To determine which of the choices given represents a valid mole ratio from the balanced chemical equation:
[tex]\[2 Fe_2O_3 + 3 C \rightarrow 4 Fe + 3 CO_2\][/tex]
we need to look at the coefficients of each substance in the balanced equation. These coefficients give us the mole ratios between reactants and products.
Let's analyze the given ratios one by one:
1. [tex]\(\frac{4 \text{ mol } C}{3 \text{ mol } Fe}\)[/tex]:
- From the equation, carbon ([tex]\(C\)[/tex]) has a coefficient of 3 and iron ([tex]\(Fe\)[/tex]) has a coefficient of 4, so this ratio should be [tex]\(\frac{3 \text{ mol C}}{4 \text{ mol Fe}}\)[/tex]. Therefore, [tex]\(\frac{4 \text{ mol C}}{3 \text{ mol Fe}}\)[/tex] is incorrect.
2. [tex]\(\frac{2 \text{ mol } Fe_2O_3}{4 \text{ mol } C}\)[/tex]:
- From the equation, [tex]\(Fe_2O_3\)[/tex] has a coefficient of 2 and carbon ([tex]\(C\)[/tex]) has a coefficient of 3, so this ratio should be [tex]\(\frac{2 \text{ mol Fe}_2O_3}{3 \text{ mol C}}\)[/tex]. Therefore, [tex]\(\frac{2 \text{ mol } Fe_2O_3}{4 \text{ mol } C}\)[/tex] is incorrect.
3. [tex]\(\frac{3 \text{ mol } C}{4 \text{ mol } CO_2}\)[/tex]:
- From the equation, carbon ([tex]\(C\)[/tex]) has a coefficient of 3 and carbon dioxide ([tex]\(CO_2\)[/tex]) has a coefficient of 3, so comparing these two, we get [tex]\(\frac{3 \text{ mol C}}{3 \text{ mol CO}_2}\)[/tex], which simplifies to [tex]\(\frac{1 \text{ mol C}}{1 \text{ mol CO}_2}\)[/tex]. Hence, [tex]\(\frac{3 \text{ mol } C}{4 \text{ mol } CO_2}\)[/tex] is incorrect.
4. [tex]\(\frac{4 \text{ mol } Fe}{3 \text{ mol } CO_2}\)[/tex]:
- From the equation, iron ([tex]\(Fe\)[/tex]) has a coefficient of 4 and carbon dioxide ([tex]\(CO_2\)[/tex]) has a coefficient of 3, so this ratio is [tex]\(\frac{4 \text{ mol Fe}}{3 \text{ mol CO}_2}\)[/tex], which matches what we have in the balanced equation and is therefore correct.
Therefore, the valid mole ratio from the balanced chemical equation is:
[tex]\[
\boxed{\frac{4 \text{ mol Fe}}{3 \text{ mol CO}_2}}
\][/tex]
