To find the coefficient for [tex]\( O_2 \)[/tex] in the balanced equation for the combustion of pentane ([tex]\( C_5H_{12} \)[/tex]), we start by setting up the unbalanced chemical equation:
[tex]\[ C_5H_{12} + O_2 \rightarrow CO_2 + H_2O \][/tex]
Next, follow these steps to balance the equation:
### Step 1: Balance the carbon atoms
Pentane ([tex]\( C_5H_{12} \)[/tex]) has 5 carbon atoms. Therefore, we need 5 molecules of carbon dioxide ([tex]\( CO_2 \)[/tex]) to balance the carbon atoms:
[tex]\[ C_5H_{12} + O_2 \rightarrow 5CO_2 + H_2O \][/tex]
### Step 2: Balance the hydrogen atoms
Pentane ([tex]\( C_5H_{12} \)[/tex]) has 12 hydrogen atoms. Therefore, we need 6 molecules of water ([tex]\( H_2O \)[/tex]) to balance the hydrogen atoms (since each water molecule has 2 hydrogen atoms):
[tex]\[ C_5H_{12} + O_2 \rightarrow 5CO_2 + 6H_2O \][/tex]
### Step 3: Balance the oxygen atoms
Now we need to balance the oxygen atoms. On the right side of the equation, we have:
- From [tex]\( 5CO_2 \)[/tex]: [tex]\( 5 \times 2 = 10 \)[/tex] oxygen atoms
- From [tex]\( 6H_2O \)[/tex]: [tex]\( 6 \times 1 = 6 \)[/tex] oxygen atoms
So, the total number of oxygen atoms needed on the right side is [tex]\( 10 + 6 = 16 \)[/tex] atoms. On the left side, [tex]\( O_2 \)[/tex] is in the diatomic form, so each molecule of [tex]\( O_2 \)[/tex] provides 2 oxygen atoms. We need enough [tex]\( O_2 \)[/tex] molecules to provide 16 oxygen atoms:
[tex]\[ 2x = 16 \implies x = 8 \][/tex]
Therefore, we need 8 molecules of [tex]\( O_2 \)[/tex].
### Balanced Equation
The balanced equation for the combustion of pentane ([tex]\( C_5H_{12} \)[/tex]) is:
[tex]\[ C_5H_{12} + 8O_2 \rightarrow 5CO_2 + 6H_2O \][/tex]
Thus, the coefficient for [tex]\( O_2 \)[/tex] when the equation for the combustion of [tex]\( C_5H_{12} \)[/tex] to [tex]\( CO_2 \)[/tex] and [tex]\( H_2O \)[/tex] is balanced is [tex]\( \boxed{8} \)[/tex].
