Step 1: Understanding the Concept:
Atom economy is a vital metric in green chemistry that measures the proportion of reactant atoms that become part of the desired product in a chemical reaction. It evaluates the efficiency of a synthetic route in terms of minimizing waste.
Step 2: Key Formula or Approach:
The formula for calculating percentage atom economy is:
\[ % \text{ Atom Economy} = \left( \frac{\text{Molecular mass of desired product}}{\text{Sum of molecular masses of all reactants}} \right) \times 100 \]
Step 3: Detailed Explanation:
From the problem description, the reaction is the synthesis of aspirin from salicylic acid and acetic anhydride.
The relevant molecular masses are provided directly in the question:
Mass of Salicylic acid (Reactant 1) = $138 \text{ u}$
Mass of Acetic anhydride (Reactant 2) = $102 \text{ u}$
Mass of Aspirin (Desired Product) = $180 \text{ u}$
First, calculate the sum of the masses of the reactants to find the total theoretical mass input:
\[ \text{Total mass of reactants} = 138 \text{ u} + 102 \text{ u} = 240 \text{ u} \]
Next, substitute these values into the atom economy formula:
\[ % \text{ Atom Economy} = \left( \frac{180}{240} \right) \times 100 \]
Simplify the fraction:
\[ % \text{ Atom Economy} = \left( \frac{3}{4} \right) \times 100 \]
\[ % \text{ Atom Economy} = 0.75 \times 100 = 75% \]
Step 4: Final Answer:
The percentage atom economy for this reaction is 75%.