To determine the correct equilibrium expression for the reaction:
[tex]\[ \text{Cd}_{(aq)}^{2+} + 4 \text{Br}_{(aq)}^{-} \longleftrightarrow \text{CdBr}_{4(aq)}^{2-} \][/tex]
we need to consider the different provided forms of the equilibrium constant (K):
1. [tex]\( K = \frac{[\text{CdBr}_4^{2-}]}{[\text{Cd}^{2+}][\text{Br}^-]^4} \)[/tex]
2. [tex]\( K = \frac{[\text{CdBr}_4^{2-}]}{[\text{Cd}^{2+}]} \)[/tex]
3. [tex]\( K = \frac{[\text{CdBr}_4^{2-}]}{[\text{Br}^-]^4} \)[/tex]
4. [tex]\( K = \frac{[\text{Cd}^{2+}][\text{Br}^-]^4}{[\text{CdBr}_4^{2-}]} \)[/tex]
Below, I'll evaluate these options:
### Option 1:
[tex]\[ K = \frac{[\text{CdBr}_4^{2-}]}{[\text{Cd}^{2+}][\text{Br}^-]^4} \][/tex]
This expression represents the standard form for a balanced equilibrium reaction where the equilibrium constant [tex]\(K\)[/tex] is the ratio of the product concentration to the reactant concentrations, each raised to the power of their respective coefficients in the balanced equation.
### Option 2:
[tex]\[ K = \frac{[\text{CdBr}_4^{2-}]}{[\text{Cd}^{2+}]} \][/tex]
This implies the equilibrium constant depends only on the concentration of [tex]\(\text{Cd}^{2+}\)[/tex] and ignores the presence of [tex]\(\text{Br}^-\)[/tex]. This is not correct for our balanced equation because [tex]\(\text{Br}^-\)[/tex] ions play a significant role with a coefficient of 4.
### Option 3:
[tex]\[ K = \frac{[\text{CdBr}_4^{2-}]}{[\text{Br}^-]^4} \][/tex]
This implies the equilibrium constant depends only on [tex]\(\text{Br}^-\)[/tex] and ignores the presence of [tex]\(\text{Cd}^{2+}\)[/tex]. This is not correct for our balanced equation because [tex]\(\text{Cd}^{2+}\)[/tex] has a crucial role in the reaction.
### Option 4:
[tex]\[ K = \frac{[\text{Cd}^{2+}][\text{Br}^-]^4}{[\text{CdBr}_4^{2-}]} \][/tex]
This represents an inverse relationship of what the equilibrium constant should be. It shows the reactants over the products, which is the reverse reaction’s equilibrium constant.
### Conclusion
The standard equilibrium expression for the given reaction is correctly represented in Option 1:
[tex]\[ K = \frac{[\text{CdBr}_4^{2-}]}{[\text{Cd}^{2+}][\text{Br}^-]^4} \][/tex]
This means that, for the given reaction:
[tex]\[ \text{Cd}_{(aq)}^{2+} + 4 \text{Br}_{(aq)}^{-} \longleftrightarrow \text{CdBr}_{4(aq)}^{2-} \][/tex]
the equilibrium constant [tex]\(K\)[/tex] is quantified as the concentration of [tex]\(\text{CdBr}_4^{2-}\)[/tex] divided by the product of [tex]\(\text{Cd}^{2+}\)[/tex] concentration and the fourth power of [tex]\(\text{Br}^-\)[/tex] concentration. Thus, the correct expression is:
[tex]\[ K = \frac{[\text{CdBr}_4^{2-}]}{[\text{Cd}^{2+}][\text{Br}^-]^4} \][/tex]
Given that [tex]\(K_1 = 1.0\)[/tex], we recognize that this equilibrium expression correctly describes the situation.
