For the following mechanism, `P + Qoverset(K_(A))underset(K_(B))hArr PQ`
`overset(K_(C))underset(K_(D))hArr` R at equilibrium `([R])/([P][Q])` is: [K represents rate constant]

Options

A. `(K_(A).K_(B))/(K_(C).K_(D))`
B. `(K_(A).K_(D))/(K_(B).K_(C))`
C. `(K_(B).K_(D))/(K_(A).K_(C))`
D. `(K_(A).K_(C))/(K_(B)K_(D))`

Explanation

To solve the problem, we need to analyze the given reaction mechanism step by step and derive the expression for \(\frac{[R]}{[P][Q]}\).

### Step 1: Write the reactions and their equilibrium constants
The first reaction is:
\[ P + Q \overset{K_A}{\underset{K_B}{\rightleftharpoons}} PQ \]
The equilibrium constant \(K_1\) for this reaction can be defined as:
\[ K_1 = \frac{[PQ]}{[P][Q]} = \frac{K_A}{K_B} \]

The second reaction is:
\[ PQ \overset{K_C}{\underset{K_D}{\rightleftharpoons}} R \]
The equilibrium constant \(K_2\) for this reaction can be defined as:
\[ K_2 = \frac{[R]}{[PQ]} = \frac{K_C}{K_D} \]

### Step 2: Combine the equilibrium constants
To find the overall relationship between \([R]\), \([P]\), and \([Q]\), we can multiply the two equilibrium constants \(K_1\) and \(K_2\):
\[ K_1 \times K_2 = \left(\frac{[PQ]}{[P][Q]}\right) \times \left(\frac{[R]}{[PQ]}\right) \]

### Step 3: Simplify the equation
When we multiply the two equations, the \([PQ]\) terms cancel out:
\[ K_1 \times K_2 = \frac{[R]}{[P][Q]} \]

### Step 4: Substitute the values of \(K_1\) and \(K_2\)
Substituting the expressions for \(K_1\) and \(K_2\):
\[ \frac{K_A}{K_B} \times \frac{K_C}{K_D} = \frac{[R]}{[P][Q]} \]

### Step 5: Rearranging the equation
Rearranging the equation gives us:
\[ \frac{[R]}{[P][Q]} = \frac{K_A \cdot K_C}{K_B \cdot K_D} \]

### Final Answer
Thus, the expression for \(\frac{[R]}{[P][Q]}\) is:
\[ \frac{[R]}{[P][Q]} = \frac{K_A \cdot K_C}{K_B \cdot K_D} \]