What is the relation between `K_p and K_c` for a general reaction,
`aA+bB iff cC+dD`?

Options

A. `K_p = K_c (RT)^(Delta n)`
B. `K_c = K_p (RT)^(Delta n)`
C. `K_p = 1/( K_c (RT)^(Delta n))`
D. `K_c = 1/(K_p (RT)^(Delta n))`

Answer

To find the relation between \( K_p \) and \( K_c \) for a general reaction of the form:

\[ aA + bB \rightleftharpoons cC + dD \]

we will derive the relationship step by step.

### Step 1: Define \( K_c \) and \( K_p \)

The equilibrium constant \( K_c \) is defined in terms of the concentrations of the products and reactants:

\[ K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]

where \([C]\), \([D]\), \([A]\), and \([B]\) are the molar concentrations of the respective species at equilibrium.

The equilibrium constant \( K_p \) is defined in terms of the partial pressures of the products and reactants:

\[ K_p = \frac{P_C^c P_D^d}{P_A^a P_B^b} \]

where \( P_C \), \( P_D \), \( P_A \), and \( P_B \) are the partial pressures of the respective gases at equilibrium.

### Step 2: Relate Concentration and Partial Pressure

Using the ideal gas law, we can relate the concentration \( C \) of a gas to its partial pressure \( P \):

\[ P = C \cdot RT \]

where \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin. Thus, we can express the concentrations in terms of partial pressures:

\[ C_A = \frac{P_A}{RT}, \quad C_B = \frac{P_B}{RT}, \quad C_C = \frac{P_C}{RT}, \quad C_D = \frac{P_D}{RT} \]

### Step 3: Substitute Concentrations into \( K_c \)

Substituting these expressions into the equation for \( K_c \):

\[ K_c = \frac{\left(\frac{P_C}{RT}\right)^c \left(\frac{P_D}{RT}\right)^d}{\left(\frac{P_A}{RT}\right)^a \left(\frac{P_B}{RT}\right)^b} \]

### Step 4: Simplify the Equation

This simplifies to:

\[ K_c = \frac{P_C^c P_D^d}{P_A^a P_B^b} \cdot \frac{1}{(RT)^{c+d}} \cdot (RT)^{a+b} \]

Rearranging gives:

\[ K_c = K_p \cdot \frac{1}{(RT)^{c+d}} \cdot (RT)^{a+b} \]

### Step 5: Combine Terms

Combining the terms gives:

\[ K_p = K_c \cdot (RT)^{\Delta n} \]

where \( \Delta n = (c + d) - (a + b) \) is the change in the number of moles of gas.

### Final Relation

Thus, the relation between \( K_p \) and \( K_c \) is:

\[ K_p = K_c \cdot RT^{\Delta n} \]