For the reaction,
` 2NO_2 (g) hArr 2NO(g) +O_2(g), `
` (K_c= 1.8 xx 10 ^(-6) at 184^(@) C )`
` (R= 0.0083 kJ) // (" mol K " ) `
When `K_p and K_c ` are compared at `184^(@)C ` it is found that

Options

A. Whether `K_p ` is greater than , less than or equal to `K_c` depends upon the total gas pressure.
B. `K_p = K_c`
C. ` K_ p ` is less than ` K_c`
D. ` K_P ` is greater than ` K_c`

Answer

To solve the problem, we need to compare the equilibrium constants \( K_p \) and \( K_c \) for the reaction:

\[ 2NO_2 (g) \rightleftharpoons 2NO(g) + O_2(g) \]

Given:
- \( K_c = 1.8 \times 10^{-6} \) at \( 184^\circ C \)
- \( R = 0.0083 \, \text{kJ} \, \text{mol}^{-1} \, \text{K}^{-1} \)

### Step 1: Calculate \( \Delta n \)

First, we need to calculate \( \Delta n \), which is the change in the number of moles of gas during the reaction.

\[ \Delta n = \text{moles of products} - \text{moles of reactants} \]

From the balanced equation:
- Products: \( 2 \, \text{moles of NO} + 1 \, \text{mole of O}_2 = 3 \, \text{moles} \)
- Reactants: \( 2 \, \text{moles of NO}_2 = 2 \, \text{moles} \)

So,

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

### Step 2: Convert Temperature to Kelvin

Next, we convert the temperature from Celsius to Kelvin:

\[ T(K) = 184 + 273 = 457 \, K \]

### Step 3: Use the Relation Between \( K_p \) and \( K_c \)

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

\[ K_p = K_c \times (R \times T)^{\Delta n} \]

Substituting the known values:

- \( K_c = 1.8 \times 10^{-6} \)
- \( R = 0.0083 \, \text{kJ} \, \text{mol}^{-1} \, \text{K}^{-1} = 8.314 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1} \) (since \( 1 \, \text{kJ} = 1000 \, \text{J} \))
- \( T = 457 \, K \)
- \( \Delta n = 1 \)

### Step 4: Calculate \( K_p \)

Now we can calculate \( K_p \):

\[ K_p = K_c \times (R \times T)^1 \]

Calculating \( R \times T \):

\[ R \times T = 8.314 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1} \times 457 \, K = 3806.918 \, \text{J} \, \text{mol}^{-1} \]

Now substituting back into the equation for \( K_p \):

\[ K_p = 1.8 \times 10^{-6} \times 3806.918 \]

Calculating \( K_p \):

\[ K_p \approx 6.82 \times 10^{-6} \]

### Step 5: Compare \( K_p \) and \( K_c \)

Now we compare \( K_p \) and \( K_c \):

- \( K_c = 1.8 \times 10^{-6} \)
- \( K_p \approx 6.82 \times 10^{-6} \)

Since \( K_p > K_c \), we conclude that:

\[ K_p \text{ is greater than } K_c \]

### Final Answer

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

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