Automotive airbags inflate when sodium azide, $NaN_{3}$ , rapidly decomposes to its constituent elements. The equation for the chemical reaction is
$\quad\quad\quad\quad 2NaN_{3}(s) → 2Na(s) + 3N_{2}(g)$
The gaseous $N_2$ so generated inflates the airbag (see Figure 6.10). How many moles of $NaN_{3}$ would have to decompose in order to generate 253 million (2.53 × $10^{8}$ ) molecules of $N_2$ ?

Question

Automotive airbags inflate when sodium azide, [latex]NaN_{3}[/latex], rapidly decomposes to its constituent elements. The equation for the chemical reaction is
[latex]\quad\quad\quad\quad 2NaN_{3}(s) → 2Na(s) + 3N_{2}(g)[/latex]
The gaseous [latex]N_2[/latex] so generated inflates the airbag (see Figure 6.10). How many moles of [latex]NaN_{3}[/latex] would have to decompose in order to generate 253 million (2.53 × [latex]10^{8}[/latex]) molecules of [latex]N_2[/latex]?

Accepted Answer

Although a calculation of this type does not have a lot of practical significance, it tests your understanding of the problem-solving relationships discussed in this section of the text.
Step 1: The given quantity is 2.53 × [latex]10^{8}[/latex] molecules of [latex]N_2[/latex], and the desired quantity is moles of [latex]NaN_{3}[/latex].
[latex]\quad\quad\quad\quad[/latex]2.53 × [latex]10^{8}[/latex] molecules [latex]N_2[/latex] = ? moles [latex]NaN_{3}[/latex]
In terms of Figure 6.9, this is a “particles of A” to “moles of B” problem.
Step 2: Using Figure 6.9 as a road map, we determine that the pathway for this problem is

[latex]\quad\quad\quad \boxed{Particles of A} \xrightarrow[number]{Avogadro’s} \boxed{Moles of A} \xrightarrow[coefficients]{Equation} \boxed{Moles of B}[/latex]

The mathematical setup is

[latex]\quad\quad 2.53 imes 10^{8} \cancel{molecules N_{2}} imes (\frac{1 \cancel{mole N_{2}}}{6.02 imes 10^{23} \cancel{molecules N_{2}}}) imes (\frac{2 \cancel{moles NaN_{3}}}{3 \cancel{moles N_{2}}})[/latex]

Avogadro’s number is present in the first conversion factor. The 2 and 3 in the second conversion factor are the coefficients, respectively, of [latex]NaN_{3}[/latex] and [latex]N_2[/latex] in the balanced chemical equation.
Step 3: The solution to the problem, obtained by doing the arithmetic after all the numerical factors have been collected, is

[latex]\quad\quad\quad (\frac{2.53 imes 10^{8} imes 1 imes 2}{6.02 imes 10^{23} imes 3}) mole NaN_{3}=2.80 imes 10^{-16} mole NaN_{3}[/latex]

Question 6.10: Automotive airbags inflate when sodium azide, NaN3, rapidly ......

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