Consider the following reactions in which all the reactants and the products are in gaseous state.
`2PQ hArr P_(2)=Q_(2), K_(1)=2.5xx10^(5)`
`PQ +1//2 R_(2) hArr PQR, K_92) =5xx10^(-3)`
The value of `K_(2)` for the equilibrium
`1//2P_(2)+1//2Q_(2) +1//2R_(2) hArr PQR,` is

Options

A. `2.5xx10^(3)`
B. `5xx10^(3)`
C. `5xx10^(-3)`
D. `1.0xx10^(-5)`

Answer

To solve the problem, we need to manipulate the given reactions and their equilibrium constants to find the equilibrium constant \( K_5 \) for the reaction:

\[ \frac{1}{2} P_2 + \frac{1}{2} Q_2 + \frac{1}{2} R_2 \rightleftharpoons PQR \]

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

1. \( 2PQ \rightleftharpoons P_2 + Q_2 \) with \( K_1 = 2.5 \times 10^5 \)
2. \( PQ + \frac{1}{2} R_2 \rightleftharpoons PQR \) with \( K_2 = 5 \times 10^{-3} \)

### Step 2: Reverse the first reaction.

When we reverse the first reaction, the equilibrium constant for the reverse reaction \( K_3 \) is given by:

\[ K_3 = \frac{1}{K_1} = \frac{1}{2.5 \times 10^5} = 4 \times 10^{-6} \]

### Step 3: Divide the reversed reaction by 2.

Now, we divide the reversed reaction by 2:

\[ \frac{1}{2} P_2 + \frac{1}{2} Q_2 \rightleftharpoons PQ \]

The equilibrium constant \( K_4 \) for this reaction is:

\[ K_4 = \sqrt{K_3} = \sqrt{4 \times 10^{-6}} = 2 \times 10^{-3} \]

### Step 4: Write down the second reaction.

The second reaction remains as:

\[ PQ + \frac{1}{2} R_2 \rightleftharpoons PQR \]

with \( K_2 = 5 \times 10^{-3} \).

### Step 5: Add the two reactions.

Now, we add the two reactions:

\[ \frac{1}{2} P_2 + \frac{1}{2} Q_2 + \frac{1}{2} R_2 \rightleftharpoons PQR \]

### Step 6: Calculate the equilibrium constant for the combined reaction.

The equilibrium constant \( K_5 \) for the overall reaction is the product of the equilibrium constants of the individual reactions:

\[ K_5 = K_4 \times K_2 = (2 \times 10^{-3}) \times (5 \times 10^{-3}) = 10 \times 10^{-6} = 1 \times 10^{-5} \]

### Final Answer:

Thus, the value of \( K_5 \) for the equilibrium

\[ \frac{1}{2} P_2 + \frac{1}{2} Q_2 + \frac{1}{2} R_2 \rightleftharpoons PQR \]

is

\[ K_5 = 1 \times 10^{-5} \]