The rocket boosters of the space shuttle Discovery, launched on July \(26,2005,\) used a fuel mixture containing primarily solid ammonium perchlorate, \(\mathrm{NH}_{4} \mathrm{ClO}_{4}(\mathrm{s}),\) and aluminum metal. The unbalanced chemical equation for the reaction is given below. \(\mathrm{Al}(\mathrm{s})+\mathrm{NH}_{4} \mathrm{ClO}_{4}(\mathrm{s}) \longrightarrow\) $$ \mathrm{Al}_{2} \mathrm{O}_{3}(\mathrm{s})+\mathrm{AlCl}_{3}(\mathrm{s})+\mathrm{H}_{2} \mathrm{O}(\mathrm{l})+\mathrm{N}_{2}(\mathrm{g}) $$ What is the minimum mass of \(\mathrm{NH}_{4} \mathrm{ClO}_{4}\) consumed, per kilogram of \(\mathrm{Al}\), by the reaction of \(\mathrm{NH}_{4} \mathrm{ClO}_{4}\) and Al?[Hint: Balance the elements in the order \(\mathrm{Cl}, \mathrm{H},\) \(\mathrm{O}, \mathrm{Al}, \mathrm{N} .\)]

Step by step solution

01

Balancing The Chemical Equation

The unbalanced equation is: \[Al(s) + NH_{4}ClO_{4}(s) \rightarrow Al_{2}O_{3}(s) + AlCl_{3}(s) + H_{2}O(l) + N_{2}(g)\] Balance the elements in the order: \(Cl, H, O, Al, N\) \[6Al(s) + 10NH_{4}ClO_{4}(s) \rightarrow 5Al_{2}O_{3}(s) + 10AlCl_{3}(s) + 20H_{2}O(l) + 10N_{2}(g)\]

02

Considering the Molar Masses

We need to take the molar masses of Aluminum (Al) and ammonium perchlorate (\(NH_{4}ClO_{4}\)) into consideration in order to convert from molecular scale to macroscopic scale. The molar mass of Aluminum (Al) is approximately 26.98 g/mol and that of ammonium perchlorate (\(NH_{4}ClO_{4}\)) is approximately 117.49 g/mol.

03

Calculating the Mass

In the now balanced equation, we see that the ratio between Aluminum (Al) and ammonium perchlorate (\(NH_{4}ClO_{4}\)) is 6:10 or 3:5. Therefore, for every 3 moles of Al, we need 5 moles of \(NH_{4}ClO_{4}\). To calculate the mass of \(NH_{4}ClO_{4}\) per mass of Al -- Use the molar masses to convert from moles to grams. Thus, \[(117.49 \, g/mol * 5 \, moles) / (26.98 \, g/mol * 3 \, moles)\] Calculate the above expression to get the mass of \(NH_{4}ClO_{4}\) per gram of Al.

04

Converting to Kilograms

The previous step provides the mass per gram of Al but the asked quantity in the task is per kilogram. Since there are 1000 grams in a kilogram, multiply the result calculated in step 3 by 1000 to get the result.

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