For a gaseous reaction
`aA(g)+bB(g)hArrcC(g)+dD(g)`
equilibrium constants `K_(c),K_(p)` and `K_(x)` are
`K_(c)=([C]^(c)[D]^(d))/([A]^(a)[B]^(b)),K_(p)=(Pc^(c).P_(D)^(d))/P_(A)^(a)//P_(b)^(b)` and `Kx=(x_(C)^(c).x_(D)^(d))/(x_(A)^(a).x_(B)^(b)`
where `[A]``rarr` molar concentration of `A,p_(A)` `rarr` partial pressure of A and P `rarr` total pressure, `x_(A)` `rarr` mole fraction of A
Select the write option

Options

A. (a) `K_(p)=K_(c)(RT)^(Deltang),K_(x)=K_(p)(RT)^(Deltang)`
B. (b) `K_(c)=K_(c)(RT)^(Deltang),K_(p)=K_(x)P^(Deltang)`
C. (c) `K_(c)=K_(x)P^(Deltang),K_(p)=K_(x)P^(Deltang)`
D. (d) `K_(c)=K_(p)(RT)^(-Deltang),K_(x)=K_(p)(RT)^(Deltang)`

Answer

To solve the problem, we need to analyze the relationships between the equilibrium constants \( K_c \), \( K_p \), and \( K_x \) for the given gaseous reaction:

\[ aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g) \]

### Step 1: Write the expressions for \( K_c \), \( K_p \), and \( K_x \)

1. **Equilibrium constant \( K_c \)**:
\[ K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]

2. **Equilibrium constant \( K_p \)**:
\[ K_p = \frac{P_C^c P_D^d}{P_A^a P_B^b} \]

3. **Equilibrium constant \( K_x \)**:
\[ K_x = \frac{x_C^c x_D^d}{x_A^a x_B^b} \]

Where:
- \([A]\), \([B]\), \([C]\), and \([D]\) are the molar concentrations.
- \(P_A\), \(P_B\), \(P_C\), and \(P_D\) are the partial pressures.
- \(x_A\), \(x_B\), \(x_C\), and \(x_D\) are the mole fractions.

### Step 2: Relate \( K_p \) and \( K_c \)

Using the ideal gas law, we know that:
\[ P = nRT/V \quad \text{and} \quad [A] = \frac{n_A}{V} \]

Thus, we can express the partial pressures in terms of concentrations:
\[ P_A = [A]RT, \quad P_B = [B]RT, \quad P_C = [C]RT, \quad P_D = [D]RT \]

Substituting these into the expression for \( K_p \):
\[ K_p = \frac{([C]RT)^c ([D]RT)^d}{([A]RT)^a ([B]RT)^b} \]

This simplifies to:
\[ K_p = K_c (RT)^{\Delta n_g} \]
where \( \Delta n_g = (c + d) - (a + b) \) is the change in the number of moles of gas.

### Step 3: Relate \( K_p \) and \( K_x \)

Using the definition of partial pressure in terms of mole fraction:
\[ P_A = x_A P, \quad P_B = x_B P, \quad P_C = x_C P, \quad P_D = x_D P \]

Substituting these into the expression for \( K_p \):
\[ K_p = \frac{(x_C P)^c (x_D P)^d}{(x_A P)^a (x_B P)^b} \]

This simplifies to:
\[ K_p = K_x P^{\Delta n_g} \]

### Step 4: Summary of Relationships

From the above derivations, we have the following relationships:
1. \( K_p = K_c (RT)^{\Delta n_g} \)
2. \( K_p = K_x P^{\Delta n_g} \)

### Step 5: Select the Correct Option

Given the relationships derived:
- The first option states \( K_p = K_c RT^{\Delta n_g} \) which is correct.
- The second option states \( K_p = K_c RT^{\Delta n_g} \) and \( K_p = K_x P^{\Delta n_g} \) which is also correct.

Thus, the correct answer is the second option.