- Assign coefficients \(x, y, z\) to the unbalanced equation \(\text{Fe}_2\text{O}_3 \rightarrow \text{Fe} + \text{O}_2\).
- Use the conservation of mass to establish the relationships \(y = 2x\) and \(z = \frac{3x}{2}\) by balancing Fe and O atoms.
- Choose the smallest integer \(x=2\) that results in integer coefficients for all components.
- The final balanced equation is \(2 \text{Fe}_2\text{O}_3(s) \rightarrow 4 \text{Fe}(s) + 3 \text{O}_2(g)\), with coefficients \(\boxed{2, 4, 3}\).
Explanation
- Analyzing the Chemical Equation
To balance the chemical equation \(\text{Fe}_2\text{O}_3(s) \rightarrow \text{Fe}(s) + \text{O}_2(g)\), we need to ensure that the same number of each type of atom is present on both the left (reactants) and the right (products) side of the equation. We introduce coefficients \(x, y, z\) to represent the number of molecules involved:
\[x \text{Fe}_2\text{O}_3(s) \rightarrow y \text{Fe}(s) + z \text{O}_2(g)\]
where \(x, y,\) and \(z\) are the smallest possible positive integers.
- Setting up Conservation Equations
We use the law of conservation of mass to set up equations for each element:
- For Iron (Fe): There are \(2x\) iron atoms in the reactants and \(y\) iron atoms in the products. Thus, we have the equation:
\[2x = y\]
- For Oxygen (O): There are \(3x\) oxygen atoms in the reactants and \(2z\) oxygen atoms in the products. Thus, we have the equation:
\[3x = 2z\]
- Finding Smallest Integer Coefficients
We now express \(y\) and \(z\) in terms of \(x\):
- From the first equation, \(y = 2x\).
- From the second equation, \(2z = 3x\), so \(z = \frac{3}{2}x\).
To obtain the smallest possible integers for \(x, y,\) and \(z\), we choose the smallest integer for \(x\) that makes \(z\) an integer. Setting \(x = 2\) gives:
- \(y = 2(2) = 4\)
- \(z = \frac{3}{2}(2) = 3\)
Thus, our coefficients are \(x=2\), \(y=4\), and \(z=3\).
- Verifying the Balanced Equation
We substitute these coefficients back into the equation:
\[2 \text{Fe}_2\text{O}_3(s) \rightarrow 4 \text{Fe}(s) + 3 \text{O}_2(g)\]
Checking the atom counts:
- Iron (Fe): \(2 \times 2 = 4\) atoms on the left, and \(4\) atoms on the right.
- Oxygen (O): \(2 \times 3 = 6\) atoms on the left, and \(3 \times 2 = 6\) atoms on the right.
The equation is now balanced with the smallest whole number coefficients \(\boxed{2, 4, 3}\).
Examples
Balancing chemical equations is essential in fields like industrial manufacturing, pharmaceuticals, and environmental science. For instance, when designing processes to extract pure iron from iron ore, engineers must calculate the exact amount of reactant (iron oxide) needed to yield a specific amount of metal product. By using balanced chemical equations, they can predict the precise mass of iron produced and manage waste gases like oxygen, ensuring safety, efficiency, and resource optimization in the production line.