Aspirin \(\left(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\right)\) is synthesized by reacting salicylic acid $\left(\mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{3}\right)\( with acetic anhydride \)\left(\mathrm{C}_{4} \mathrm{H}_{6} \mathrm{O}_{3}\right) .$ The balanced equation is $$ \mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{3}+\mathrm{C}_{4} \mathrm{H}_{6} \mathrm{O}_{3} \longrightarrow \mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}+\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2} $$ a. What mass of acetic anhydride is needed to completely consume $1.00 \times 10^{2}$ g salicylic acid? b. What is the maximum mass of aspirin (the theoretical yield) that could be produced in this reaction?

Step by step solution

01

Calculate moles of given salicylic acid

We are given the mass of salicylic acid to be \(1.00 \times 10^2\) grams. To determine the moles of the salicylic acid, we will use its molar mass: $$\mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{3} \rightarrow 7(12.01) + 6(1.01)+3(16.00) = 138.12 \, \mathrm{g/mol}$$ Now, calculate the moles of salicylic acid: $$n=\frac{m}{M} = \frac{1.00\times10^2\, \text{g}}{138.12\, \text{g/mol}}$$

02

Calculate moles of acetic anhydride required to react completely with salicylic acid

According to the stoichiometry of the balanced equation, 1 mole of salicylic acid reacts completely with 1 mole of acetic anhydride. Therefore, the moles of acetic anhydride needed are equal to the moles of salicylic acid calculated in step 1.

03

Calculate the mass of acetic anhydride needed

Now, we need to convert the moles of acetic anhydride to its mass. First, we find the molar mass of acetic anhydride: $$\mathrm{C}_{4} \mathrm{H}_{6} \mathrm{O}_{3} \rightarrow 4(12.01) + 6(1.01)+3(16.00) = 102.09 \, \mathrm{g/mol}$$ Now, we will multiply the moles of acetic anhydride calculated in step 2 by its molar mass to get the mass of acetic anhydride: $$\text{mass} = n \times M$$

04

Determine the maximum mass of aspirin that can be produced

According to stoichiometry, 1 mole of salicylic acid will produce 1 mole of aspirin. Therefore, the moles of aspirin produced will be equal to moles of salicylic acid. Next, we will find the molar mass of aspirin: $$\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4} \rightarrow 9(12.01) + 8(1.01)+4(16.00) = 180.16 \, \mathrm{g/mol}$$ Now, we will multiply the moles of aspirin by its molar mass to get the maximum mass of aspirin (theoretical yield) that can be produced. $$\text{mass} = n \times M$$

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