To find the coefficients that balance the chemical equation:
[tex]\[ \mathrm{Fe_2O_3 + CO \rightarrow Fe + CO_2} \][/tex]
we will follow a step-by-step process to ensure that the number of atoms for each element is the same on both sides of the equation.
### Step 1: Write down the unbalanced equation
[tex]\[ \mathrm{Fe_2O_3 + CO \rightarrow Fe + CO_2} \][/tex]
### Step 2: List the number of atoms for each element on both sides of the equation
- Left side:
- [tex]\(\mathrm{Fe}\)[/tex]: 2 from [tex]\(\mathrm{Fe_2O_3}\)[/tex]
- [tex]\(\mathrm{O}\)[/tex]: 3 from [tex]\(\mathrm{Fe_2O_3}\)[/tex], 1 from [tex]\(\mathrm{CO}\)[/tex] (total: 4)
- [tex]\(\mathrm{C}\)[/tex]: 1 from [tex]\(\mathrm{CO}\)[/tex]
- Right side:
- [tex]\(\mathrm{Fe}\)[/tex]: 1
- [tex]\(\mathrm{O}\)[/tex]: 2 from [tex]\(\mathrm{CO_2}\)[/tex]
- [tex]\(\mathrm{C}\)[/tex]: 1 from [tex]\(\mathrm{CO_2}\)[/tex]
### Step 3: Identify discrepancies and start balancing the equation
- To balance the iron ([tex]\(\mathrm{Fe}\)[/tex]) atoms, we can place the coefficient 2 in front of [tex]\(\mathrm{Fe}\)[/tex]:
[tex]\[ \mathrm{Fe_2O_3 + CO \rightarrow 2Fe + CO_2} \][/tex]
- Now let's count the atoms again:
- Left side:
- [tex]\(\mathrm{Fe}\)[/tex]: 2 from [tex]\(\mathrm{Fe_2O_3}\)[/tex]
- [tex]\(\mathrm{O}\)[/tex]: 3 from [tex]\(\mathrm{Fe_2O_3}\)[/tex], 1 from [tex]\(\mathrm{CO}\)[/tex] (total: 4)
- [tex]\(\mathrm{C}\)[/tex]: 1 from [tex]\(\mathrm{CO}\)[/tex]
- Right side:
- [tex]\(\mathrm{Fe}\)[/tex]: 2
- [tex]\(\mathrm{O}\)[/tex]: 2 from [tex]\(\mathrm{CO_2}\)[/tex]
- [tex]\(\mathrm{C}\)[/tex]: 1 from [tex]\(\mathrm{CO_2}\)[/tex]
- To balance the oxygen ([tex]\(\mathrm{O}\)[/tex]) atoms, we notice that we need more [tex]\(\mathrm{CO_2}\)[/tex]. Let’s try 3 in front of [tex]\(\mathrm{CO}\)[/tex] and 3 in front of [tex]\(\mathrm{CO_2}\)[/tex]:
[tex]\[ \mathrm{Fe_2O_3 + 3CO \rightarrow 2Fe + 3CO_2} \][/tex]
- Now, recount the atoms:
- Left side:
- [tex]\(\mathrm{Fe}\)[/tex]: 2 from [tex]\(\mathrm{Fe_2O_3}\)[/tex]
- [tex]\(\mathrm{O}\)[/tex]: 3 from [tex]\(\mathrm{Fe_2O_3}\)[/tex], 3 from [tex]\(\mathrm{3CO}\)[/tex] (total: 6)
- [tex]\(\mathrm{C}\)[/tex]: 3 from [tex]\(\mathrm{3CO}\)[/tex]
- Right side:
- [tex]\(\mathrm{Fe}\)[/tex]: 2
- [tex]\(\mathrm{O}\)[/tex]: 6 from [tex]\(\mathrm{3CO_2}\)[/tex]
- [tex]\(\mathrm{C}\)[/tex]: 3 from [tex]\(\mathrm{3CO_2}\)[/tex]
- The atoms now balance on both sides.
### Step 4: Write the balanced equation with coefficients
[tex]\[ \mathrm{Fe_2O_3 + 3CO \rightarrow 2Fe + 3CO_2} \][/tex]
### Step 5: Verify the balanced equation
Check that all atoms match on both sides again:
- [tex]\(\mathrm{Fe}\)[/tex]: 2 (left and right)
- [tex]\(\mathrm{O}\)[/tex]: 6 (left and right)
- [tex]\(\mathrm{C}\)[/tex]: 3 (left and right)
Everything matches, so the balanced equation is correct. Thus, the coefficients required are:
[tex]\[ \boxed{1, 3, 2, 3} \][/tex]
Therefore, the correct answer is:
[tex]\[ 1, 3, 2, 3 \][/tex]
