To balance the chemical equation [tex]\( \text{C}_3\text{H}_8 + \text{O}_2 \rightarrow 3 \text{CO}_2 + 4 \text{H}_2\text{O} \)[/tex], we need to balance the number of each type of atom on both sides of the equation. Here is a step-by-step solution:
1. Identify and Write Down the Number of Atoms for Each Element:
On the left side (reactants):
- Carbon (C): 3 (from [tex]\( \text{C}_3\text{H}_8 \)[/tex])
- Hydrogen (H): 8 (from [tex]\( \text{C}_3\text{H}_8 \)[/tex])
- Oxygen (O): unknown (because it comes from [tex]\( \text{O}_2 \)[/tex])
On the right side (products):
- Carbon (C): 3 (from 3 [tex]\( \text{CO}_2 \)[/tex])
- Hydrogen (H): 8 (from 4 [tex]\( \text{H}_2\text{O} \)[/tex])
- Oxygen (O): determined from both [tex]\( \text{CO}_2 \)[/tex] and [tex]\( \text{H}_2\text{O} \)[/tex]
- [tex]\( 3 \text{CO}_2 \)[/tex] contributes [tex]\( 3 \times 2 = 6 \)[/tex] oxygen atoms
- [tex]\( 4 \text{H}_2\text{O} \)[/tex] contributes [tex]\( 4 \times 1 = 4 \)[/tex] oxygen atoms
- Total oxygen atoms on the right side: 6 + 4 = 10
2. Balance Each Element One at a Time:
- Carbon: Both sides already have 3 carbon atoms. No need to add more carbon-containing molecules.
- Hydrogen: Both sides already have 8 hydrogen atoms. No need to add more hydrogen-containing molecules.
- Oxygen: The left side needs to have the same number of oxygen atoms as the right side. We have a total of 10 oxygen atoms on the right side and we need them on the left side using [tex]\( \text{O}_2 \)[/tex].
3. Determine the Coefficient for [tex]\( \text{O}_2 \)[/tex]:
- Each [tex]\( \text{O}_2 \)[/tex] molecule contributes 2 oxygen atoms.
- To get 10 oxygen atoms on the left side, we need [tex]\( \frac{10}{2} = 5 \)[/tex] molecules of [tex]\( \text{O}_2 \)[/tex].
Therefore, the coefficient of [tex]\( \text{O}_2 \)[/tex] needed to balance the equation is:
C. 5
