To determine the equilibrium constant [tex]\( K \)[/tex] for the given reaction [tex]\( O_2(g) + 2 SO_2(g) \rightleftarrows 2 SO_3(g) \)[/tex], we will use the equilibrium concentrations provided in the table.
The general formula for the equilibrium constant [tex]\( K \)[/tex] for a reaction of the form:
[tex]\[ aA + bB \rightleftarrows cC + dD \][/tex]
is given by:
[tex]\[ K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \][/tex]
For our specific reaction:
[tex]\[ O_2(g) + 2 SO_2(g) \rightleftarrows 2 SO_3(g) \][/tex]
The equilibrium constant expression [tex]\( K \)[/tex] is:
[tex]\[ K = \frac{[SO_3]^2}{[SO_2]^2 [O_2]} \][/tex]
From the given equilibrium concentrations:
- [tex]\([ O_2 ] = 0.024 \, \text{M}\)[/tex]
- [tex]\([ SO_2 ] = 0.015 \, \text{M}\)[/tex]
- [tex]\([ SO_3 ] = 0.26 \, \text{M}\)[/tex]
Substitute these values into the equilibrium constant expression:
[tex]\[ K = \frac{(0.26)^2}{(0.015)^2 \cdot 0.024} \][/tex]
Now, let's compute each part of the expression:
1. Calculate the numerator [tex]\([SO_3]^2\)[/tex]:
[tex]\[ (0.26)^2 = 0.0676 \][/tex]
2. Calculate the denominator [tex]\([SO_2]^2 [O_2]\)[/tex]:
[tex]\[ (0.015)^2 = 0.000225 \][/tex]
[tex]\[ 0.000225 \cdot 0.024 = 0.0000054 \][/tex]
3. Divide the numerator by the denominator:
[tex]\[ K = \frac{0.0676}{0.0000054} = 12518.51851851852 \][/tex]
Thus, the equilibrium constant [tex]\( K \approx 1.3 \times 10^4 \)[/tex].
To determine whether the reaction is reactant favored or product favored, we compare [tex]\(K\)[/tex] to 1:
- If [tex]\( K < 1 \)[/tex], the reaction is reactant favored.
- If [tex]\( K > 1 \)[/tex], the reaction is product favored.
Since [tex]\( K = 1.3 \times 10^4 \)[/tex] is much larger than 1, the reaction is product favored.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{D. } K = 1.3 \times 10^4; \text{ product favored}} \][/tex]
