To balance the reaction:
[tex]\[
\text{CO}_2 + \text{H}_2\text{O} \rightarrow \text{C}_6\text{H}_{12}\text{O}_6 + \text{O}_2
\][/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.
Step-by-Step Solution:
1. Identify the molecules involved and their elemental composition:
- Carbon Dioxide ([tex]\(\text{CO}_2\)[/tex]): 1 Carbon (C), 2 Oxygen (O).
- Water ([tex]\(\text{H}_2\text{O}\)[/tex]): 2 Hydrogen (H), 1 Oxygen (O).
- Glucose ([tex]\(\text{C}_6\text{H}_{12}\text{O}_6\)[/tex]): 6 Carbons (C), 12 Hydrogens (H), 6 Oxygens (O).
- Oxygen ([tex]\(\text{O}_2\)[/tex]): 2 Oxygens (O).
2. Balance Carbons:
- On the product side, there are 6 carbons in one molecule of glucose ([tex]\(\text{C}_6\text{H}_{12}\text{O}_6\)[/tex]).
- Therefore, we need 6 molecules of carbon dioxide ([tex]\(\text{CO}_2\)[/tex]) to supply these 6 carbons.
[tex]\[
6 \text{CO}_2 + \text{H}_2\text{O} \rightarrow \text{C}_6\text{H}_{12}\text{O}_6 + \text{O}_2
\][/tex]
3. Balance Hydrogens:
- On the product side, there are 12 hydrogens in one molecule of glucose ([tex]\(\text{C}_6\text{H}_{12}\text{O}_6\)[/tex]).
- To match these 12 hydrogens on the reactant side, we need 6 molecules of water ([tex]\(\text{H}_2\text{O}\)[/tex]).
[tex]\[
6 \text{CO}_2 + 6 \text{H}_2\text{O} \rightarrow \text{C}_6\text{H}_{12}\text{O}_6 + \text{O}_2
\][/tex]
4. Balance Oxygens:
- On the product side, there are 6 oxygens in one molecule of glucose ([tex]\(\text{C}_6\text{H}_{12}\text{O}_6\)[/tex]) and 2 more from the oxygen molecule ([tex]\(\text{O}_2\)[/tex]), making a total of 8 oxygens.
- On the reactant side, we have 6 molecules of [tex]\(\text{CO}_2\)[/tex], contributing [tex]\(6 \times 2 = 12\)[/tex] oxygens, and 6 molecules of [tex]\(\text{H}_2\text{O}\)[/tex], contributing [tex]\(6 \times 1 = 6\)[/tex] oxygens, making a total of [tex]\(12 + 6 = 18\)[/tex] oxygens.
- Since we need 18 oxygens on the reactant side to balance, we already have 12 oxygens in the [tex]\(\text{CO}_2\)[/tex]. After accounting for the 6 oxygens in water ([tex]\(\text{H}_2\text{O}\)[/tex]), we need additional 6 oxygens on the product side to balance. Thus we need 6 molecules of [tex]\(\text{O}_2\)[/tex].
[tex]\[
6 \text{CO}_2 + 6 \text{H}_2\text{O} \rightarrow \text{C}_6\text{H}_{12}\text{O}_6 + 6 \text{O}_2
\][/tex]
Thus, the balanced chemical equation is:
[tex]\[
\boxed{6} \text{CO}_2 + \boxed{6} \text{H}_2\text{O} \rightarrow \boxed{1} \text{C}_6\text{H}_{12}\text{O}_6 + \boxed{6} \text{O}_2
\][/tex]
So the coefficients are [tex]\(6, 6, 1, 6\)[/tex].
