To solve this problem, we will use the concept of the equilibrium constant, denoted as [tex]\( K \)[/tex]. The equilibrium constant expression depends on the balanced chemical equation of the reaction.
Given the reaction equation:
[tex]\[ H_2(g) + I_2(g) \rightleftharpoons 2 HI(g) \][/tex]
The equilibrium constant expression for this reaction is:
[tex]\[ K = \frac{[HI]^2}{[H_2][I_2]} \][/tex]
Where:
- [tex]\([HI]\)[/tex] is the concentration of hydrogen iodide at equilibrium,
- [tex]\([H_2]\)[/tex] is the concentration of hydrogen gas at equilibrium,
- [tex]\([I_2]\)[/tex] is the concentration of iodine gas at equilibrium.
The given equilibrium concentrations are:
- [tex]\([HI] = 4.4 \, \text{mol/L}\)[/tex]
- [tex]\([H_2] = 3.2 \, \text{mol/L}\)[/tex]
- [tex]\([I_2] = 1.5 \, \text{mol/L}\)[/tex]
Substituting these values into the equilibrium constant expression gives us:
[tex]\[ K = \frac{(4.4)^2}{(3.2)(1.5)} \][/tex]
Now, calculate the numerator and denominator separately:
- The numerator is [tex]\((4.4)^2 = 19.36\)[/tex]
- The denominator is [tex]\((3.2) \times (1.5) = 4.8\)[/tex]
Thus:
[tex]\[ K = \frac{19.36}{4.8} \approx 4.033333333333333 \][/tex]
Since the calculated value of [tex]\(K\)[/tex] is approximately [tex]\(4.03\)[/tex], the equilibrium constant for this system matches the value of 4.0 from the choices given.
Therefore, the correct answer is:
[tex]\[ \boxed{4.0} \][/tex]
