To balance the combustion equation for ethane ([tex]\(\text{C}_2\text{H}_6\)[/tex]), we need to ensure that the number of each type of atom on the reactant side is equal to the number on the product side. The equation is:
[tex]\[a \, \text{C}_2\text{H}_6 + b \, \text{O}_2 \rightarrow c \, \text{CO}_2 + d \, \text{H}_2\text{O}\][/tex]
Let's break this down step-by-step:
1. Balance the Carbon atoms:
[tex]\[
\text{C}_2\text{H}_6 \rightarrow 2 \text{C atoms}
\][/tex]
[tex]\[
\text{CO}_2 \rightarrow \text{C atoms on product side}
\][/tex]
To balance the carbon atoms, we need 2 CO[tex]\(_2\)[/tex] molecules on the product side:
[tex]\[
c = 2
\][/tex]
The equation now looks like:
[tex]\[
\text{C}_2\text{H}_6 + b \, \text{O}_2 \rightarrow 2 \, \text{CO}_2 + d \, \text{H}_2\text{O}
\][/tex]
2. Balance the Hydrogen atoms:
[tex]\[
\text{C}_2\text{H}_6 \rightarrow 6 \text{H atoms}
\][/tex]
[tex]\[
\text{H}_2\text{O} \rightarrow 2 \text{H atoms per molecule}
\][/tex]
To balance the hydrogen atoms, we need [tex]\( \frac{6}{2} = 3 \)[/tex] H[tex]\(_2\)[/tex]O molecules:
[tex]\[
d = 3
\][/tex]
The equation now looks like:
[tex]\[
\text{C}_2\text{H}_6 + b \, \text{O}_2 \rightarrow 2 \, \text{CO}_2 + 3 \, \text{H}_2\text{O}
\][/tex]
3. Balance the Oxygen atoms:
[tex]\[
\text{O}_2 \rightarrow 2 \text{O atoms per molecule}
\][/tex]
[tex]\[
\text{CO}_2 \rightarrow 2 \times 2 = 4 \text{O atoms}
\][/tex]
[tex]\[
\text{H}_2\text{O} \rightarrow 3 \times 1 = 3 \text{O atoms}
\][/tex]
In total, we need [tex]\(4 + 3 = 7\)[/tex] Oxygen atoms on the reactant side:
[tex]\[
b = \frac{7}{2} = 3.5
\][/tex]
So the equation becomes:
[tex]\[
\text{C}_2\text{H}_6 + 3.5 \, \text{O}_2 \rightarrow 2 \, \text{CO}_2 + 3 \, \text{H}_2\text{O}
\][/tex]
To avoid fractions, multiply the entire equation by 2:
[tex]\[
2 \text{C}_2\text{H}_6 + 7 \text{O}_2 \rightarrow 4 \text{CO}_2 + 6 \text{H}_2\text{O}
\][/tex]
Hence, the balanced equation is:
[tex]\[2 \, \text{C}_2\text{H}_6 + 7 \, \text{O}_2 \rightarrow 4 \, \text{CO}_2 + 6 \, \text{H}_2\text{O}\][/tex]
Therefore, the coefficients are:
[tex]\[ \boxed{2} \, \text{C}_2\text{H}_6 + \boxed{7} \, \text{O}_2 \rightarrow \boxed{4} \, \text{CO}_2 + \boxed{6} \, \text{H}_2\text{O} \][/tex]
