`K_p` for the reaction :
`N_2O_4(g) hArr 2NO_2(g)` is 0.157 atm at `27^@C` and 1 atm pressure . Calculate `K_c` for the reaction.

Explanation

To calculate \( K_c \) for the reaction \( N_2O_4(g) \rightleftharpoons 2NO_2(g) \), we can use the relationship between \( K_p \) and \( K_c \):

### Step 1: Identify the given values
- \( K_p = 0.157 \, \text{atm} \)
- Temperature \( T = 27^\circ C = 273 + 27 = 300 \, \text{K} \)
- The pressure is given as \( 1 \, \text{atm} \).

### Step 2: Determine \( \Delta n_g \)
The change in the number of moles of gas (\( \Delta n_g \)) is calculated as follows:
- For the products: \( 2 \, \text{moles of } NO_2 \)
- For the reactants: \( 1 \, \text{mole of } N_2O_4 \)

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

### Step 3: Use the relationship between \( K_p \) and \( K_c \)
The relationship between \( K_p \) and \( K_c \) is given by the equation:
\[ K_p = K_c \cdot R^T \cdot \Delta n_g \]
where \( R \) (the gas constant) is \( 0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1} \).

### Step 4: Substitute the known values into the equation
Substituting the known values into the equation:
\[ 0.157 = K_c \cdot (0.0821)^{300} \cdot 1 \]

### Step 5: Calculate \( K_c \)
First, calculate \( R \cdot T \):
\[ R \cdot T = 0.0821 \cdot 300 = 24.63 \, \text{L atm/mol} \]

Now, substitute this back into the equation:
\[ 0.157 = K_c \cdot 24.63 \]

Now, solve for \( K_c \):
\[ K_c = \frac{0.157}{24.63} \approx 0.00637 \, \text{mol/L} \]

### Final Answer
Thus, the value of \( K_c \) for the reaction is approximately \( 0.00637 \, \text{mol/L} \).

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