For a gaseous reaction
`aA(g)+bB(g)hArrcC(g)+dD(g)`
equilibrium constants `K_(c),K_(p)` and `K_(x)` are represented by the following reation
`K_(c)=([C]^(c)[D]^(d))/([A]^(a)[B]^(b)),K_(p)=(Pc^(c).P_(D)^(d))/P_(A)^(a)` and `Kx=(x_(C)^(c).x_(D)^(d))/(x_(A)^(a).x_(B)^(b)`
where `[A]` represents molar concentrationof `A,p_(A)` represents partial pressure of A and P represents total pressure, `x_(A)` represents mole fraction of For the following equilibrium relation betwen `K_(c)` and `K_(c)` (in terms of mole fraction) is
`PCl_(3)(g)+Cl_(2)(g)hArrPCl_(5)(g)`

Options

A. `K_(c)=K_(x)(RT)^(-1)`
B. `K_(c)=K_(x)(RT)`
C. `K_(c)=K_(x)((RT)/P)`
D. `K_(c)=K_(x)((P)/(RT))`

Answer

To solve the problem, we need to establish the relationship between the equilibrium constants \( K_c \) and \( K_x \) for the given reaction:

\[ \text{PCl}_3(g) + \text{Cl}_2(g) \rightleftharpoons \text{PCl}_5(g) \]

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

1. **For \( K_c \)** (concentration equilibrium constant):
\[ K_c = \frac{[\text{PCl}_5]}{[\text{PCl}_3][\text{Cl}_2]} \]

2. **For \( K_p \)** (pressure equilibrium constant):
\[ K_p = \frac{P_{\text{PCl}_5}}{P_{\text{PCl}_3} P_{\text{Cl}_2}} \]

3. **For \( K_x \)** (mole fraction equilibrium constant):
\[ K_x = \frac{x_{\text{PCl}_5}}{x_{\text{PCl}_3} x_{\text{Cl}_2}} \]

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

Using the ideal gas law, we know that:
\[ P = C \cdot RT \]
where \( P \) is the pressure, \( C \) is the concentration, \( R \) is the gas constant, and \( T \) is the temperature.

From this, we can express \( K_p \) in terms of \( K_c \):
\[ K_p = K_c \cdot (RT)^{\Delta n_g} \]
where \( \Delta n_g \) is the change in the number of moles of gas during the reaction.

### Step 3: Calculate \( \Delta n_g \)

For the reaction:
\[ \text{PCl}_3(g) + \text{Cl}_2(g) \rightleftharpoons \text{PCl}_5(g) \]
- Moles of products = 1 (from PCl5)
- Moles of reactants = 2 (from PCl3 and Cl2)

Thus,
\[ \Delta n_g = \text{(moles of products)} - \text{(moles of reactants)} = 1 - 2 = -1 \]

### Step 4: Relate \( K_x \) and \( K_p \)

Using the relationship between mole fractions and partial pressures:
\[ P_i = x_i \cdot P_{\text{total}} \]
We can express \( K_x \) in terms of \( K_p \):
\[ K_x = \frac{x_{\text{PCl}_5}}{x_{\text{PCl}_3} x_{\text{Cl}_2}} = \frac{\frac{P_{\text{PCl}_5}}{P_{\text{total}}}}{\frac{P_{\text{PCl}_3}}{P_{\text{total}}} \cdot \frac{P_{\text{Cl}_2}}{P_{\text{total}}}} = \frac{P_{\text{PCl}_5}}{P_{\text{PCl}_3} P_{\text{Cl}_2}} \cdot \frac{1}{P_{\text{total}}^2} \]
Thus,
\[ K_x = \frac{K_p}{P_{\text{total}}^2} \]

### Step 5: Combine the relationships

From the previous steps, we have:
1. \( K_p = K_c \cdot (RT)^{-1} \)
2. \( K_x = \frac{K_p}{P_{\text{total}}^2} \)

Substituting \( K_p \) into the equation for \( K_x \):
\[ K_x = \frac{K_c \cdot (RT)^{-1}}{P_{\text{total}}^2} \]

### Step 6: Rearranging to find the relationship between \( K_c \) and \( K_x \)

Rearranging gives:
\[ K_c = K_x \cdot P_{\text{total}}^2 \cdot RT \]

### Final Relationship

Thus, the relationship between \( K_c \) and \( K_x \) is:
\[ K_c = K_x \cdot RT \cdot P_{\text{total}}^2 \]