- Assign variables to the coefficients and set up a system of equations based on the conservation of mass for each element (Si, O, Ca, C).
- Express all coefficients in terms of one variable, yielding the relationship \(a = c = \frac{5}{2}b\), \(d = b\), and \(e = 2b\).
- Choose the smallest integer for \(b\) that makes all coefficients whole numbers, which is \(b = 2\).
- The balanced equation is \(5 \text{SiO}_2 + 2 \text{CaC}_2 \rightarrow 5 \text{Si} + 2 \text{CaO} + 4 \text{CO}_2\), so the answer is \(\boxed{5, 2 \rightarrow 5, 2, 4}\).
Explanation
- Problem Analysis and Setup
To balance the chemical equation \(\Box \text{SiO}_2 + \Box \text{CaC}_2 \rightarrow \Box \text{Si} + \Box \text{CaO} + \Box \text{CO}_2\), we must ensure that the number of atoms of each element is the same on both sides of the reaction. Let the coefficients be \(a, b, c, d,\) and \(e\) respectively, so the equation is \(a \text{SiO}_2 + b \text{CaC}_2 \rightarrow c \text{Si} + d \text{CaO} + e \text{CO}_2\).
- Formulating the System of Equations
We track the number of atoms for each element involved:
- Silicon (Si): The number of atoms on the left is \(a\), and on the right is \(c\). Thus, \(a = c\).
- Oxygen (O): The number of atoms on the left is \(2a\), and on the right is \(d + 2e\). Thus, \(2a = d + 2e\).
- Calcium (Ca): The number of atoms on the left is \(b\), and on the right is \(d\). Thus, \(b = d\).
- Carbon (C): The number of atoms on the left is \(2b\), and on the right is \(e\). Thus, \(2b = e\).
- Solving for Coefficients
We can now express all coefficients in terms of \(b\):
- From the equations, we have \(d = b\) and \(e = 2b\).
- Substitute these into the oxygen equation: \(2a = d + 2e = b + 2(2b) = b + 4b = 5b\).
- Therefore, \(a = \frac{5}{2}b\).
- Since \(c = a\), we have \(c = \frac{5}{2}b\).
To get the smallest positive integers, let \(b = 2\). Then:
- \(b = 2\)
- \(a = \frac{5}{2}(2) = 5\)
- \(c = 5\)
- \(d = 2\)
- \(e = 2(2) = 4\)
- Verification and Final Answer
Substituting these values back into our original equation, we get \(5 \text{SiO}_2 + 2 \text{CaC}_2 \rightarrow 5 \text{Si} + 2 \text{CaO} + 4 \text{CO}_2\). We verify the atom counts:
- Silicon: 5 (left) = 5 (right). Correct.
- Oxygen: \(2 \times 5 = 10\) (left) = \(2 + (4 \times 2) = 10\) (right). Correct.
- Calcium: 2 (left) = 2 (right). Correct.
- Carbon: \(2 \times 2 = 4\) (left) = 4 (right). Correct.
The balanced equation is \(5 \text{SiO}_2 + 2 \text{CaC}_2 \rightarrow 5 \text{Si} + 2 \text{CaO} + 4 \text{CO}_2\), and the coefficients are \(5, 2, 5, 2,\) and \(4\). The final answer is \(\boxed{5 \text{SiO}_2 + 2 \text{CaC}_2 \rightarrow 5 \text{Si} + 2 \text{CaO} + 4 \text{CO}_2}\).
Examples
Chemical balancing is essential in industrial processes like smelting, where precise proportions of raw materials must be combined to yield the desired products efficiently. For example, in the production of silicon from silica, balancing the equation ensures that no excess reactants are wasted and that the theoretical maximum of the product is achieved. Mastering this technique helps scientists and engineers scale up reactions safely for real-world applications.