To determine the correct coefficients for balancing the reaction equation, given as:
[tex]\[
\boxed{\square} \, Al_2O_3 + \boxed{\square} \, Fe \rightarrow \boxed{\square} \, Fe_3O_4 + \boxed{\square} \, Al
\][/tex]
we need to ensure that the number of atoms for each element is the same on both sides of the equation.
Here are the steps to find the correct coefficients:
1. Analyze the initial reactants and products:
- Reactants: [tex]\(Al_2O_3\)[/tex] and [tex]\(Fe\)[/tex]
- Products: [tex]\(Fe_3O_4\)[/tex] and [tex]\(Al\)[/tex]
2. Set up the initial counts for each atom in the unbalanced equation:
- [tex]\(Al_2O_3\)[/tex]:
- 2 Aluminum (Al) atoms
- 3 Oxygen (O) atoms
- [tex]\(Fe_3O_4\)[/tex]:
- 3 Iron (Fe) atoms
- 4 Oxygen (O) atoms
3. Balance the elements one by one:
- First balance for aluminum (Al):
- On the left side, [tex]\(Al_2O_3\)[/tex] has 2 Al atoms.
- On the right side, Al needs to appear as a separate element. To balance Al atoms, we need 2 Al atoms on the right side.
- So, coefficient of Al on the right side should be 2.
- Now balance oxygen (O):
- From [tex]\(Al_2O_3\)[/tex], we have 3 oxygen atoms on the left side.
- From [tex]\(Fe_3O_4\)[/tex], we have 4 oxygen atoms. To balance 3 oxygen atoms from [tex]\(Al_2O_3\)[/tex] with 4 oxygen atoms from [tex]\(Fe_3O_4\)[/tex], we will look for simplifying the least common multiple (12 are the least common multiple of 3 and 4):
- We will balance 3[tex]\(Al_2O_3\)[/tex] on left side giving us total of [tex]\(3 \times 3 = 9\)[/tex] oxygen atoms.
- We will balance 3[tex]\(Fe_3O_4\)[/tex] on the right side giving us total of [tex]\(3 \times 4 = 12\)[/tex] oxygen atoms.
- Now, we will adjust the oxygen atoms by multiplying another compound of Fe.
- Balance iron (Fe):
- For every 3 [tex]\( Fe\)[/tex], we produce [tex]\(Fe_3O_4\)[/tex] having compound with 3 Fe.
- Therefore, balance by ensuring three Fe atoms combine giving 9 Fe on left side providing us :
- Fe appears on left side as (total [tex]\(9 Fe\)[/tex] required).
4. Summarize coefficients:
By balancing Ai, O, and Fe as per step you get this summarized step:
Looking at these set of steps again step by step your correct coefficient form is:
[tex]\[
\boxed{1}, \boxed{3}, \boxed{1}, \boxed{2}
\][/tex]
Therefore, the correct set of coefficients is:
B. [tex]\(1, 3, 1, 2\)[/tex]
The balanced equation would be:
[tex]\[
\boxed{1} Al_2O_3 + \boxed{3} Fe \rightarrow \boxed{1} Fe_3O_4 + \boxed{2} Al
\][/tex]
Thus, option B is the correct answer.
