To determine how many grams of copper (Cu) are needed to react with 2.0 mol of HNO3 according to the reaction:
\[ 3 \text{Cu} + 8 \text{HNO}_3 \rightarrow 3 \text{Cu(NO}_3)_2 + 2 \text{NO} + 4 \text{H}_2\text{O} \]
we start by identifying the stoichiometric relationship given in the balanced equation between Cu and HNO3.
From the equation, we see that 3 moles of Cu react with 8 moles of HNO3. We can set up a conversion based on this ratio to find out how many moles of Cu are required for 2.0 moles of HNO3.
Using the ratio:
\[ \frac{3 \text{ mol Cu}}{8 \text{ mol HNO}_3} \]
We can determine the moles of Cu needed for 2.0 moles of HNO3:
\[ \text{moles of Cu} = 2.0 \text{ mol HNO}_3 \times \frac{3 \text{ mol Cu}}{8 \text{ mol HNO}_3} \]
Calculating this gives:
\[ \text{moles of Cu} = 2.0 \times \frac{3}{8} = \frac{6.0}{8} = 0.75 \text{ mol Cu} \]
Now, we need to convert moles of Cu into grams. The molar mass of Cu is approximately 63.55 g/mol. Therefore, we can calculate the mass of Cu required:
\[ \text{mass of Cu} = \text{moles of Cu} \times \text{molar mass of Cu} \]
\[ \text{mass of Cu} = 0.75 \text{ mol} \times 63.55 \text{ g/mol} \approx 47.6625 \text{ g} \]
Rounding to three significant figures:
\[ \text{mass of Cu} \approx 47.7 \text{ g} \]
Thus, approximately 47.7 grams of Cu would be needed to react with 2.0 mol of HNO3.