Certainly! Let's balance the chemical equation step-by-step.
Given the unbalanced equation:
[tex]\[ \text{Al} + \text{H}_2\text{O} \rightarrow \text{Al}_2\text{O}_3 + \text{H}_2 \][/tex]
We need to balance this equation by finding the correct coefficients for each compound, ensuring that the number of atoms of each element is the same on both sides of the equation.
### Step-by-Step Solution:
1. Write down the number of each type of atom for the reactants and products:
- Reactants:
- Aluminum (Al): 1 atom
- Hydrogen (H): 2 atoms
- Oxygen (O): 1 atom
- Products:
- Aluminum (Al): 2 atoms (in Al\(_2\)O\(_3\))
- Oxygen (O): 3 atoms (in Al\(_2\)O\(_3\))
- Hydrogen (H): 2 atoms
2. Set up the coefficients for each substance:
Let's denote:
- The coefficient of \(\text{Al}\) as \(a\)
- The coefficient of \(\text{H}_2\text{O}\) as \(b\)
- The coefficient of \(\text{Al}_2\text{O}_3\) as \(c\)
- The coefficient of \(\text{H}_2\) as \(d\)
The balanced equation will look like:
[tex]\[ a \text{Al} + b \text{H}_2\text{O} \rightarrow c \text{Al}_2\text{O}_3 + d \text{H}_2 \][/tex]
3. Make equations based on the conservation of each type of atom:
- Aluminum atoms:
[tex]\[ a = 2c \][/tex]
- Oxygen atoms:
[tex]\[ b = \frac{3c}{2} \][/tex]
- Hydrogen atoms:
[tex]\[ 2b = 2d \][/tex]
4. Solve these equations for the coefficients:
- From the Aluminum equation:
[tex]\[ a = 2c \][/tex]
- From the Oxygen equation:
[tex]\[ b = \frac{3c}{2} \][/tex]
- From the Hydrogen equation:
[tex]\[ b = d \][/tex]
Thus, \(2b = 2d\), so \(d = b\).
Let's choose \(d = 1\) (for simplicity), then:
- From the equation \(a = 2c\):
[tex]\[ a = 2c\][/tex]
- From the equation \(b = \frac{3c}{2}\) and \(b = d\):
[tex]\[ \frac{3c}{2} = 1 \][/tex]
Solve for \(c\):
[tex]\[ c = \frac{2}{3} \][/tex]
- Find \( a \):
[tex]\[ a = 2c = 2 \times \frac{2}{3} = \frac{4}{3} \][/tex]
- Find \( b \):
[tex]\[ b = d = 1 \][/tex]
5. Find the lowest common multiple to get whole numbers for the coefficients:
The coefficients we have found are fractional:
[tex]\[a = \frac{4}{3}, \; b = 1, \; c = \frac{2}{3}, \; d = 1 \][/tex]
To convert these to whole numbers, we multiply each coefficient by 3 (the lowest common multiple of the denominators):
- \(a = \frac{4}{3} \times 3 = 4\)
- \(b = 1 \times 3 = 3\)
- \(c = \frac{2}{3} \times 3 = 2\)
- \(d = 1 \times 3 = 3\)
6. Write the balanced equation:
[tex]\[ 4\text{Al} + 3\text{H}_2\text{O} \rightarrow 2\text{Al}_2\text{O}_3 + 3\text{H}_2 \][/tex]
### Conclusion:
The balanced chemical equation is:
[tex]\[ 4\text{Al} + 3\text{H}_2\text{O} \rightarrow 2\text{Al}_2\text{O}_3 + 3\text{H}_2 \][/tex]
