To determine which equation accurately represents the balanced combustion reaction for isobutane ([tex]$C_4H_{10}$[/tex]), we need to account for all atoms of each element involved. The combustion of hydrocarbons generally yields carbon dioxide ([tex]$CO_2$[/tex]) and water ([tex]$H_2O$[/tex]). Let's proceed step-by-step to achieve a balanced equation.
1. Write the unbalanced equation:
[tex]\[
C_4H_{10} + O_2 \rightarrow CO_2 + H_2O
\][/tex]
2. Balance carbon atoms:
Isobutane has 4 carbon atoms. Therefore, we need 4 molecules of [tex]$CO_2$[/tex] to balance the carbons:
[tex]\[
C_4H_{10} + O_2 \rightarrow 4CO_2 + H_2O
\][/tex]
3. Balance hydrogen atoms:
Isobutane contains 10 hydrogen atoms. Hence, we require 5 molecules of [tex]$H_2O$[/tex] to balance the hydrogens:
[tex]\[
C_4H_{10} + O_2 \rightarrow 4CO_2 + 5H_2O
\][/tex]
4. Balance oxygen atoms:
On the right side, there are:
- [tex]\(4 \times 2 = 8\)[/tex] oxygen atoms from [tex]$CO_2$[/tex],
- [tex]\(5 \times 1 = 5\)[/tex] oxygen atoms from [tex]$H_2O$[/tex].
A total of [tex]\(8 + 5 = 13\)[/tex] oxygen atoms are required on the left side. Since each molecule of [tex]$O_2$[/tex] contains 2 oxygen atoms, we need [tex]\(\frac{13}{2} = 6.5\)[/tex] molecules of [tex]$O_2$[/tex] to balance the equation:
[tex]\[
C_4H_{10} + 6.5O_2 \rightarrow 4CO_2 + 5H_2O
\][/tex]
After balancing, the combustion reaction should be:
[tex]\[
C_4H_{10} + 6.5O_2 \rightarrow 4CO_2 + 5H_2O + \text{heat}
\][/tex]
Thus, the correct balanced equation for the combustion of isobutane is option C:
[tex]\[
\boxed{C_4H_{10} + 6.5O_2 \rightarrow 4CO_2 + 5H_2O + \text{heat}}
\][/tex]
