- Represent the equation as \(aC + bH_2 \rightarrow cC_2H_6\).
- Use atom conservation to derive: \(a = 2c\) and \(2b = 6c \rightarrow b = 3c\).
- Assign \(c = 1\) to find \(a = 2\) and \(b = 3\).
- Final balanced equation: \(2C(s) + 3H_2(g) \rightarrow C_2H_6(g)\), with coefficients \(\boxed{2, 3, 1}\).
Explanation
- Problem Analysis and Setup
To balance the chemical equation \(C(s) + H_2(g) \rightarrow C_2H_6(g)\), we need to ensure that the number of atoms of each element is the same on both sides of the equation. We represent the equation as \(aC + bH_2 \rightarrow cC_2H_6\), where \(a, b,\) and \(c\) are the stoichiometric coefficients we need to determine.
- Conservation of Atoms
We apply the principle of conservation of mass, which states that atoms are neither created nor destroyed during a chemical reaction.
- For carbon (C) atoms: The number of C atoms on the left is \(a\), and on the right is \(2c\). So, we have the equation: \(a = 2c\).
- For hydrogen (H) atoms: The number of H atoms on the left is \(2b\), and on the right is \(6c\). So, we have the equation: \(2b = 6c\), which simplifies to \(b = 3c\).
- Solving for Coefficients
To find the smallest whole number coefficients, we assign a value to \(c\). Let's set \(c = 1\).
- Using \(a = 2c\), we find \(a = 2(1) = 2\).
- Using \(b = 3c\), we find \(b = 3(1) = 3\).
The coefficients \(a=2, b=3, c=1\) are already the smallest whole numbers.
- Final Balanced Equation
Substituting the values of \(a, b,\) and \(c\) back into the original equation gives:
\[2C(s) + 3H_2(g) \rightarrow C_2H_6(g)\]
Counting the atoms:
- Left side: 2 C atoms and \(3 \times 2 = 6\) H atoms.
- Right side: 2 C atoms and 6 H atoms.
Since the atoms are balanced, the smallest whole number coefficients are 2, 3, and 1.
Examples
Chemical balancing is essential in real-world applications like industrial manufacturing of chemicals, such as ethane for plastics. By knowing the precise amount of carbon and hydrogen needed to create a specific amount of ethane, scientists can minimize waste and reduce production costs, ensuring a more sustainable process. Understanding stoichiometry is key to chemical engineering and resource management.