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 determine the correct 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: Write the expression for \( K_p \) and \( K_c \)

The equilibrium constant \( K_c \) is expressed in terms of concentrations, while \( K_p \) is expressed in terms of partial pressures. For the given reaction:

\[ K_c = \frac{[NH_3]^2}{[N_2][H_2]^3} \]

\[ K_p = \frac{(P_{NH_3})^2}{(P_{N_2})(P_{H_2})^3} \]

### Step 2: Determine \( \Delta N_g \)

Next, we need to calculate \( \Delta N_g \), which is the change in the number of moles of gas during the reaction. This is calculated as:

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

From the reaction:

- Moles of gaseous products (NH3) = 2
- Moles of gaseous reactants (N2 + 3H2) = 1 + 3 = 4

So,

\[ \Delta N_g = 2 - 4 = -2 \]

### Step 3: Use the relationship between \( K_p \) and \( K_c \)

The relationship between \( K_p \) and \( K_c \) is given by the formula:

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

Substituting \( \Delta N_g \):

\[ K_p = K_c (RT)^{-2} \]

### Step 4: Analyze the relationship

From the equation \( K_p = K_c (RT)^{-2} \), we can see that:

\[ K_p = \frac{K_c}{(RT)^2} \]

This implies that \( K_p \) is less than \( K_c \) because \( (RT)^2 \) is a positive number. Thus, we can conclude:

\[ K_p < K_c \]

### Conclusion

The correct answer is that \( K_p \) is less than \( K_c \).

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