The relation between equilibrium constant `K_p and K_c` is

Options

A. `K_c=K_p(RT)^(Deltan)`
B. `K_p=K_c(RT)^(Deltan)`
C. `K_p=((K_c)/(RT))^(Deltan)`
D. `K_p-K_c=(RT)^(Deltan)`

Explanation

To derive the relationship between the equilibrium constants \( K_p \) and \( K_c \), we can follow these steps:

### Step 1: Understand the Definitions
- **\( K_c \)** is the equilibrium constant expressed in terms of molar concentrations (moles per liter) of reactants and products.
- **\( K_p \)** is the equilibrium constant expressed in terms of partial pressures of gaseous reactants and products.

### Step 2: Write the General Reaction
Consider a general reaction:
\[ aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g) \]

### Step 3: Write the Expressions for \( K_c \) and \( K_p \)
- The expression for \( K_c \) is given by:
\[ K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]
- The expression for \( K_p \) is given by:
\[ K_p = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b} \]
where \( P \) represents the partial pressures of the gases.

### Step 4: Relate Concentration and Pressure
For gases, the relationship between concentration and pressure is given by the ideal gas law:
\[ P = [C]RT \]
where \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin. Therefore, we can express concentrations in terms of pressures:
\[ [C] = \frac{P_C}{RT}, \quad [D] = \frac{P_D}{RT}, \quad [A] = \frac{P_A}{RT}, \quad [B] = \frac{P_B}{RT} \]

### Step 5: Substitute Concentrations into \( K_c \)
Substituting these expressions into the \( K_c \) equation, we get:
\[ K_c = \frac{\left(\frac{P_C}{RT}\right)^c \left(\frac{P_D}{RT}\right)^d}{\left(\frac{P_A}{RT}\right)^a \left(\frac{P_B}{RT}\right)^b} \]

### Step 6: Simplify the Expression
This simplifies to:
\[ K_c = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b} \cdot \frac{1}{(RT)^{c+d-a-b}} \]
Thus, we can write:
\[ K_c = K_p \cdot \frac{1}{(RT)^{\Delta n}} \]
where \( \Delta n = (c + d) - (a + b) \) is the change in the number of moles of gas.

### Step 7: Rearranging the Equation
Rearranging gives us the relationship:
\[ K_p = K_c (RT)^{\Delta n} \]

### Final Result
Thus, the relationship between \( K_p \) and \( K_c \) is:
\[ K_p = K_c (RT)^{\Delta n} \]