- Set up the conservation equations for K, Cl, and O: \(a = b + c\) and \(3a = 4b\).
- Solve the system of linear equations to express variables \(a\) and \(b\) in terms of \(c\).
- Choose the smallest positive integer for \(c\) (\(c=1\)) to find the corresponding coefficients \(b=3\) and \(a=4\).
- The balanced equation is: \(4 KClO_3 \rightarrow 3 KClO_4 + 1 KCl\).
- The final coefficients are \(\boxed{4, 3, 1}\).
Explanation
- Analysis of the Equation
To balance the chemical equation \(a KClO_3 \rightarrow b KClO_4 + c KCl\), we apply the Law of Conservation of Mass, which states that the number of atoms of each element must be equal on both sides of the equation. We identify the elements involved: Potassium (K), Chlorine (Cl), and Oxygen (O).
- Setting Up the System of Equations
We write the conservation equations based on the number of atoms for each element:
- Potassium (K): \(a = b + c\)
- Chlorine (Cl): \(a = b + c\)
- Oxygen (O): \(3a = 4b\)
We observe that the Potassium and Chlorine equations are identical. Thus, we have the system:
1) \(a = b + c\)
2) \(3a = 4b\)
- Solving the Equations
From the oxygen equation, we can express \(a\) in terms of \(b\):
\(a = \frac{4}{3}b\)
Substitute this into the first equation:
\(\frac{4}{3}b = b + c\)
\(\frac{4}{3}b - b = c\)
\(\frac{1}{3}b = c\)
\(b = 3c\)
To obtain the smallest positive integer coefficients, we set \(c = 1\). Then:
\(b = 3(1) = 3\)
\(a = b + c = 3 + 1 = 4\)
- Verification and Final Balanced Equation
We have found the coefficients: \(a = 4\), \(b = 3\), and \(c = 1\).
Checking the balance:
- Potassium: \(4\) atoms on the left, \(3 + 1 = 4\) atoms on the right.
- Chlorine: \(4\) atoms on the left, \(3 + 1 = 4\) atoms on the right.
- Oxygen: \(4 \times 3 = 12\) atoms on the left, \(3 \times 4 = 12\) atoms on the right.
All elements are balanced. The balanced equation is \(4 KClO_3 \rightarrow 3 KClO_4 + 1 KCl\).
Examples
Balancing chemical equations is essential in chemistry for predicting the amounts of products formed from a given amount of reactants. In a laboratory setting, if you were performing this specific decomposition of potassium chlorate, knowing that you need 4 moles of KClO_3 to produce 3 moles of KClO_4 and 1 mole of KCl allows you to calculate the precise masses of chemicals to measure for your experiment. This ensures that you don't waste reagents and helps you understand the stoichiometric ratios of the reaction.