To determine the equilibrium constant for the reaction [tex]\(SO_2(g) + NO_2(g) = SO_3(g) + NO(g)\)[/tex], we need to write the expression for the equilibrium constant [tex]\( K_{\text{eq}} \)[/tex] in terms of the concentrations of the reactants and products.
For a general chemical reaction of the form:
[tex]\[
\text{aA} + \text{bB} \rightleftharpoons \text{cC} + \text{dD}
\][/tex]
the equilibrium constant [tex]\( K_{\text{eq}} \)[/tex] is given by:
[tex]\[
K_{\text{eq}} = \frac{[\text{C}]^c [\text{D}]^d}{[\text{A}]^a [\text{B}]^b}
\][/tex]
Here, [tex]\( \left[ \text{species} \right] \)[/tex] denotes the concentration of the species.
For the given reaction:
[tex]\[
SO_2(g) + NO_2(g) \rightleftharpoons SO_3(g) + NO(g)
\][/tex]
we can identify:
- [tex]\( \text{A} \)[/tex] as [tex]\(SO_2\)[/tex] with [tex]\(a = 1\)[/tex]
- [tex]\( \text{B} \)[/tex] as [tex]\(NO_2\)[/tex] with [tex]\(b =1\)[/tex]
- [tex]\( \text{C} \)[/tex] as [tex]\(SO_3\)[/tex] with [tex]\(c = 1\)[/tex]
- [tex]\( \text{D} \)[/tex] as [tex]\(NO\)[/tex] with [tex]\(d = 1\)[/tex]
Thus, the equilibrium constant [tex]\( K_{\text{eq}} \)[/tex] can be written as:
[tex]\[
K_{\text{eq}} = \frac{[\text{SO}_3][\text{NO}]}{[\text{SO}_2][\text{NO}_2]}
\][/tex]
This expression indicates the ratio of the products' concentrations to the reactants' concentrations at equilibrium.
Given the multiple-choice options:
A. [tex]\(K_{\text {eq }}=\frac{\left[ SO _3\right]\left[ SO _2\right]}{[ NO ]\left[ NO _2\right]}\)[/tex]
B. [tex]\(K_{\text {eq }}=\frac{\left[ SO _3 \right] [ NO ]}{\left[ SO _2 \right]^2 \left[ NO _2 \right]^2}\)[/tex]
C. [tex]\(K_{\text {eq }}=\frac{\left[ SO _2 \right]\left[ NO _2 \right]}{\left[ SO _3 \right] [ NO ]}\)[/tex]
D. [tex]\(K_{\text {eq }}=\frac{\left[ SO _3 \right] [ NO ]}{\left[ SO _2 \right] \left[ NO _2 \right]}\)[/tex]
The correct expression matches option D:
[tex]\[
K_{\text{eq }}=\frac{[\text{SO}_3][\text{NO}]}{[\text{SO}_2][\text{NO}_2]}
\][/tex]
Therefore, the correct answer is option D.
