To determine the coefficient of [tex]\( O_2 \)[/tex] needed to balance the equation [tex]\( C_3H_8 + O_2 \rightarrow 3CO_2 + 4H_2O \)[/tex], we need to balance all the atoms on both sides of the equation.
Step-by-step process:
1. Carbon atoms (C):
- On the left side, we have 3 carbon atoms in [tex]\( C_3H_8 \)[/tex].
- On the right side, we have 3 carbon atoms in the 3 molecules of [tex]\( CO_2 \)[/tex] (since each [tex]\( CO_2 \)[/tex] contains 1 carbon atom).
Therefore, carbon is already balanced.
2. Hydrogen atoms (H):
- On the left side, we have 8 hydrogen atoms in [tex]\( C_3H_8 \)[/tex].
- On the right side, we have 8 hydrogen atoms in the 4 molecules of [tex]\( H_2O \)[/tex] (since each [tex]\( H_2O \)[/tex] contains 2 hydrogen atoms).
Therefore, hydrogen is already balanced.
3. Oxygen atoms (O):
- On the left side, we have [tex]\( x \)[/tex] molecules of [tex]\( O_2 \)[/tex], each [tex]\( O_2 \)[/tex] molecule containing 2 oxygen atoms, so there are [tex]\( 2x \)[/tex] oxygen atoms.
- On the right side, we have the oxygen atoms from both [tex]\( CO_2 \)[/tex] and [tex]\( H_2O \)[/tex]:
- 3 molecules of [tex]\( CO_2 \)[/tex] contribute [tex]\( 3 \times 2 = 6 \)[/tex] oxygen atoms.
- 4 molecules of [tex]\( H_2O \)[/tex] contribute [tex]\( 4 \times 1 = 4 \)[/tex] oxygen atoms.
Therefore, the total oxygen atoms on the right side are [tex]\( 6 + 4 = 10 \)[/tex].
To balance the oxygen atoms, we set up the equation:
[tex]\[ 2x = 10 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{10}{2} \][/tex]
[tex]\[ x = 5 \][/tex]
Therefore, the coefficient of [tex]\( O_2 \)[/tex] needed to balance the equation is [tex]\( \boxed{5} \)[/tex].
Hence, the correct answer is:
C. 5
