To determine how many atoms of carbon are needed to produce 0.45 mol of aluminum (Al), we will use the balanced chemical equation given and Avogadro's number.
The balanced chemical equation is:
[tex]\[ 3 \text{C} + 2 \text{Al}_2\text{O}_3 \rightarrow 4 \text{Al} + 3 \text{CO}_2 \][/tex]
From this equation, we see that 3 moles of carbon (C) are needed to produce 4 moles of aluminum (Al). This gives us the mole ratio of carbon to aluminum:
[tex]\[ \text{Moles of C} : \text{Moles of Al} = 3 : 4 \][/tex]
Since we need to produce 0.45 moles of aluminum, we can set up a proportion to find out how many moles of carbon are needed:
[tex]\[ \frac{3 \text{ moles of C}}{4 \text{ moles of Al}} = \frac{x \text{ moles of C}}{0.45 \text{ moles of Al}} \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{3}{4} \times 0.45 \][/tex]
[tex]\[ x = 0.3375 \text{ moles of carbon} \][/tex]
Next, we use Avogadro’s number to convert moles of carbon to atoms of carbon. Avogadro’s number is approximately [tex]\( 6.022 \times 10^{23} \)[/tex] atoms per mole.
Thus, the number of atoms of carbon needed:
[tex]\[ \text{Number of atoms of C} = 0.3375 \text{ moles of C} \times 6.022 \times 10^{23} \text{ atoms/mole} \][/tex]
[tex]\[ \text{Number of atoms of C} \approx 2.032425 \times 10^{23} \text{ atoms} \][/tex]
Thus, the correct answer is:
D. [tex]\( 2.0 \times 10^{23} \)[/tex] atoms
