{O} + 4\ \mathrm{Na} \rightarrow 1\ \mathrm{NaN}_{3}\]
We found that sodium runs out before all the available dinitrogen monoxide can react.

• First, calculate the available moles of each reactant.
• Determine how many moles of one reactant are needed to completely react with the other.
• The reactant with the fewest complete moles is the limiting reactant.

Recognizing the limiting reactant allows us to correctly compute the amount of product without wasting resources.

Resonance Structures

Resonance structures are different possible configurations of electrons in a molecule where electrons can be placed in various positions. These structures depict potential distributions of electron pairs within a molecule or ion, which are crucial for understanding reactivity and properties.
In the azide ion \(\mathrm{N}_{3}^{-}\), resonance structures help explain its stability and chemical behavior. Possible resonance structures include:

• \([N \equiv N - N]^{-}\)
• \([N = N = N]^{-}\)
• \([N - N \equiv N]^{-}\)

The most likely and stable structure is \([N \equiv N - N]^{-}\), where the triple bond provides greater thermodynamic stability. Each structure contributes to the overall resonance hybrid, influencing the linear shape of the azide ion.

Ideal Gas Law

The ideal gas law is a fundamental principle that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as:\[PV = nRT\]Where:

• \(P\) is pressure in atmospheres (atm)
• \(V\) is volume in liters (L)
• \(n\) is the number of moles of gas
• \(R\) is the ideal gas constant (0.0821 L atm/mol K)
• \(T\) is temperature in Kelvin (K)

In our problem, we used the ideal gas law to calculate the moles of dinitrogen monoxide (\(\mathrm{N}_2\mathrm{O}\)) present in the flask. By rearranging the formula to solve for \(n = \frac{PV}{RT}\), we found the moles of gas given its pressure, volume, and temperature. This is essential for determining how much oxygen is available to react with sodium. This law is particularly useful when dealing with reactions involving gases where temperature and pressure conditions change.