Write the relationship between `K_p` and `K_c` for the reaction
`2SO_2(g) +O_2(g) hArr 2SO_3(g)`

Answer

To find the relationship between \( K_p \) and \( K_c \) for the reaction

\[ 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \]

we can follow these steps:

### Step 1: Identify the reaction and its components
The given reaction involves the following:
- Reactants: 2 moles of \( SO_2 \) and 1 mole of \( O_2 \)
- Products: 2 moles of \( SO_3 \)

### Step 2: Write the expression for \( K_c \)
The equilibrium constant \( K_c \) is expressed in terms of the concentrations of the reactants and products:

\[ K_c = \frac{[SO_3]^2}{[SO_2]^2 \cdot [O_2]} \]

### Step 3: Determine the change in moles (\( \Delta N_g \))
To relate \( K_p \) and \( K_c \), we need to calculate \( \Delta N_g \), which is the change in the number of moles of gas:

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

From the reaction:
- Moles of products = 2 (from \( 2SO_3 \))
- Moles of reactants = 2 (from \( 2SO_2 \)) + 1 (from \( O_2 \)) = 3

Thus,

\[ \Delta N_g = 2 - 3 = -1 \]

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

\[ K_p = K_c \cdot R^{\Delta N_g} \]

Substituting \( \Delta N_g = -1 \):

\[ K_p = K_c \cdot R^{-1} \]

This can also be written as:

\[ K_p = \frac{K_c}{R} \]

### Final Answer
The relationship between \( K_p \) and \( K_c \) for the reaction \( 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \) is:

\[ K_p = \frac{K_c}{R} \]