For the reaction
`CuSO_(4).5H_(2)O(s) hArr CuSO_(4).3H_(2)O(s)+2H_(2)O(g)`
Which one is the correct representation?

Options

A. `K_(p)=[p_(H_(2)O)]^(2)`
B. `K_(c)=[H_(2)O]^(2)`
C. `K_(p)=K_(c)(RT)^(2)`
D. All are correct

Answer

To determine the correct representation for the reaction:

\[ \text{CuSO}_4 \cdot 5\text{H}_2\text{O (s)} \rightleftharpoons \text{CuSO}_4 \cdot 3\text{H}_2\text{O (s)} + 2\text{H}_2\text{O (g)} \]

we need to analyze the equilibrium expressions for this reaction.

### Step 1: Identify the components of the reaction
In this reaction, we have:
- Reactants: \(\text{CuSO}_4 \cdot 5\text{H}_2\text{O (s)}\) (solid)
- Products: \(\text{CuSO}_4 \cdot 3\text{H}_2\text{O (s)}\) (solid) and \(2\text{H}_2\text{O (g)}\) (gas)

### Step 2: Write the equilibrium constant expression
The equilibrium constant expression, \(K_p\), is defined for gaseous and aqueous species only. Solids and pure liquids do not appear in the expression because their activities are considered to be 1.

For this reaction, the equilibrium constant in terms of partial pressures (\(K_p\)) is given by:

\[ K_p = \frac{(P_{\text{H}_2\text{O}})^2}{1} = P_{\text{H}_2\text{O}}^2 \]

### Step 3: Write the equilibrium constant in terms of concentration
Similarly, the equilibrium constant in terms of concentration (\(K_c\)) is given by:

\[ K_c = [\text{H}_2\text{O}]^2 \]

### Step 4: Relate \(K_p\) and \(K_c\)
The relationship between \(K_p\) and \(K_c\) is given by the equation:

\[ K_p = K_c \cdot R^T \cdot \Delta n_g \]

where \(\Delta n_g\) is the change in the number of moles of gas, calculated as:

\[ \Delta n_g = \text{(moles of gaseous products)} - \text{(moles of gaseous reactants)} = 2 - 0 = 2 \]

Thus, we can express \(K_p\) as:

\[ K_p = K_c \cdot R^T \cdot 2 \]

### Conclusion
From the analysis:
1. \(K_p = P_{\text{H}_2\text{O}}^2\)
2. \(K_c = [\text{H}_2\text{O}]^2\)
3. \(K_p = K_c \cdot R^T \cdot 2\)

Therefore, all three options provided in the question are correct, leading us to conclude that the correct representation is option 4, which states that all of the above are correct.