- Analyze the atom count for each element: reactant side has 1 Na, 2 O; product side has 2 Na, 2 O.
- Assign stoichiometric coefficients \(a, b,\) and \(c\) to the equation \(aNa + bO_2 \rightarrow cNa_2O_2\).
- Solve the system of equations derived from the conservation of mass: \(a = 2c\) and \(2b = 2c\).
- Set \(c = 1\) to find the smallest integer coefficients, yielding \(a = 2\) and \(b = 1\), resulting in the final balanced equation \(\boxed{2Na(s) + O_2(g) \rightarrow Na_2O_2(s)}\).
Explanation
- Problem Analysis and Law of Conservation of Mass
To balance a chemical equation, we must satisfy the Law of Conservation of Mass, which states that the total number of atoms of each element must be equal on both the reactant and product sides. The unbalanced equation is:
\[Na(s) + O_2(g) \rightarrow Na_2O_2(s)\]
We can see that the reactant side has one sodium atom (\(Na\)) and two oxygen atoms (\(O\)), while the product side has two sodium atoms and two oxygen atoms. We need to assign stoichiometric coefficients \(a, b,\) and \(c\) such that:
\[a \cdot Na + b \cdot O_2 \rightarrow c \cdot Na_2O_2\]
- Setting up and Solving the Balancing Equations
By counting the atoms of each element on both sides, we obtain the following system of equations:
- For Sodium (\(Na\)): \(a = 2c\)
- For Oxygen (\(O\)): \(2b = 2c\)
We seek the smallest positive integers for \(a, b,\) and \(c\). Let's set \(c = 1\) as a starting point.
Substituting \(c = 1\) into our equations:
- \(a = 2(1) = 2\)
- \(2b = 2(1) \implies 2b = 2 \implies b = 1\)
Thus, we have found that \(a = 2, b = 1, c = 1\).
- Final Balanced Equation and Verification
Substituting the coefficients back into the equation, we get:
\[2Na(s) + 1O_2(g) \rightarrow 1Na_2O_2(s)\]
By convention, we omit the coefficient 1 in the final balanced equation:
\[2Na(s) + O_2(g) \rightarrow Na_2O_2(s)\]
Verifying the atoms:
- Reactants: \(2 \times 1 = 2\) Na atoms, \(1 \times 2 = 2\) O atoms.
- Products: \(1 \times 2 = 2\) Na atoms, \(1 \times 2 = 2\) O atoms.
The equation is now balanced with the smallest whole-number coefficients.
Examples
Balancing chemical equations is essential in fields like industrial chemistry, where engineers calculate the exact amount of raw materials needed to produce a specific quantity of a product. For instance, in the production of sodium peroxide, which is used as a bleaching agent in the textile industry, understanding the stoichiometric relationship between sodium metal and oxygen allows manufacturers to minimize waste and maximize yield. This ensures that resources are used efficiently while maintaining precise control over the chemical reaction.