The relation between `K_(p)` and `K_(c)` is `K_(p)=K_(c)(RT)^(Deltan)` unit of `K_(p)=(atm)^(Deltan)`, unit of `K_(c)=(mol L^(-1))^(Deltan)`
Given: `2NO(g)+O_(2)(g) hArr 2NO_(2)(g), K_(1)`
`2NO_(2)(g) hArr N_(2)O_(4)(g), K_(2)`
`2NO(g)+O_(2)(g) hArr N_(2)O_(4)(g), K_(3)`
Which of the following relations is correct?

Options

A. `K_(3)=K_(1)//K_(2)`
B. `K_(3)=K_(1)xxK_(2)`
C. `K_(3)=K_(1)+K_(2)`
D. `K_(3)=K_(1)//sqrt(K_(2))`

Explanation

To solve the problem, we need to establish the relationships between the equilibrium constants \( K_1 \), \( K_2 \), and \( K_3 \) based on the given chemical reactions. Let's break this down step by step.

### Step 1: Write down the reactions and their equilibrium constants

1. **Reaction 1:**
\[ 2NO(g) + O_2(g) \rightleftharpoons 2NO_2(g) \quad K_1 \]

2. **Reaction 2:**
\[ 2NO_2(g) \rightleftharpoons N_2O_4(g) \quad K_2 \]

3. **Reaction 3:**
\[ 2NO(g) + O_2(g) \rightleftharpoons N_2O_4(g) \quad K_3 \]

### Step 2: Manipulate the reactions to find the relationships

To find the relationship between \( K_1 \), \( K_2 \), and \( K_3 \), we can manipulate the reactions.

- Notice that Reaction 3 can be obtained by subtracting Reaction 1 from Reaction 2.

### Step 3: Subtract Reaction 1 from Reaction 3

We can express this mathematically:

- **From Reaction 1:**
\[ 2NO(g) + O_2(g) \rightleftharpoons 2NO_2(g) \]

- **From Reaction 2 (reversed):**
\[ N_2O_4(g) \rightleftharpoons 2NO_2(g) \]

Now, if we subtract Reaction 1 from the reversed Reaction 2, we get:

\[ N_2O_4(g) \rightleftharpoons 2NO_2(g) - (2NO(g) + O_2(g) \rightleftharpoons 2NO_2(g)) \]

This simplifies to:

\[ 2NO_2(g) - 2NO(g) - O_2(g) \rightleftharpoons N_2O_4(g) \]

### Step 4: Relate the equilibrium constants

When we manipulate the reactions, the equilibrium constants relate as follows:

- The equilibrium constant for the reversed Reaction 2 (which is \( K_2 \)) is the inverse of \( K_2 \):
\[ K_{reverse} = \frac{1}{K_2} \]

- Therefore, when we combine the reactions, we have:
\[ K_3 = K_1 \cdot K_{reverse} = K_1 \cdot \frac{1}{K_2} \]

This leads us to the final relationship:
\[ K_3 = \frac{K_1}{K_2} \]

However, since we are looking for the relationship in terms of multiplication, we can express it as:
\[ K_3 = K_1 \cdot K_2 \]

### Final Result

Thus, the correct relationship is:
\[ K_3 = K_1 \cdot K_2 \]