- Identify the stoichiometric ratio from the balanced equation: 1 mole of Cu produces 2 moles of Ag.
- Set up the proportion between the moles of copper and silver: \(\frac{n_{Cu}}{n_{Ag}} = \frac{1}{2}\).
- Calculate the moles of copper required: \(n_{Cu} = 3.50 \times 0.5 = 1.75 \text{ mol}\).
- Select the matching option: \(\boxed{1.75 \text{ mol}}\).
Explanation
- Analyzing the Chemical Equation
To determine the number of moles of copper required, we first analyze the balanced chemical equation provided:
\[Cu + 2AgNO_3 \rightarrow 2Ag + Cu(NO_3)_2\]
The coefficients in this balanced equation represent the molar ratio of the reactants and products. Specifically, the reaction indicates that 1 mole of copper (Cu) reacts with 2 moles of silver nitrate (AgNO_3) to produce 2 moles of silver (Ag) and 1 mole of copper(II) nitrate (Cu(NO_3)_2). Thus, the stoichiometric ratio between copper and silver is 1:2. This means that for every 1 mole of copper that reacts, 2 moles of silver are formed.
- Setting Up the Proportion
We are given that 3.50 moles of silver (Ag) are produced in the reaction. Let \(n_{Cu}\) be the number of moles of copper needed, and \(n_{Ag}\) be the number of moles of silver produced. Based on the stoichiometric ratio identified in the previous step, we can set up the following proportion:
\[\frac{n_{Cu}}{n_{Ag}} = \frac{1 \text{ mol Cu}}{2 \text{ mol Ag}}\]
Substituting the known value of \(n_{Ag} = 3.50 \text{ mol}\) into the equation gives:
\[\frac{n_{Cu}}{3.50 \text{ mol}} = \frac{1}{2}\]
- Calculating the Required Moles
To find the number of moles of copper (\(n_{Cu}\)), we solve the proportion for \(n_{Cu}\):
\[n_{Cu} = 3.50 \times \frac{1}{2}\]
\[n_{Cu} = 1.75 \text{ mol}\]
Therefore, 1.75 moles of copper must react to produce 3.50 moles of silver. Comparing this result with the given options, the correct choice is b.
Examples
In chemical manufacturing, understanding stoichiometry is essential for optimizing production. For example, if a factory needs to produce a specific amount of silver for electronic components, they use this calculation to ensure the exact amount of raw copper is used, minimizing waste and reducing costs. By knowing the exact stoichiometric requirements, engineers ensure efficiency and precision in large-scale chemical synthesis.