To determine the equilibrium constant [tex]\( K_{\text{eq}} \)[/tex] for the reaction [tex]\( 2 \text{SO}_3(g) \rightleftharpoons 2 \text{SO}_2(g) + \text{O}_2(g) \)[/tex], we follow these steps:
1. Write the balanced chemical equation:
[tex]\[
2 \text{SO}_3(g) \rightleftharpoons 2 \text{SO}_2(g) + \text{O}_2(g)
\][/tex]
2. Understand the form of the equilibrium constant expression:
The general form for the equilibrium constant [tex]\( K_{\text{eq}} \)[/tex] is given by the ratio of the concentrations of products raised to their stoichiometric coefficients to the concentrations of reactants raised to their stoichiometric coefficients.
3. Identify the concentrations involved:
- The concentration of [tex]\(\text{SO}_2\)[/tex] is [tex]\([ \text{SO}_2 ]\)[/tex]
- The concentration of [tex]\(\text{O}_2\)[/tex] is [tex]\([ \text{O}_2 ]\)[/tex]
- The concentration of [tex]\(\text{SO}_3\)[/tex] is [tex]\([ \text{SO}_3 ]\)[/tex]
4. Write the equilibrium constant expression for this reaction:
[tex]\[
K_{\text{eq}} = \frac{[\text{SO}_2]^2 [\text{O}_2]}{[\text{SO}_3]^2}
\][/tex]
This expression takes into account the stoichiometric coefficients from the balanced equation:
- [tex]\(\text{SO}_2\)[/tex] has a coefficient of 2, so its concentration is squared.
- [tex]\(\text{O}_2\)[/tex] has a coefficient of 1, so its concentration is raised to the power of one.
- [tex]\(\text{SO}_3\)[/tex] has a coefficient of 2, so its concentration is also squared.
By analyzing the provided options:
A. [tex]\( K_{\text{eq}} = \frac{2[\text{SO}_2][\text{O}_2]}{2[\text{SO}_3]} \)[/tex]
B. [tex]\( K_{\text{eq}} = \frac{2[\text{SO}_3]}{2[\text{SO}_2][\text{O}_2]} \)[/tex]
C. [tex]\( K_{\text{eq}} = \frac{[\text{SO}_2]^2 [\text{O}_2]}{[\text{SO}_3]^2} \)[/tex]
D. [tex]\( K_{\text{eq}} = \frac{[\text{SO}_3]^2}{[\text{SO}_2]^2 [\text{O}_2]} \)[/tex]
Option C matches the correct equilibrium constant expression:
[tex]\[
K_{\text{eq}} = \frac{[\text{SO}_2]^2 [\text{O}_2]}{[\text{SO}_3]^2}
\][/tex]
Therefore, the correct answer is:
C. [tex]\( K_{\text{eq}} = \frac{[\text{SO}_2]^2 [\text{O}_2]}{[\text{SO}_3]^2} \)[/tex]
