For the reaction: `Cu(s) + 2Ag_((aq))^+ Leftrightarrow Cu_((aq))^(2+) +2Ag_((s))`, the equilibrium constant is given by

Options

A. `([Cu^(2+)] [Ag]^(2+))/([Cu] [Ag^(+)]^(2))`
B. `([Cu^(2+)] [Ag]^(2))/([Cu] [Ag^(+)]^(2))`
C. `([Cu^(2+)])/([Ag^(+)]^(2))`
D. `([Ag^(+)]^(2))/([Cu^(2+)])`

Answer

To derive the equilibrium constant expression for the reaction:

\[ \text{Cu(s)} + 2\text{Ag}^+_{(aq)} \leftrightarrow \text{Cu}^{2+}_{(aq)} + 2\text{Ag(s)} \]

we will follow these steps:

### Step 1: Write the balanced chemical equation
The balanced equation is already provided:
\[ \text{Cu(s)} + 2\text{Ag}^+_{(aq)} \leftrightarrow \text{Cu}^{2+}_{(aq)} + 2\text{Ag(s)} \]

### Step 2: Identify the phases of the reactants and products
- **Reactants**:
- Cu is a solid (s)
- \( \text{Ag}^+ \) is in aqueous solution (aq)
- **Products**:
- \( \text{Cu}^{2+} \) is in aqueous solution (aq)
- Ag is a solid (s)

### Step 3: Write the expression for the equilibrium constant (K)
The equilibrium constant expression is derived from the concentrations of the products raised to the power of their coefficients divided by the concentrations of the reactants raised to the power of their coefficients.

The general form is:
\[ K = \frac{[\text{Products}]}{[\text{Reactants}]} \]

### Step 4: Apply the equilibrium constant expression to the reaction
For the given reaction, the equilibrium constant expression will be:
\[ K = \frac{[\text{Cu}^{2+}] \cdot [\text{Ag}]^2}{[\text{Ag}^+]^2 \cdot [\text{Cu}]} \]

### Step 5: Consider the activity of solids
In equilibrium expressions, the activities of pure solids and pure liquids are considered to be 1. Therefore, we can ignore the terms for Cu(s) and Ag(s) in the expression.

### Step 6: Simplify the equilibrium constant expression
Since the activities of Cu(s) and Ag(s) are both 1, we simplify the expression to:
\[ K = \frac{[\text{Cu}^{2+}]}{[\text{Ag}^+]^2} \]

### Final Expression
Thus, the expression for the equilibrium constant \( K \) for the reaction is:
\[ K = \frac{[\text{Cu}^{2+}]}{[\text{Ag}^+]^2} \]