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Sagot :
To balance the given chemical equation for the oxidation of aluminum [tex]\(\text{Al} + \text{O}_2 \rightarrow \text{Al}_2\text{O}_3\)[/tex], we need to ensure there are the same number of each type of atom on both sides of the equation. Let's go through the steps to balance it.
1. Identify the number of atoms of each element on both sides of the unbalanced equation:
[tex]\[ \text{Al} + \text{O}_2 \rightarrow \text{Al}_2\text{O}_3 \][/tex]
- Reactants: 1 aluminum (Al) atom, 2 oxygen (O) atoms
- Products: 2 aluminum (Al) atoms, 3 oxygen (O) atoms
2. Balance aluminum (Al) atoms:
- The right side has 2 aluminum atoms (in [tex]\(\text{Al}_2\text{O}_3\)[/tex]), so we need 2 aluminum atoms on the left side.
- Place a coefficient of 2 in front of Al on the left side:
[tex]\[ 2\text{Al} + \text{O}_2 \rightarrow \text{Al}_2\text{O}_3 \][/tex]
3. Balance oxygen (O) atoms:
- The right side has 3 oxygen atoms (in [tex]\(\text{Al}_2\text{O}_3\)[/tex]), and the left side has 2 oxygen atoms (in [tex]\(\text{O}_2\)[/tex]).
- To balance the oxygens, we need 3 oxygen atoms on the left side. Since each [tex]\(\text{O}_2\)[/tex] molecule provides 2 oxygen atoms, we need [tex]\(\frac{3}{2}\)[/tex] (1.5) molecules of [tex]\(\text{O}_2\)[/tex].
- Place a coefficient of [tex]\(\frac{3}{2}\)[/tex] in front of [tex]\(\text{O}_2\)[/tex] on the left side:
[tex]\[ 2\text{Al} + \frac{3}{2}\text{O}_2 \rightarrow \text{Al}_2\text{O}_3 \][/tex]
4. Eliminate fractional coefficients by multiplying the entire equation by 2:
- This ensures we have whole numbers for all coefficients:
[tex]\[ 2 \times 2\text{Al} + 2 \times \frac{3}{2}\text{O}_2 \rightarrow 2 \times \text{Al}_2\text{O}_3 \][/tex]
Simplifies to:
[tex]\[ 4\text{Al} + 3\text{O}_2 \rightarrow 2\text{Al}_2\text{O}_3 \][/tex]
5. Confirm the final balanced equation:
- Reactants: 4 aluminum (Al) atoms, 6 oxygen (O) atoms (3 molecules of [tex]\(\text{O}_2\)[/tex])
- Products: 4 aluminum (Al) atoms (in 2 ([tex]\(\text{Al}_2\text{O}_3\)[/tex]), 6 oxygen (O) atoms (in 2 ([tex]\(\text{Al}_2\text{O}_3\)[/tex])
Therefore, the sequence of coefficients that balances the equation is [tex]\(4, 3, 2\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{4, 3, 2} \][/tex]
1. Identify the number of atoms of each element on both sides of the unbalanced equation:
[tex]\[ \text{Al} + \text{O}_2 \rightarrow \text{Al}_2\text{O}_3 \][/tex]
- Reactants: 1 aluminum (Al) atom, 2 oxygen (O) atoms
- Products: 2 aluminum (Al) atoms, 3 oxygen (O) atoms
2. Balance aluminum (Al) atoms:
- The right side has 2 aluminum atoms (in [tex]\(\text{Al}_2\text{O}_3\)[/tex]), so we need 2 aluminum atoms on the left side.
- Place a coefficient of 2 in front of Al on the left side:
[tex]\[ 2\text{Al} + \text{O}_2 \rightarrow \text{Al}_2\text{O}_3 \][/tex]
3. Balance oxygen (O) atoms:
- The right side has 3 oxygen atoms (in [tex]\(\text{Al}_2\text{O}_3\)[/tex]), and the left side has 2 oxygen atoms (in [tex]\(\text{O}_2\)[/tex]).
- To balance the oxygens, we need 3 oxygen atoms on the left side. Since each [tex]\(\text{O}_2\)[/tex] molecule provides 2 oxygen atoms, we need [tex]\(\frac{3}{2}\)[/tex] (1.5) molecules of [tex]\(\text{O}_2\)[/tex].
- Place a coefficient of [tex]\(\frac{3}{2}\)[/tex] in front of [tex]\(\text{O}_2\)[/tex] on the left side:
[tex]\[ 2\text{Al} + \frac{3}{2}\text{O}_2 \rightarrow \text{Al}_2\text{O}_3 \][/tex]
4. Eliminate fractional coefficients by multiplying the entire equation by 2:
- This ensures we have whole numbers for all coefficients:
[tex]\[ 2 \times 2\text{Al} + 2 \times \frac{3}{2}\text{O}_2 \rightarrow 2 \times \text{Al}_2\text{O}_3 \][/tex]
Simplifies to:
[tex]\[ 4\text{Al} + 3\text{O}_2 \rightarrow 2\text{Al}_2\text{O}_3 \][/tex]
5. Confirm the final balanced equation:
- Reactants: 4 aluminum (Al) atoms, 6 oxygen (O) atoms (3 molecules of [tex]\(\text{O}_2\)[/tex])
- Products: 4 aluminum (Al) atoms (in 2 ([tex]\(\text{Al}_2\text{O}_3\)[/tex]), 6 oxygen (O) atoms (in 2 ([tex]\(\text{Al}_2\text{O}_3\)[/tex])
Therefore, the sequence of coefficients that balances the equation is [tex]\(4, 3, 2\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{4, 3, 2} \][/tex]
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