For the reaction:
`4NH_(3)(g)+7O_(2(g))hArr4NO_(2(g))+6H_(2)O_(g).K_(p)` is related to `K_(c)` by

Options

A. `K_(p)=K_(c)(RT)`
B. `K_(p)=K_(c)`
C. `K_(p)=K_(c)(RT)^(3)`
D. `K_(p)=K_(c)|(RT)`.

Answer

To solve the problem of how \( K_p \) is related to \( K_c \) for the given reaction:

\[ 4NH_3(g) + 7O_2(g) \rightleftharpoons 4NO_2(g) + 6H_2O(g) \]

we can follow these steps:

### Step 1: Identify the reaction and its components
We start with the balanced chemical equation. The reaction involves:
- Reactants: \( 4 \) moles of \( NH_3 \) and \( 7 \) moles of \( O_2 \)
- Products: \( 4 \) moles of \( NO_2 \) and \( 6 \) moles of \( H_2O \)

### Step 2: Calculate the total number of moles of gaseous products and reactants
- **Total moles of gaseous products** = \( 4 \) (from \( NO_2 \)) + \( 6 \) (from \( H_2O \)) = \( 10 \)
- **Total moles of gaseous reactants** = \( 4 \) (from \( NH_3 \)) + \( 7 \) (from \( O_2 \)) = \( 11 \)

### Step 3: Calculate \( \Delta n_g \)
\[ \Delta n_g = \text{(moles of products)} - \text{(moles of reactants)} = 10 - 11 = -1 \]

### Step 4: Use the relationship between \( K_p \) and \( K_c \)
The relationship between \( K_p \) and \( K_c \) is given by the formula:
\[ K_p = K_c \cdot R^T \cdot \Delta n_g \]
Substituting the value of \( \Delta n_g \):
\[ K_p = K_c \cdot R^T \cdot (-1) \]

### Step 5: Rearranging the equation
This can be rearranged to express \( K_p \) in terms of \( K_c \):
\[ K_p = \frac{K_c}{R^T} \]

### Conclusion
Thus, the relationship between \( K_p \) and \( K_c \) for the given reaction is:
\[ K_p = \frac{K_c}{R^T} \]