Which of the following is correct for the reaction? `N_(2)(g) + 3H_(2)(g) rarr 2NH_(3)(g)`

Options

A. `K_(P) = K_(c)`
B. `K_(P) lt K_(c)`
C. `K_(P) gt K_(c)`
D. Pressure is required to predict the correlation

Explanation

To solve the problem regarding the relationship between \( K_p \) and \( K_c \) for the reaction \( N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \), we will follow these steps:

### Step 1: Identify the Reaction
The given reaction is:
\[ N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \]

### Step 2: Determine the Change in Moles of Gas (\( \Delta N_g \))
To find \( \Delta N_g \), we need to calculate the number of moles of gaseous products and reactants.

- **Products**: There are 2 moles of \( NH_3 \).
- **Reactants**: There is 1 mole of \( N_2 \) and 3 moles of \( H_2 \), totaling 4 moles.

Now, we can calculate \( \Delta N_g \):
\[ \Delta N_g = \text{moles of products} - \text{moles of reactants} = 2 - 4 = -2 \]

### Step 3: Use the Relationship Between \( K_p \) and \( K_c \)
The relationship between \( K_p \) and \( K_c \) is given by the formula:
\[ K_p = K_c \cdot (RT)^{\Delta N_g} \]
Substituting \( \Delta N_g = -2 \):
\[ K_p = K_c \cdot (RT)^{-2} \]
This can be rewritten as:
\[ K_p = \frac{K_c}{(RT)^2} \]

### Step 4: Analyze the Relationship
Since \( R \) (the gas constant) and \( T \) (temperature) are both positive quantities, \( (RT)^2 \) is also positive. Therefore, we can conclude that:
\[ K_c = K_p \cdot (RT)^2 \]
This implies that \( K_c \) is greater than \( K_p \):
\[ K_c > K_p \]

### Conclusion
Thus, the correct relationship for the reaction is:
\[ K_c > K_p \]