To balance the chemical equation:
[tex]\[
CH_3(CH_2)_2CH_3(g) + O_2(g) \rightarrow CO_2(g) + H_2O(g)
\][/tex]
We'll use the stoichiometric coefficients to ensure that the number of atoms of each element on the reactants side equals the number on the products side.
### Step-by-Step Solution:
1. Write the unbalanced equation:
[tex]\[
CH_3(CH_2)_2CH_3 + O_2 \rightarrow CO_2 + H_2O
\][/tex]
2. Identify the number of each type of atom in the unbalanced equation:
- For [tex]\(CH_3(CH_2)_2CH_3\)[/tex]:
- Carbon (C): [tex]\(3\)[/tex] from [tex]\(CH_3\)[/tex] + [tex]\(2 \times 2\)[/tex] from [tex]\((CH_2)\)[/tex] + [tex]\(1\)[/tex] from [tex]\(CH_3\)[/tex] = [tex]\(8\)[/tex] carbons.
- Hydrogen (H): [tex]\(3\)[/tex] from first [tex]\(CH_3\)[/tex] + [tex]\(2 \times 2\)[/tex] from two middle [tex]\(CH_2\)[/tex] + [tex]\(3\)[/tex] from last [tex]\(CH_3\)[/tex] = [tex]\(18\)[/tex] hydrogens.
- For [tex]\(O_2\)[/tex]:
- Oxygen (O): [tex]\(2\)[/tex] oxygens.
3. Write the number of atoms of each element needed to balance:
- For [tex]\(CO_2\)[/tex]:
- Each [tex]\(CO_2\)[/tex] has [tex]\(1\)[/tex] carbon and [tex]\(2\)[/tex] oxygens.
- For [tex]\(H_2O\)[/tex]:
- Each [tex]\(H_2O\)[/tex] has [tex]\(2\)[/tex] hydrogens and [tex]\(1\)[/tex] oxygen.
4. Set up the equations based on atom count balance:
To balance the carbon atoms:
[tex]\[
8 \ \textrm{carbons} \rightarrow 8\ CO_2
\][/tex]
To balance the hydrogen atoms:
[tex]\[
18 \ \textrm{hydrogens} \rightarrow 9\ H_2O
\][/tex]
5. Determine the number of oxygen atoms required on the products side:
- From [tex]\(8\ CO_2\)[/tex], we have [tex]\(8 \times 2 = 16\)[/tex] oxygens.
- From [tex]\(9\ H_2O\)[/tex], we have [tex]\(9 \times 1 = 9\)[/tex] oxygens.
Total oxygens needed on the products side:
[tex]\[
16 + 9 = 25\ \textrm{oxygens}
\][/tex]
Therefore:
[tex]\[
\frac{25}{2}\ O_2
\][/tex]
6. Write the balanced equation using these coefficients:
The coefficients we have are [tex]\(1\)[/tex] for [tex]\(CH_3(CH_2)_2CH_3\)[/tex], [tex]\(\frac{25}{2}\)[/tex] for [tex]\(O_2\)[/tex], [tex]\(8\)[/tex] for [tex]\(CO_2\)[/tex], and [tex]\(9\)[/tex] for [tex]\(H_2O\)[/tex].
7. Convert fraction coefficients to whole numbers by multiplying through by [tex]\(2\)[/tex]:
[tex]\[
2 \left[ CH_3(CH_2)_2CH_3 + \frac{25}{2} O_2 \rightarrow 8 CO_2 + 9 H_2O \right]
\][/tex]
Yields:
[tex]\[
2\ CH_3(CH_2)_2CH_3 + 25\ O_2 \rightarrow 16\ CO_2 + 18\ H_2O
\][/tex]
Thus, the balanced equation is:
[tex]\[
2\ CH_3(CH_2)_2CH_3 + 25\ O_2 \rightarrow 16\ CO_2 + 18\ H_2O
\][/tex]
